Weighted Fractional Hermite–Hadamard Integral Inequalities for up and down Ԓ-Convex Fuzzy Mappings over Coordinates

Abstract

Due to its significant influence on numerous areas of mathematics and practical sciences, the theory of integral inequality has attracted a lot of interest. Convexity has undergone several improvements, generalizations, and extensions over time in an effort to produce more accurate variations of known findings. This article’s main goal is to introduce a new class of convexity as well as to prove several Hermite–Hadamard type interval-valued integral inequalities in the fractional domain. First, we put forth the new notion of generalized convexity mappings, which is defined as $\mathsf{U}\mathsf{D}$-$\mathrm{Ԓ}$-convexity on coordinates with regard to fuzzy-number-valued mappings and the up and down ($\mathsf{U}\mathsf{D}$) fuzzy relation. The generic qualities of this class make it novel. By taking into account different values for $\mathrm{Ԓ}$, we produce several known classes of convexity. Additionally, we create some new fractional variations of the Hermite–Hadamard ($\mathsf{H}\mathsf{H}$) and Pachpatte types of inequalities using the concepts of coordinated $\mathsf{U}\mathsf{D}$-$\mathrm{Ԓ}$-convexity and double Riemann–Liouville fractional operators. The results attained here are the most cohesive versions of previous findings. To demonstrate the importance of the key findings, we offer a number of concrete examples.

1. Introduction

The most fundamental area of mathematical analysis is known as convex analysis; see [1,2]. Due to its significant contribution to the advancement of both pure and applied mathematics, it has attracted considerable attention. A problem can be solved geometrically and analytically using convexity and its effects. Convexity plays a crucial role in topology, functional analysis, specifically separation axioms, fixed-point theory, engineering, and economics. First, by proposing the idea of convex mappings based on a convex set in 1905, Jensen greatly increased the appeal of the theory of convex functions. Since a positive second derivative denotes the convexity of functions, one would wish to describe it in terms of functions and their derivatives. It has a close relationship with optimization theory, particularly with linear programming. Convex mappings frequently offer distinctive minima and are used to derive a workable solution; see [3,4,5].

The theory of mathematical inequalities has various applications in many branches of physics and engineering. This theory is closely connected to fields such as approximation theory, probability theory, and information theory. The importance of this topic will increase in the future due to its impact on applied mathematics. The theory of inequalities has greatly benefited from the study of the theory of convex functions. By using the idea of convexity, it is possible to directly obtain many inequalities, like Jensen’s inequality, the $\mathsf{H}\mathsf{H}$ inequality, Young’s inequality, etc. In this regard, we are reminded of the well-known inequality resulting from Hermite and Hadamard acting independently.

Theorem 1.

Assume that the convex mapping $G:\left[\mathfrak{e},\mathfrak{g}\right]\to \mathfrak{R}$. Then, the following double-inequality holds:

$$G\left(\frac{\mathfrak{e}+\mathfrak{g}}{2}\right)\le \frac{1}{\mathfrak{g}-\mathfrak{e}}{\int }_{\mathfrak{e}}^{\mathfrak{g}}G\left(x\right)dx\le \frac{G\left(\mathfrak{g}\right)+G\left(\mathfrak{e}\right)}{2},$$

(1)

where $\mathfrak{R}$ is set of real numbers.

One can check the concavity of the mappings by using the aforementioned inequality. See [6,7] for further information about this. For more information, related to different inequalities, see [8,9,10,11,12,13,14,15,16] and the references therein.

With the help of new and creative ideas, particularly the use of weighted means, the concept of convexity has recently been improved and expanded. Examples include harmonic, geometric, and $P$-convexity, which are based on the weighted harmonic mean, the weighted geometric mean, and the generalized weighted p mean, respectively. In 2019, Wu et al. [17,18] used the quasi-arithmetic mean to investigate a new class of convexity.

We work with multi-valued functions in a set-valued analysis, and interval-valued ($\mathsf{I}\mathsf{V}$) analysis is a branch of this field. The initial method for calculating the error estimates of finite machines was an interval analysis. If we assign a single value to any variable, just like in everyday life operations, the likelihood of inaccuracy rises; to address this shortcoming, interval numbers are used in place of single numerical values. Moore authored some fascinating works on interval analysis that offered fresh approaches to putting this theory into practice and suggested some uses for it in computer programming and error analysis; see [19,20,21].

Since Moore’s outstanding and useful work, several authors have expressed interest in the area and exploited it in various ways. Investigations into the dynamic systems of differential equations, fluid mechanics, combinatorics, neural networking, and inequalities are carried out using $\mathsf{I}\mathsf{V}$ approaches (see [22]). Breckner [23] continued by advancing the concept of convexity from the standpoint of set-valued mappings.

By using the ordering relations and $\mathsf{I}\mathsf{V}$ mappings defined over interval numbers, certain inequalities have recently been improved and extended; see [24,25,26,27,28,29,30]. Regarding this, Chalco Cano et al. [31,32] computed the well-known Ostrowski’s integral inequality using $\mathsf{I}\mathsf{V}$ mappings and Hukuhara derivatives and came to the conclusion that the primary results were useful in numerical analyses. In 2017, Costa et al. [33] applied the mappings established over fuzzy numbers to investigate fresh integral inequalities. In the follow-up, Flores et al. [34] calculated new integral inequality variations related to $\mathsf{I}\mathsf{V}$ mappings. In [35], the authors looked into the preinvex-$\mathsf{I}\mathsf{V}$-mappings-related integral inequalities of $\mathsf{H}\mathsf{H}$. Jensen’s and $\mathsf{H}\mathsf{H}$-type containments involving a general class of $\mathsf{I}\mathsf{V}$ convexity, also known as the h-$\mathsf{I}\mathsf{V}$ mapping and Chebyshev-type inequality, respectively, were established by Zhao et al. [36,37]. Extremely significant contributions to the growth of integral inequality have come from fractional calculus. The first successful attempt to build fractional equivalents of $\mathsf{H}\mathsf{H}$-type inequalities was carried out in 2012 by Sarikaya et al. [38], who essentially took integral fractional operators into consideration. Following this, other inequalities have been reduced utilizing fractional methods, and this area of study is still quite active. Mohammad et al. explored the novel tempered Hermite–Hadamard-like inequalities and offered several applications in [39,40,41,42,43,44], along with fractional mid-point-like inequalities within the framework of fractional calculus. In [45], Akdemir et al. used unified fractional operators to evaluate the Chebyshev-like inequality. Inequalities involving AB-fractional integral operators and the differentiability of convex mappings were the conclusions of Set et al.’s [46] work. Budak et al. [47] investigated the $\mathsf{H}\mathsf{H}$-type inequalities in 2020 by studying interval-valued fractional operators. In order to show several $\mathsf{H}\mathsf{H}$-type inequalities, Kara et al. [48] combined novel double-fractional operators with the idea of interval-valued coordinated convexity. In order to extract some novel fractional versions of the $\mathsf{H}\mathsf{H}$-type inequalities, Bin-Mohsin et al. [49] recently presented the concept of interval-valued coordinated, harmonically convex mappings and double-fractional operators using the modified Mittag-Leffler function introduced by Raina as a kernel. Trigonometric convex functions with exponential weights were investigated by Zhou et al. [50] to create some novel $\mathsf{H}\mathsf{H}$-like inequalities. A fuzzy order relation was used to execute their discussion of trigonometric convexity and related integral containment in [51]. $\mathsf{I}\mathsf{V}$ convexity and (p, q) calculus were employed by Kalsoom et al. in [52] to establish some fresh refinements of previous findings. To create certain $\mathsf{H}\mathsf{H}$-like inequalities, the authors of [53] developed the idea of the fuzzy-interval-valued bi-convex function. For interval-coordinated convex functions and products of Hermite–Hadamard-type inequalities, the authors of [54,55], respectively, obtained fractional forms of these inequalities. They wrapped up this work in [56] with some fresh Hermite–Hadamard inequalities involving interval-valued convexity and generalized quantum calculus. See [33,57,58,59,60,61,62,63,64,65,66] for additional information and current developments.

The goal of the current study is to use AB-fractional notions to develop new generic inclusion relations of the $\mathsf{H}\mathsf{H}$ type. First, we create a novel class of convexity based on the interval analysis bi-function and monotonically continuous function g. The uniqueness of this study is in the derivation of numerous new and existing fractional counterparts using various values. Additionally, we use numerical simulations to confirm the results of our theoretical work. To the best of our knowledge, these results are more helpful for obtaining variations of $\mathsf{I}\mathsf{V}$ $\mathsf{H}\mathsf{H}$-type inequalities for some classes of convexity; see [55,59,67,68,69,70,71,72,73,74].

The work is divided into two sections. In the first half, we review some information about convexity and fractional calculus and discuss the problem’s history. The newly proposed class of convexity is introduced in the second section, along with its implications and uses in integral inequalities. Concluding observations are included later.

2. Preliminaries

We will go through the fundamental terminologies and findings in this section, which aid in comprehending the ideas behind our fresh findings.

Definition 1

([63,64]). Let ${\mathbb{F}}_0$ be a fuzzy number space. Given $\widetilde{Ɖ}\in {\mathbb{F}}_0$, the level sets or cut sets are given by ${\left[\widetilde{Ɖ}\right]}^{\gamma }=\left\{x\in \mathfrak{R}|\widetilde{Ɖ}\left(x\right)>\gamma \right\}$$\forall$$\gamma \in [0, 1]$ and by

$${\left[\widetilde{Ɖ}\right]}^0=\left\{x\in \mathfrak{R}|\widetilde{Ɖ}\left(x\right)>0\right\}.$$

(2)

These sets are known as $\gamma$-level sets or $\gamma$-cut sets of $\widetilde{Ɖ}$.

Proposition 1

([33]). Let $\widetilde{Ɖ},\widetilde{₼}\in {\mathbb{F}}_0$. Then, the relation $“{\le }_{\mathbb{F}}”$ is given on ${\mathbb{F}}_0$ by $\widetilde{Ɖ}{\le }_{\mathbb{F}}\widetilde{₼}$ when and only when ${{\left[\widetilde{Ɖ}\right]}^{\gamma }\le }_I{\left[\widetilde{₼}\right]}^{\gamma }$ for every $\gamma \in [0, 1],$ which are left- and right-order relations or just order relations.

Proposition 2

([62]). Let $\widetilde{Ɖ},\widetilde{₼}\in {\mathbb{F}}_0$. Then, the relation $“{\supseteq }_{\mathbb{F}}”$ is given on ${\mathbb{F}}_0$ by $\widetilde{Ɖ}{\supseteq }_{\mathbb{F}}\widetilde{₼}$ when and only when ${\left[\widetilde{Ɖ}\right]}^{\gamma }{\supseteq }_I{\left[\widetilde{₼}\right]}^{\gamma }$ for every $\gamma \in [0, 1],$ which is the $\mathsf{U}\mathsf{D}-$order relation on ${\mathbb{F}}_0$.

Remember the approaching notions, which are offered in the literature. If $\widetilde{Ɖ},\widetilde{₼}\in {\mathbb{F}}_0$ and $\mathcal{t}\in \mathfrak{R}$, then, for every $\gamma \in \left[0, 1\right],$ the arithmetic operations addition, “$\oplus ”$, multiplication, “$\otimes ”$, and scaler multiplication, “$\odot ”,$ are defined by

$${\left[\widetilde{Ɖ}\oplus \widetilde{₼}\right]}^{\gamma }={\left[\widetilde{Ɖ}\right]}^{\gamma }+{\left[\widetilde{₼}\right]}^{\gamma },$$

(3)

$${\left[\widetilde{Ɖ}\otimes \widetilde{₼}\right]}^{\gamma }={\left[\widetilde{Ɖ}\right]}^{\gamma }\times {\left[ \widetilde{₼}\right]}^{\gamma },$$

(4)

$${\left[\mathcal{t}\odot \widetilde{Ɖ}\right]}^{\gamma }=\mathcal{t}{\left[\widetilde{Ɖ}\right]}^{\gamma },$$

(5)

Equations (4) through to (6) have immediate consequences for these outcomes.

Theorem 2

([33]). The space ${\mathbb{F}}_0$ dealing with a supremum metric, i.e., for $\widetilde{Ɖ}, \widetilde{₼}\in {\mathbb{F}}_0$,

$$d_{\mathrm{\infty }}\left(\widetilde{\mathrm{Ɖ}},\widetilde{\mathrm{₼}}\right)=\underset{0\le \gamma \le 1}{\mathit{sup}}{d}_H\left({\left[\widetilde{\mathrm{Ɖ}}\right]}^{\gamma },{\left[\widetilde{\mathrm{₼}}\right]}^{\gamma }\right),$$

(6)

is a complete metric space, where $H$ indicates the well-known Hausdorff metric on the space of intervals.

Theorem 3

([33]). Let ${\mathbb{R}}_I^{ }$ be a set of inetervals and $\tilde{G}:[\mathfrak{u},\mathfrak{v}]\subset \mathfrak{R}\to {\mathbb{F}}_0$ be an $\mathsf{F}\mathsf{N}\mathsf{V}\mathsf{M}$; its $\mathsf{I}\mathsf{V}\mathsf{M}$s are classified according to their $\gamma$-levels, $G_{\gamma }:[\mathfrak{u},\mathfrak{v}]\subset \mathfrak{R}\to {\mathbb{R}}_I^{ }$ are given by $G_{\gamma }\left(x\right)=\left[G_{\ast }\left(x,\gamma \right), G^{\ast }\left(x,\gamma \right)\right]$$\forall$$x\in [\mathfrak{u},\mathfrak{v}]$, and $\forall$$\gamma \in (0, 1].$ Then, $\tilde{G}$ is $FA$-integrable over $[\mathfrak{u},\mathfrak{v}]$ if and only if $G_{\ast }\left(x,\gamma \right)$ and $G^{\ast }\left(x,\gamma \right)$ are both $A$-integrable over $[\mathfrak{u},\mathfrak{v}]$. Moreover, if $\tilde{G}$ is $FA$-integrable over $\left[\mathfrak{u},\mathfrak{v}\right],$ then

$$\begin{array}{cc}{\left[\left(FA\right){\int }_{\mathfrak{u}}^{\mathfrak{v}}\tilde{G}\left(x\right)dx\right]}^{\gamma }=& \left[\left(A\right){\int }_{\mathfrak{u}}^{\mathfrak{v}}{G}_{\ast }\left(x,\gamma \right)dx, \left(A\right){\int }_{\mathfrak{u}}^{\mathfrak{v}}{G}^{\ast }\left(x,\gamma \right)dx\right]\\ & =\left(IA\right){\int }_{\mathfrak{u}}^{\mathfrak{v}}{G}_{\gamma }\left(x\right)dx,\end{array}$$

(7)

$\forall$ $\gamma \in (0, 1].$ $\forall$ $\gamma \in \left(0, 1\right],$ ${\mathcal{F}\mathcal{A}}_{\left(\left[\mathfrak{u},\mathfrak{v}\right], \gamma \right)}$ denotes the collection of all $FA$-integrable $\mathsf{F}\mathsf{N}\mathsf{V}\mathsf{M}$s over $[\mathfrak{u},\mathfrak{v}]$.

Definition 2.

([67]). Let ${\mathbb{R}}_I^{+}$ be a set of positive intervals and $G:\left[\mathfrak{e},\mathfrak{g}\right]\to {\mathbb{R}}_I^{+}$ be an $\mathsf{I}\mathsf{V}\mathsf{M}$ , where $G\in {\mathcal{I}\mathcal{R}}_{\left[\mathfrak{e},\mathfrak{g}\right]}$. Then, interval Riemann–Liouville-type integrals of $G$ are defined as

$${\mathcal{I}}_{{\mathfrak{e}}^{+}}^{\alpha }G\left(y\right)=\frac{1}{\Gamma \left(\alpha \right)}{\int }_{\mathfrak{e}}^y{\left(y-\mathtt{t}\right)}^{\alpha -1}G\left(\mathtt{t}\right)d\mathtt{t}\left(y>\mathfrak{e}\right),$$

(8)

$${\mathcal{I}}_{{\mathfrak{g}}^{-}}^{\alpha }G\left(y\right)=\frac{1}{\Gamma \left(\alpha \right)}{\int }_y^{\mathfrak{g}}{\left(\mathtt{t}-y\right)}^{\alpha -1}G\left(\mathtt{t}\right)d\mathtt{t}(y<\mathfrak{g}),$$

(9)

where $\alpha >0$ and $\Gamma$ is the gamma function.

Recently, Allahviranloo et al. [68] introduced the fuzzy version of this and defined fractional integrals, resulting in the following:

Definition 3.

Let $\alpha >0$ and $L\left(\left[\mathfrak{e},\mathfrak{g}\right],{\mathbb{F}}_0\right)$ be the collection of all Lebesgue measurable $\mathsf{F}\mathsf{N}\mathsf{V}\mathsf{M}$s on $[\mathfrak{e},\mathfrak{g}]$. Then, the fuzzy left and right Riemann–Liouville fractional integrals of $\tilde{G}\in$$L\left(\left[\mathfrak{e},\mathfrak{g}\right],{\mathbb{F}}_0\right)$ with order $\alpha >0$ are defined by

$${\mathcal{I}}_{{\mathfrak{e}}^{+}}^{\alpha } \tilde{G}\left(y\right)=\frac{1}{\Gamma \left(\alpha \right)}{\int }_{\mathfrak{e}}^y{\left(y-\mathtt{t}\right)}^{\alpha -1}\tilde{G}\left(\mathtt{t}\right)d\mathtt{t},\left(y>\mathfrak{e}\right),$$

(10)

and

$${\mathcal{I}}_{{\mathfrak{g}}^{-}}^{\alpha } \tilde{G}\left(y\right)=\frac{1}{\Gamma \left(\alpha \right)}{\int }_y^{\mathfrak{g}}{\left(\mathtt{t}-y\right)}^{\alpha -1}\tilde{G}\left(\mathtt{t}\right)d\mathtt{t},(y<\mathfrak{g}),$$

(11)

respectively, where $\Gamma \left(y\right)={\int }_0^{\infty }{\mathtt{t}}^{y-1}{e}^{-\mathtt{t}}d\mathtt{t}$ is the Euler gamma function. The fuzzy left and right Riemann–Liouville fractional integral, $y$, based on the left and right endpoint functions, can be defined, that is

$$\begin{array}{c}{\left[{\mathcal{I}}_{{\mathfrak{e}}^{+}}^{\alpha }\tilde{G}\left(y\right)\right]}^{\gamma }=\frac{1}{\Gamma \left(\alpha \right)}{\int }_{\mathfrak{e}}^y{\left(y-\mathtt{t}\right)}^{\alpha -1}{G}_{\gamma }\left(\mathtt{t}\right)d\mathtt{t}\\ =\frac{1}{\Gamma \left(\alpha \right)}{\int }_{\mathfrak{e}}^y{\left(y-\mathtt{t}\right)}^{\alpha -1}\left[G_{\ast }\left(\mathtt{t},\gamma \right),G^{\ast }\left(\mathtt{t},\gamma \right)\right]d\mathtt{t},\left(y>\mathfrak{e}\right),\end{array}$$

(12)

where

$${\mathcal{I}}_{{\mathfrak{e}}^{+}}^{\alpha } G_{\ast }\left(y, \gamma \right)=\frac{1}{\Gamma \left(\alpha \right)}{\int }_{\mathfrak{e}}^y{\left(y-\mathtt{t}\right)}^{\alpha -1}{G}_{\ast }\left(\mathtt{t},\gamma \right)d\mathtt{t},\left(y>\mathfrak{e}\right),$$

(13)

and

$${\mathcal{I}}_{{\mathfrak{e}}^{+}}^{\alpha } G^{\ast }\left(y, \gamma \right)=\frac{1}{\Gamma \left(\alpha \right)}{\int }_{\mathfrak{e}}^y{\left(y-\mathtt{t}\right)}^{\alpha -1}{G}^{\ast }\left(\mathtt{t},\gamma \right)d\mathtt{t},\left(y>\mathfrak{e}\right),$$

(14)

The right Riemann–Liouville fractional integral, denoted by ${\left[{\mathcal{I}}_{{\mathfrak{g}}^{-}}^{\alpha }\tilde{G}\left(y\right)\right]}^{\gamma }$, can also be defined using the left and right endpoint functions.

Theorem 4.

([69]). Let ${\mathbb{F}}_0^{+}$ be a set of positive fuzzy numbers, $\mathrm{Ԓ} :\left[0, 1\right]\to {\mathbb{R}}^{+},$ and $\tilde{G}:\left[\mathfrak{u},\mathfrak{v}\right]\to {\mathbb{F}}_0^{+}$ be a $\mathsf{U}\mathsf{D}$-convex $\mathsf{F}\mathsf{N}\mathsf{V}\mathsf{M}$ on $\left[\mathfrak{u},\mathfrak{v}\right],$ whose $\gamma$-cuts set up the sequence of $\mathsf{I}\mathsf{V}\mathsf{M}$s $G_{\gamma }:\left[\mathfrak{u},\mathfrak{v}\right]\subset \mathbb{R}\to {{\mathbb{R}}_C}^{+}, which$ is given by $G_{\gamma }\left(y\right)=\left[G_{\ast }\left(y,\gamma \right), G^{\ast }\left(y,\gamma \right)\right]$ for all $y\in \left[\mathfrak{u},\mathfrak{v}\right]$ and for all $\gamma \in \left[0, 1\right]$. If $\tilde{G}\in L\left(\left[\mathfrak{u},\mathfrak{v}\right],{\mathbb{F}}_0\right)$; then,

$$\frac{1}{\alpha \mathrm{Ԓ}\left(\frac{1}{2}\right)}\tilde{G}\left(\frac{\mathfrak{u}+\mathfrak{v}}{2}\right){\supseteq }_{\mathbb{F}}\frac{\Gamma \left(\alpha \right)}{{\left(\mathfrak{v}-\mathfrak{u}\right)}^{\alpha }}\left[{\mathcal{I}}_{{\mathfrak{u}}^{+}}^{\alpha }\tilde{G}\left(\mathfrak{v}\right)\oplus {\mathcal{I}}_{{\mathfrak{v}}^{-}}^{\alpha }\tilde{G}\left(\mathfrak{u}\right)\right]{\supseteq }_{\mathbb{F}}\left[\tilde{G}\left(\mathfrak{u}\right)\oplus \tilde{G}\left(\mathfrak{v}\right)\right]{\int }_0^1{\tau }^{\beta -1}\left[\mathrm{Ԓ}\left(\tau \right)+\mathrm{Ԓ}\left(1-\tau \right)\right]d\tau .$$

(15)

Interval and fuzzy Aumann’s type integrals are defined as follows for the coordinated $\mathsf{I}\mathsf{V}\mathsf{M}$ $G\left(x,y\right)$ and the coordinated $\mathsf{F}\mathsf{N}\mathsf{V}\mathsf{M}$ $\tilde{G}\left(x,y\right)$:

Theorem 5.

([59]). Let $\tilde{G}:∆\left[\mathfrak{e},\mathfrak{g}\right]\times \left[\mathfrak{u},\mathfrak{v}\right]\subset {\mathbb{R}}^2\to {\mathbb{F}}_0^{+}$ be an $\mathsf{F}\mathsf{N}\mathsf{V}\mathsf{M}$ on coordinates, whose $\gamma$-cuts set up the sequence of $\mathsf{I}\mathsf{V}\mathsf{M}s$  $G_{\gamma }:∆\subset {\mathbb{R}}^2\to {\mathbb{R}}_I, which$ is given by $G_{\gamma }\left(x,y\right)=\left[G_{\ast }\left(\left(x,y\right),\gamma \right), G^{\ast }\left(\left(x,y\right),\gamma \right)\right]$ for all $\left(x,y\right)\in ∆=\left[\mathfrak{e},\mathfrak{g}\right]\times \left[\mathfrak{u},\mathfrak{v}\right]$ and for all $\gamma \in \left[0, 1\right].$ Then, $\tilde{G}$ is fuzzy double integrable ($FD$-integrable) over $∆$ if and only if $G_{\ast }\left(x,\gamma \right)$ and $G^{\ast }\left(x,\gamma \right)$ both are $D$ -integrable over $∆.$ Moreover, if $\tilde{G}$ is $FD$-integrable over $∆,$ then

$$\begin{array}{c}{\left[\left(FD\right){\int }_{\mathfrak{e}}^{\mathfrak{g}}{\int }_{\mathfrak{u}}^{\mathfrak{v}}\tilde{G}\left(x,y\right)dydx\right]}^{\gamma }=\left[\left(D\right){\int }_{\mathfrak{e}}^{\mathfrak{g}}{\int }_{\mathfrak{u}}^{\mathfrak{v}}{G}_{\ast }\left(\left(x,y\right),\gamma \right)dydx, \left(D\right){\int }_{\mathfrak{e}}^{\mathfrak{g}}{\int }_{\mathfrak{u}}^{\mathfrak{v}}{G}^{\ast }\left(\left(x,y\right),\gamma \right)dydx\right]\\ =\left(ID\right){\int }_{\mathfrak{e}}^{\mathfrak{g}}{\int }_{\mathfrak{u}}^{\mathfrak{v}}{G}_{\gamma }\left(x,y\right)dydx, \end{array}$$

(16)

for all $\gamma \in \left[0, 1\right].$

The families of all $FD$-integrable $\mathsf{F}\mathsf{N}\mathsf{V}\mathsf{M}$s over coordinates and $D$-integrable functions over coordinates are denoted by ${\mathcal{F}\mathfrak{O}}_{∆}$ and ${\mathfrak{O}}_{\left(∆, \gamma \right)}$ for all $\gamma \in \left[0, 1\right].$

Here is the main definition of a fuzzy Riemann–Liouville fractional integral on the coordinates of the function $\tilde{G}\left(x,y\right)$ by:

Definition 4

([70]). Let $\tilde{G}:∆\to {\mathbb{F}}_0$ and $\tilde{G}\in {\mathcal{F}\mathfrak{O}}_{∆}$. The double fuzzy interval Riemann–Liouville-type integrals ${\mathcal{I}}_{{\mathfrak{e}}^{+},{\mathfrak{u}}^{+} }^{\alpha , \beta }, {\mathcal{I}}_{{\mathfrak{e}}^{+},{\mathfrak{v}}^{-} }^{\alpha , \beta }$$,{\mathcal{I}}_{{\mathfrak{g}}^{-},{\mathfrak{u}}^{+} }^{\alpha , \beta },{\mathcal{I}}_{{\mathfrak{g}}^{-},{\mathfrak{v}}^{-} }^{\alpha , \beta }$ of $G$ of the order $\alpha , \beta >0$ are defined by:

$${\mathcal{I}}_{{\mathfrak{e}}^{+},{\mathfrak{u}}^{+}}^{\alpha , \beta }\tilde{G}\left(x,y\right)=\frac{1}{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{\int }_{\mathfrak{e}}^x{\int }_{\mathfrak{u}}^y{{\left(x-\mathtt{t}\right)}^{\alpha -1}\left(y-\mathtt{s}\right)}^{\beta -1}\tilde{G}\left(\mathtt{t},\mathtt{s}\right)d\mathtt{s}d\mathtt{t}, \left(x>\mathfrak{e},y>\mathfrak{u}\right),$$

(17)

$${\mathcal{I}}_{{\mathfrak{e}}^{+},{\mathfrak{v}}^{-}}^{\alpha , \beta }\tilde{G}\left(x,y\right)=\frac{1}{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{\int }_{\mathfrak{e}}^x{\int }_y^{\mathfrak{v}}{{\left(x-\mathtt{t}\right)}^{\alpha -1}\left(\mathtt{s}-y\right)}^{\beta -1}\tilde{G}\left(\mathtt{t},\mathtt{s}\right)d\mathtt{s}d\mathtt{t}, \left(x>\mathfrak{e},y<\mathfrak{v}\right),$$

(18)

$${\mathcal{I}}_{{\mathfrak{g}}^{-},{\mathfrak{u}}^{+}}^{\alpha , \beta }\tilde{G}\left(x,y\right)=\frac{1}{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{\int }_x^{\mathfrak{g}}{\int }_{\mathfrak{u}}^y{{\left(\mathtt{t}-x\right)}^{\alpha -1}\left(y-\mathtt{s}\right)}^{\beta -1}\tilde{G}\left(\mathtt{t},\mathtt{s}\right)d\mathtt{s}d\mathtt{t}, \left(x<\mathfrak{g},y>\mathfrak{u}\right),$$

(19)

$${\mathcal{I}}_{{\mathfrak{g}}^{-},{\mathfrak{v}}^{-}}^{\alpha , \beta }\tilde{G}\left(x,y\right)=\frac{1}{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{\int }_x^{\mathfrak{g}}{\int }_y^{\mathfrak{v}}{{\left(\mathtt{t}-x\right)}^{\alpha -1}\left(\mathtt{s}-y\right)}^{\beta -1}\tilde{G}\left(\mathtt{t},\mathtt{s}\right)d\mathtt{s}d\mathtt{t}, \left(x<\mathfrak{g},y<\mathfrak{v}\right).$$

(20)

Here is the newly defined concept of coordinated $\mathsf{U}\mathsf{D}$-$\mathbf{Ԓ}$-convexity over fuzzy number space in the codomain via the $\mathsf{U}\mathsf{D}$-relation given by the following:

Definition 5.

The $\mathsf{F}\mathsf{N}\mathsf{V}\mathsf{M}$$\tilde{G}:∆\to {\mathbb{F}}_0$ is referred to as a coordinated $\mathsf{U}\mathsf{D}$-$\mathrm{Ԓ}$-convex $\mathsf{F}\mathsf{N}\mathsf{V}\mathsf{M}$ on$∆$ if

$$\begin{array}{c} \tilde{G}\left(\tau \mathfrak{e}+\left(1-\tau \right)\mathfrak{g},\kappa \mathfrak{u}+\left(1-\kappa \right)\mathfrak{v}\right)\\ {\supseteq }_{\mathbb{F}}\mathrm{Ԓ}\left(\tau \right)\mathrm{Ԓ}\left(\kappa \right)\tilde{G}\left(\mathfrak{e},\mathfrak{u}\right)\oplus \mathrm{Ԓ}\left(\tau \right)\mathrm{Ԓ}\left(1-\kappa \right)\tilde{G}\left(\mathfrak{e},\mathfrak{v}\right)\oplus \mathrm{Ԓ}\left(1-\tau \right)\mathrm{Ԓ}\left(\kappa \right)\tilde{G}\left(\mathfrak{g},\mathfrak{u}\right)\oplus \mathrm{Ԓ}\left(1-\tau \right)\mathrm{Ԓ}(1-\kappa )\tilde{G}\left(\mathfrak{g},\mathfrak{v}\right),\end{array}$$

(21)

for all$\left(\mathfrak{e},\mathfrak{g}\right), \left(\mathfrak{u},\mathfrak{v}\right)\in ∆$ and $\tau ,\kappa \in \left[0, 1\right],$ where $\tilde{G}\left(x\right){\ge }_{\mathbb{F}}\tilde{0}.$ If inequality (21) is reversed, then $\tilde{G}$ is referred to as a coordinate $\mathsf{U}\mathsf{D}$-$\mathrm{Ԓ}$-concave $\mathsf{F}\mathsf{N}\mathsf{V}\mathsf{M}$ on $∆$.

Lemma 1.

Let $\tilde{G}:∆\to {\mathbb{F}}_0$ be a coordinated $\mathsf{F}\mathsf{N}\mathsf{V}\mathsf{M}$ on $∆$. Then, $\tilde{G}$ is a coordinated $\mathsf{U}\mathsf{D}$-$\mathrm{Ԓ}$-convex $\mathsf{F}\mathsf{N}\mathsf{V}\mathsf{M}$ on $∆$ if and only if there exist two coordinated $\mathsf{U}\mathsf{D}$-$\mathrm{Ԓ}$-convex $\mathsf{F}\mathsf{N}\mathsf{V}\mathsf{M}$s, ${\tilde{G}}_x:\left[\mathfrak{u},\mathfrak{v}\right]\to {\mathbb{F}}_0$, ${\tilde{G}}_x\left(w\right)=\tilde{G}\left(x,w\right)$ and ${\tilde{G}}_y:\left[\mathfrak{e},\mathfrak{g}\right]\to {\mathbb{F}}_0$, ${\tilde{G}}_y\left(z\right)=\tilde{G}\left(z,y\right)$.

Theorem 6.

Let $\tilde{G}:∆\to {\mathbb{F}}_0^{+}$ be an $\mathsf{F}\mathsf{N}\mathsf{V}\mathsf{M}$ on $∆$. Then, from $\gamma$-levels, we obtain the collection of $\mathsf{I}\mathsf{V}\mathsf{M}$s $G_{\gamma }:∆\to {\mathbb{R}}_I^{+}\subset {\mathbb{R}}_I, which$ is given by

$$G_{\gamma }\left(x,y\right)=\left[G_{\ast }\left(\left(x,y\right),\gamma \right), G^{\ast }\left(\left(x,y\right),\gamma \right)\right],$$

(22)

for all $\left(x,y\right)\in ∆$ and for all $\gamma \in \left[0, 1\right]$. Then, $\tilde{G}$ is a coordinated $\mathsf{U}\mathsf{D}$-$\mathrm{Ԓ}$-convex $\mathsf{F}\mathsf{N}\mathsf{V}\mathsf{M}$ on $∆$ if and only if for all $\gamma \in \left[0, 1\right],$ $G_{\ast }\left(\left(x,y\right), \gamma \right)$ and $G^{\ast }\left(\left(x,y\right), \gamma \right)$ are coordinated $\mathrm{Ԓ}$-convex and $\mathrm{Ԓ}$-concave functions, respectively.

Proof. 

Assume that for each $\gamma \in \left[0, 1\right],$$G_{\ast }\left(\mathcal{x},\gamma \right)$ and $G^{\ast }\left(\mathcal{x},\gamma \right)$ are coordinated $\mathbf{Ԓ}$-convex and $\mathbf{Ԓ}$-concave on $∆$, respectively. Then, from Equation (21), for all$\left(\mathfrak{e},\mathfrak{g}\right), \left(\mathfrak{u},\mathfrak{v}\right)\in ∆, \tau$ and $\kappa \in \left[0, 1\right],$ we have

$$\begin{array}{c}{G}_{\ast }\left(\left(\tau \mathfrak{e}+\left(1-\tau \right)\mathfrak{g},\kappa \mathfrak{u}+\left(1-\kappa \right)\mathfrak{v}\right), \gamma \right)\\ \le \mathrm{Ԓ}\left(\tau \right){\mathrm{Ԓ}\left(\kappa \right)G}_{\ast }\left(\left(\mathfrak{e},\mathfrak{u}\right), \gamma \right)+\mathrm{Ԓ}\left(\tau \right)\mathrm{Ԓ}\left(1-\kappa \right)G_{\ast }\left(\left(\mathfrak{e},\mathfrak{v}\right), \gamma \right)+{\mathrm{Ԓ}\left(\kappa \right)\mathrm{Ԓ}\left(1-\tau \right)G}_{\ast }\left(\left(\mathfrak{e},\mathfrak{u}\right), \gamma \right)+\\ \mathrm{Ԓ}\left(1-\tau \right)\mathrm{Ԓ}\left(1-\kappa \right)G_{\ast }\left(\left(\mathfrak{e},\mathfrak{v}\right), \gamma \right), \end{array}$$

and

$$\begin{array}{c}{G}^{\ast }\left(\left(\tau \mathfrak{e}+\left(1-\tau \right)\mathfrak{g},\kappa \mathfrak{u}+\left(1-\kappa \right)\mathfrak{v}\right), \gamma \right)\\ \ge \mathrm{Ԓ}\left(\tau \right){\mathrm{Ԓ}\left(\kappa \right)G}_{\ast }\left(\left(\mathfrak{e},\mathfrak{u}\right), \gamma \right)+\mathrm{Ԓ}\left(\tau \right)\mathrm{Ԓ}\left(1-\kappa \right)G^{\ast }\left(\left(\mathfrak{e},\mathfrak{v}\right), \gamma \right)+\mathrm{Ԓ}\left(\kappa \right)\mathrm{Ԓ}\left(1-\tau \right)G^{\ast }\left(\left(\mathfrak{e},\mathfrak{u}\right), \gamma \right)+\\ \mathrm{Ԓ}\left(1-\tau \right)\mathrm{Ԓ}\left(1-\kappa \right)G^{\ast }\left(\left(\mathfrak{e},\mathfrak{v}\right), \gamma \right),\end{array}$$

Then, from Equations (3), (5), and (22), we obtain

$$\begin{array}{c}{G}_{\gamma }\left(\left(\tau \mathfrak{e}+\left(1-\tau \right)\mathfrak{g},\kappa \mathfrak{u}+\left(1-\kappa \right)\mathfrak{v}\right)\right)\hfill \\ =\left[G_{\ast }\left(\left(\tau \mathfrak{e}+\left(1-\tau \right)\mathfrak{g},\kappa \mathfrak{u}+\left(1-\kappa \right)\mathfrak{v}\right), \gamma \right), G^{\ast }\left(\left(\tau \mathfrak{e}+\left(1-\tau \right)\mathfrak{g},\kappa \mathfrak{u}+\left(1-\kappa \right)\mathfrak{v}\right), \gamma \right)\right]\hfill \\ {\supseteq }_I\mathrm{Ԓ}\left(\tau \right)\mathrm{Ԓ}\left(\kappa \right)\left[G_{\ast }\left(\left(\mathfrak{e},\mathfrak{u}\right), \gamma \right), G^{\ast }\left(\left(\mathfrak{e},\mathfrak{u}\right), \gamma \right)\right]+\mathrm{Ԓ}\left(\tau \right)\mathrm{Ԓ}\left(1-\kappa \right)[G_{\ast }((\mathfrak{e},\mathfrak{v}), \gamma ), G^{\ast }((\mathfrak{e},\mathfrak{v}), \gamma )]+\hfill \\ \mathrm{Ԓ}\left(\kappa \right)\mathrm{Ԓ}\left(1-\tau \right)\left[G_{\ast }\left(\left(\mathfrak{e},\mathfrak{u}\right), \gamma \right), G^{\ast }\left(\left(\mathfrak{e},\mathfrak{u}\right), \gamma \right)\right]+\mathrm{Ԓ}\left(1-\tau \right)\mathrm{Ԓ}\left(1-\kappa \right)\left[G_{\ast }\left(\left(\mathfrak{e},\mathfrak{v}\right), \gamma \right), G^{\ast }\left(\left(\mathfrak{e},\mathfrak{v}\right), \gamma \right)\right]\hfill \end{array}$$

That is

$$\begin{array}{c}\tilde{G}\left(\tau \mathfrak{e}+\left(1-\tau \right)\mathfrak{g},\kappa \mathfrak{u}+\left(1-\kappa \right)\mathfrak{v}\right)\\ {\supseteq }_{\mathbb{F}}\mathrm{Ԓ}\left(\tau \right)\mathrm{Ԓ}\left(\kappa \right)\tilde{G}\left(\mathfrak{e},\mathfrak{u}\right)\oplus \mathrm{Ԓ}\left(\tau \right)\mathrm{Ԓ}\left(1-\kappa \right)\tilde{G}\left(\mathfrak{e},\mathfrak{v}\right)\oplus \mathrm{Ԓ}\left(1-\tau \right)\mathrm{Ԓ}\left(1-\kappa \right)\tilde{G}\left(\mathfrak{g},\mathfrak{u}\right)\oplus \\ \mathrm{Ԓ}\left(1-\tau \right)\mathrm{Ԓ}\left(1-\kappa \right)\tilde{G}\left(\mathfrak{g},\mathfrak{v}\right),\end{array}$$

and hence, $\tilde{G}$ is a coordinated $\mathsf{U}\mathsf{D}$-$\mathbf{Ԓ}$-convex $\mathsf{F}\mathsf{N}\mathsf{V}\mathsf{M}$ on $∆.$

Conversely, let $\tilde{G}$ be a coordinated $\mathsf{U}\mathsf{D}$-$\mathbf{Ԓ}$-convex $\mathsf{F}\mathsf{N}\mathsf{V}\mathsf{M}$ on $∆.$ Then, for all$\left(\mathfrak{e},\mathfrak{g}\right), \left(\mathfrak{u},\mathfrak{v}\right)\in ∆, \tau$ and $\kappa \in \left[0, 1\right],$ we have

$$\begin{array}{c}\tilde{G}\left(\tau \mathfrak{e}+\left(1-\tau \right)\mathfrak{g},\kappa \mathfrak{u}+\left(1-\kappa \right)\mathfrak{v}\right)\\ {\supseteq }_{\mathbb{F}}\mathrm{Ԓ}\left(\tau \right)\mathrm{Ԓ}\left(\kappa \right)\tilde{G}\left(\mathfrak{e},\mathfrak{u}\right)\oplus \mathrm{Ԓ}\left(\tau \right)\mathrm{Ԓ}\left(1-\kappa \right)\tilde{G}\left(\mathfrak{e},\mathfrak{v}\right)\oplus \mathrm{Ԓ}\left(1-\tau \right)\mathrm{Ԓ}\left(\kappa \right)\tilde{G}\left(\mathfrak{g},\mathfrak{u}\right)\oplus \mathrm{Ԓ}(1-\tau )\mathrm{Ԓ}\left(1-\kappa \right)\tilde{G}\left(\mathfrak{g},\mathfrak{v}\right).\end{array}$$

Therefore, again from Equation (22), for each $\gamma \in \left[0, 1\right]$, we have

$$\begin{array}{c}{G}_{\gamma }\left(\left(\tau \mathfrak{e}+\left(1-\tau \right)\mathfrak{g},\kappa \mathfrak{u}+\left(1-\kappa \right)\mathfrak{v}\right)\right)\\ =\left[G_{\ast }\left(\left(\tau \mathfrak{e}+\left(1-\tau \right)\mathfrak{g},\kappa \mathfrak{u}+\left(1-\kappa \right)\mathfrak{v}\right), \gamma \right), G^{\ast }\left(\left(\tau \mathfrak{e}+\left(1-\tau \right)\mathfrak{g},\kappa \mathfrak{u}+\left(1-\kappa \right)\mathfrak{v}\right), \gamma \right)\right].\end{array}$$

Again, from Equations (3) and (5), we obtain

$$\begin{array}{c}\mathrm{Ԓ}\left(\tau \right)\mathrm{Ԓ}\left(\kappa \right)G_{\gamma }\left(\mathfrak{e},\mathfrak{u}\right)+\mathrm{Ԓ}\left(\tau \right)\mathrm{Ԓ}\left(1-\kappa \right)G_{\gamma }\left(\mathfrak{e},\mathfrak{v}\right)+\mathrm{Ԓ}\left(1-\tau \right)\mathrm{Ԓ}\left(\kappa \right)G_{\gamma }\left(\mathfrak{g},\mathfrak{u}\right)+\mathrm{Ԓ}\left(1-\tau \right)\mathrm{Ԓ}\left(1-\kappa \right)G_{\gamma }\left(\mathfrak{g},\mathfrak{v}\right)\\ =\mathrm{Ԓ}\left(\tau \right)\mathrm{Ԓ}\left(\kappa \right)\left[G_{\ast }\left(\left(\mathfrak{e},\mathfrak{u}\right), \gamma \right), G^{\ast }\left(\left(\mathfrak{e},\mathfrak{u}\right), \gamma \right)\right]+\mathrm{Ԓ}\left(\tau \right)\mathrm{Ԓ}\left(1-\kappa \right)\left[G_{\ast }\left(\left(\mathfrak{e},\mathfrak{v}\right), \gamma \right), G^{\ast }\left(\left(\mathfrak{e},\mathfrak{v}\right), \gamma \right)\right]\\ +\mathrm{Ԓ}\left(\kappa \right)\mathrm{Ԓ}\left(1-\tau \right)\left[G_{\ast }\left(\left(\mathfrak{e},\mathfrak{u}\right), \gamma \right), G^{\ast }\left(\left(\mathfrak{e},\mathfrak{u}\right), \gamma \right)\right]+\mathrm{Ԓ}\left(1-\tau \right)\mathrm{Ԓ}\left(1-\kappa \right)\left[G_{\ast }\left(\left(\mathfrak{e},\mathfrak{v}\right), \gamma \right), G^{\ast }\left(\left(\mathfrak{e},\mathfrak{v}\right), \gamma \right)\right],\end{array}$$

for all $\mathcal{x},\omega \in ∆$ and $\tau \in \left[0, 1\right].$ Then, through the coordinated $\mathsf{U}\mathsf{D}$-$\mathbf{Ԓ}$-convexity of $\tilde{G}$, we have, for all $\mathcal{x},\omega \in ∆$ and $\tau \in \left[0, 1\right],$ that

$$\begin{array}{c}{G}_{\ast }\left(\left(\tau \mathfrak{e}+\left(1-\tau \right)\mathfrak{g},\kappa \mathfrak{u}+\left(1-\kappa \right)\mathfrak{v}\right), \gamma \right)\\ \le \mathrm{Ԓ}\left(\tau \right)\mathrm{Ԓ}\left(\kappa \right)G_{\ast }\left(\mathfrak{e},\mathfrak{u}\right)+\mathrm{Ԓ}\left(\tau \right)\mathrm{Ԓ}\left(1-\kappa \right)G_{\ast }\left(\mathfrak{e},\mathfrak{v}\right)+\mathrm{Ԓ}\left(1-\tau \right)\mathrm{Ԓ}\left(\kappa \right)G_{\ast }\left(\mathfrak{g},\mathfrak{u}\right)+\mathrm{Ԓ}\left(1-\tau \right)\mathrm{Ԓ}\left(1-\kappa \right)G_{\ast }\left(\mathfrak{g},\mathfrak{v}\right), \end{array}$$

and

$$\begin{array}{c}{G}^{\ast }\left(\left(\tau \mathfrak{e}+\left(1-\tau \right)\mathfrak{g},\kappa \mathfrak{u}+\left(1-\kappa \right)\mathfrak{v}\right), \gamma \right)\\ \ge \mathrm{Ԓ}\left(\tau \right)\mathrm{Ԓ}\left(\kappa \right)G^{\ast }\left(\mathfrak{e},\mathfrak{u}\right)+\mathrm{Ԓ}\left(\tau \right)\mathrm{Ԓ}\left(1-\kappa \right)G^{\ast }\left(\mathfrak{e},\mathfrak{v}\right)+\mathrm{Ԓ}\left(1-\tau \right)\mathrm{Ԓ}\left(\kappa \right)G^{\ast }\left(\mathfrak{g},\mathfrak{u}\right)+\mathrm{Ԓ}\left(1-\tau \right)\mathrm{Ԓ}\left(1-\kappa \right)G^{\ast }\left(\mathfrak{g},\mathfrak{v}\right), \end{array}$$

for each $\gamma \in \left[0, 1\right].$ Hence, the result follows. □

Example 1.

We consider the $\mathsf{F}\mathsf{N}\mathsf{V}\mathsf{M}$  $\tilde{G}:\left[0, 1\right]\times \left[0, 1\right]\to {\mathbb{F}}_0$ defined by

$$G\left(x\right)\left(\sigma \right)=\left\{\begin{array}{c}\frac{\sigma -xy}{5-xy}, \sigma \in \left[xy, 5\right]\\ \frac{\left(6+e^x\right)\left(6+e^y\right)-\sigma }{\left(6+e^x\right)\left(6+e^y\right)-5}, \sigma \in \left(5, \left(6+e^x\right)\left(6+e^y\right)\right]\\ 0 , otherwise,\end{array}\right.$$

(23)

Then, for each $\gamma \in \left[0, 1\right],$ we have $G_{\gamma }\left(x\right)=\left[\left(1-\gamma \right)xy+5\gamma ,\left(1-\gamma \right)\left(6+e^x\right)\left(6+e^y\right)+5\gamma \right]$. Since the endpoint functions $G_{\ast }\left(\left(x,y\right),\gamma \right)$ and $G^{\ast }\left(\left(x,y\right),\gamma \right)$ are coordinate $\mathrm{Ԓ}$-concave functions for each $\gamma \in \left[0, 1\right]$,$\tilde{G}\left(x,y\right)$ is a coordinate $\mathsf{U}\mathsf{D}$-$\mathrm{Ԓ}$-convex $\mathsf{F}\mathsf{N}\mathsf{V}\mathsf{M}$.

From Lemma 1 and Example 1, we can easily note that each $\mathsf{U}\mathsf{D}$-$\mathrm{Ԓ}$-convex $\mathsf{F}\mathsf{N}\mathsf{V}\mathsf{M}$ is a coordinated $\mathsf{U}\mathsf{D}$-$\mathrm{Ԓ}$-convex $\mathsf{F}\mathsf{N}\mathsf{V}\mathsf{M}$. But the converse is not true.

Remark 1.

If one assumes that$\mathrm{Ԓ}\left(\tau \right)=\tau , \mathrm{Ԓ}\left(\kappa \right)=\kappa$ and $G_{\ast }\left(\left(x,y\right),\gamma \right)=G^{\ast }\left(\left(x,y\right),\gamma \right)$ with $\gamma =1$, then $G$ is referred to as a coordinated convex function if $G$ meets the stated inequality here:

$$\begin{array}{c}G\left(\tau \mathfrak{e}+\left(1-\tau \right)\mathfrak{g},\kappa \mathfrak{u}+\left(1-\kappa \right)\mathfrak{v}\right)\\ \le \tau \kappa G\left(\mathfrak{e},\mathfrak{u}\right)+\tau \left(1-\kappa \right)G\left(\mathfrak{e},\mathfrak{v}\right)+\left(1-\tau \right)\kappa G\left(\mathfrak{g},\mathfrak{u}\right)+\left(1-\tau \right)\left(1-\kappa \right)G\left(\mathfrak{g},\mathfrak{v}\right).\end{array}$$

(24)

Let one assume that $\mathrm{Ԓ}\left(\tau \right)=\tau , \mathrm{Ԓ}\left(\kappa \right)=\kappa$ and $G_{\ast }\left(\left(x,y\right),\gamma \right)\ne G^{\ast }\left(\left(x,y\right),\gamma \right)$ with $\gamma =1$, $G_{\ast }\left(\left(x,y\right),\gamma \right)$ is an affine function, and $G^{\ast }\left(\left(x,y\right),\gamma \right)$ is a concave function. Then, the stated inequality here, (see [68])

$$\begin{array}{c}G\left(\tau \mathfrak{e}+\left(1-\tau \right)\mathfrak{g},\kappa \mathfrak{u}+\left(1-\kappa \right)\mathfrak{v}\right)\\ \supseteq \tau \kappa G\left(\mathfrak{e},\mathfrak{u}\right)+\tau \left(1-\kappa \right)G\left(\mathfrak{e},\mathfrak{v}\right)+\left(1-\tau \right)\kappa G\left(\mathfrak{g},\mathfrak{u}\right)+\left(1-\tau \right)\left(1-\kappa \right)G\left(\mathfrak{g},\mathfrak{v}\right),\end{array}$$

(25)

is true.

Definition 6.

Let $\tilde{G}:∆\to {\mathbb{F}}_0$ be an $\mathsf{F}\mathsf{N}\mathsf{V}\mathsf{M}$ on $∆$. Then, from $\gamma$-levels, we obtain that the collection of IVMs $G_{\gamma }:∆\to {\mathbb{R}}_I^{+}\subset {\mathbb{R}}_I$ is given by

$$G_{\gamma }\left(x,y\right)=\left[G_{\ast }\left(\left(x,y\right),\gamma \right), G^{\ast }\left(\left(x,y\right),\gamma \right)\right],$$

(26)

for all $\left(x,y\right)\in ∆$ and for all $\gamma \in \left[0, 1\right]$. Then, $\tilde{G}$ is a coordinated left-$\mathsf{U}\mathsf{D}$-$\mathrm{Ԓ}$-convex (concave) $\mathsf{F}\mathsf{N}\mathsf{V}\mathsf{M}$ on $∆$ if and only if for all $\gamma \in \left[0, 1\right],$ $G_{\ast }\left(\left(x,y\right), \gamma \right)$ and $G^{\ast }\left(\left(x,y\right), \gamma \right)$ are coordinated $\mathrm{Ԓ}$-convex (concave) and affine functions on $∆$, respectively.

Definition 7.

Let $\tilde{G}:∆\to {\mathbb{F}}_0$ be an $\mathsf{F}\mathsf{N}\mathsf{V}\mathsf{M}$ on $∆$. Then, from $\gamma$-levels, we obtain that the collection of IVMs $G_{\gamma }:∆\to {\mathbb{R}}_I^{+}\subset {\mathbb{R}}_I$ is given by

$$G_{\gamma }\left(x,y\right)=\left[G_{\ast }\left(\left(x,y\right),\gamma \right), G^{\ast }\left(\left(x,y\right),\gamma \right)\right],$$

for all $\left(x,y\right)\in ∆$ and for all $\gamma \in \left[0, 1\right]$. Then, $\tilde{G}$ is a coordinated right-$\mathsf{U}\mathsf{D}$-$\mathrm{Ԓ}$-convex (concave) $\mathsf{F}\mathsf{N}\mathsf{V}\mathsf{M}$ on $∆$ if and only if for all $\gamma \in \left[0, 1\right],$ $G_{\ast }\left(\left(x,y\right), \gamma \right)$ and $G^{\ast }\left(\left(x,y\right), \gamma \right)$ are coordinated $\mathrm{Ԓ}$-affine and $\mathrm{Ԓ}$-convex (concave) functions on $∆$, respectively.

Theorem 7.

Let $∆$ be a coordinated convex set, and let $\tilde{G}:∆\to {\mathbb{F}}_0^{+}$ be an $\mathsf{F}\mathsf{N}\mathsf{V}\mathsf{M}$. Then, from $\gamma$-levels, we obtain that the collection of IVMs $G_{\gamma }:∆\to {\mathbb{R}}_I^{+}\subset {\mathbb{R}}_I$ is given by

$$G_{\gamma }\left(x,y\right)=\left[G_{\ast }\left(\left(x,y\right),\gamma \right), G^{\ast }\left(\left(x,y\right),\gamma \right)\right],$$

for all $\left(x,y\right)\in ∆$ and for all $\gamma \in \left[0, 1\right]$. Then, $\tilde{G}$ is a coordinated $\mathsf{U}\mathsf{D}$-$\mathrm{Ԓ}$-concave $\mathsf{F}\mathsf{N}\mathsf{V}\mathsf{M}$ on $∆$ if and only if for all $\gamma \in \left[0, 1\right],$ $G_{\ast }\left(\left(x,y\right), \gamma \right)$ and $G^{\ast }\left(\left(x,y\right), \gamma \right)$ are coordinated $\mathrm{Ԓ}$-concave and $\mathrm{Ԓ}$-convex functions, respectively.

Proof. 

The demonstration of the proof of Theorem 7 is similar to the demonstration of the proof of Theorem 6. □

Example 2.

We consider the $\mathsf{F}\mathsf{N}\mathsf{V}\mathsf{M}$s $\tilde{G}:\left[0, 1\right]\times \left[0, 1\right]\to {\mathbb{F}}_0^{+}$ defined by

$$\tilde{G}\left(x,y\right)\left(\sigma \right)=\left\{\begin{array}{c}\frac{\sigma -\left(6-e^x\right)\left(6-e^y\right)}{\left(6-e^x\right)\left(6-e^y\right)-25}, \sigma \in \left[\left(6-e^x\right)\left(6-e^y\right), 25\right]\\ \frac{35xy-\sigma }{35xy-25}, \sigma \in \left(25, 35xy\right]\\ 0, otherwise.\end{array}\right.$$

(27)

Then, for each $\gamma \in \left[0, 1\right],$ we have $G_{\gamma }\left(x,y\right)=\left[\left(1-\gamma \right)\left(6-e^x\right)\left(6-e^y\right)+25\gamma ,35\left(1-\gamma \right)xy+25\gamma \right]$. Since the endpoint functions $G_{\ast }\left(\left(x,y\right),\gamma \right) and$ $G^{\ast }\left(\left(x,y\right),\gamma \right)$ are coordinated $\mathrm{Ԓ}$-concave and $\mathrm{Ԓ}$-convex functions for each $\gamma \in \left[0, 1\right]$,$\tilde{G}\left(x,y\right)$ is a coordinated $\mathsf{U}\mathsf{D}$-$\mathrm{Ԓ}$-concave $\mathsf{F}\mathsf{N}\mathsf{V}\mathsf{M}$.

3. Main Results

Here is the first result of the coordinated integral inequalities of the Hermite–Hadamard type using fuzzy fractional operators via coordinated $\mathsf{U}\mathsf{D}$-$\mathbf{Ԓ}$-concave $\mathsf{F}\mathsf{N}\mathsf{V}\mathsf{M}$s.

Theorem 8.

Let $\tilde{G}:∆\to {\mathbb{F}}_0^{+}$ be a coordinated $\mathsf{U}\mathsf{D}$-$\mathrm{Ԓ}$-convex $\mathsf{F}\mathsf{N}\mathsf{V}\mathsf{M}$ on $∆,$ and let $\mathrm{Ԓ} :\left[0, 1\right]\to {\mathbb{R}}^{+}$. Then, from $\gamma$-cuts, we set up the sequence of $\mathsf{I}\mathsf{V}\mathsf{M}s$  $G_{\gamma }:∆\to {\mathbb{R}}_I^{+}, which$ is given by $G_{\gamma }\left(x,y\right)=\left[G_{\ast }\left(\left(x,y\right),\gamma \right), G^{\ast }\left(\left(x,y\right),\gamma \right)\right]$ for all $\left(x,y\right)\in ∆$ and for all $\gamma \in \left[0, 1\right]$. If ${\tilde{G}\in \mathcal{F}\mathfrak{O}}_{∆}$, then the following inequalities hold:

$$\begin{array}{c}\frac{1}{{\mathrm{Ԓ}}^2\left(\frac{1}{2}\right)}\tilde{G}\left(\frac{\mathfrak{e}+\mathfrak{g}}{2}, \frac{\mathfrak{u}+\mathfrak{v}}{2}\right)\\ {\supseteq }_{\mathbb{F}}\frac{\Gamma \left(\alpha +1\right)}{{2\mathrm{Ԓ}\left(\frac{1}{2}\right)\left(\mathfrak{g}-\mathfrak{e}\right)}^{\alpha }}\left[{\mathcal{I}}_{{\mathfrak{e}}^{+}}^{\alpha }\tilde{G}\left(\mathfrak{g},\frac{\mathfrak{u}+\mathfrak{v}}{2}\right)\oplus {\mathcal{I}}_{{\mathfrak{g}}^{-}}^{\alpha }\tilde{G}\left(\mathfrak{e},\frac{\mathfrak{u}+\mathfrak{v}}{2}\right)\right]\oplus \frac{\Gamma \left(\beta +1\right)}{{2\mathrm{Ԓ}\left(\frac{1}{2}\right)\left(\mathfrak{v}-\mathfrak{u}\right)}^{\beta }}\left[{\mathcal{I}}_{{\mathfrak{u}}^{+}}^{\beta }\tilde{G}\left(\frac{\mathfrak{e}+\mathfrak{g}}{2},\mathfrak{v}\right)\oplus {\mathcal{I}}_{{\mathfrak{v}}^{-}}^{\beta }\tilde{G}\left(\frac{\mathfrak{e}+\mathfrak{g}}{2},\mathfrak{u}\right)\right]\\ {\supseteq }_{\mathbb{F}}\frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{{\left(\mathfrak{g}-\mathfrak{e}\right)}^{\alpha }{\left(\mathfrak{v}-\mathfrak{u}\right)}^{\beta }}\left[{\mathcal{I}}_{{\mathfrak{e}}^{+},{\mathfrak{u}}^{+}}^{\alpha , \beta }\tilde{G}\left(\mathfrak{g},\mathfrak{v}\right)\oplus {\mathcal{I}}_{{\mathfrak{e}}^{+},{\mathfrak{v}}^{-}}^{\alpha , \beta }\tilde{G}\left(\mathfrak{g},\mathfrak{u}\right)\oplus {\mathcal{I}}_{{\mathfrak{g}}^{-},{\mathfrak{u}}^{+}}^{\alpha , \beta }\tilde{G}\left(\mathfrak{e},\mathfrak{v}\right)\oplus {\mathcal{I}}_{{\mathfrak{g}}^{-},{\mathfrak{v}}^{-}}^{\alpha , \beta }\tilde{G}\left(\mathfrak{e},\mathfrak{u}\right)\right]\\ {\supseteq }_{\mathbb{F}}\frac{\beta \Gamma \left(\alpha +1\right)}{{\left(\mathfrak{g}-\mathfrak{e}\right)}^{\alpha }}\left[{\mathcal{I}}_{{\mathfrak{e}}^{+}}^{\alpha }\tilde{G}\left(\mathfrak{g},\mathfrak{u}\right)\oplus {\mathcal{I}}_{{\mathfrak{e}}^{+}}^{\alpha }\tilde{G}\left(\mathfrak{g},\mathfrak{v}\right)\oplus {\mathcal{I}}_{{\mathfrak{g}}^{-}}^{\alpha }\tilde{G}\left(\mathfrak{e},\mathfrak{u}\right)\oplus {\mathcal{I}}_{{\mathfrak{g}}^{-}}^{\alpha }\tilde{G}\left(\mathfrak{e},\mathfrak{v}\right)\right]\times {\int }_0^1{\kappa }^{\beta -1}\left[\mathrm{Ԓ}\left(\kappa \right)+\mathrm{Ԓ}\left(1-\kappa \right)\right]d\kappa \\ \oplus \frac{\alpha \Gamma \left(\beta +1\right)}{{\left(\mathfrak{v}-\mathfrak{u}\right)}^{\beta }}\left[{\mathcal{I}}_{{\mathfrak{u}}^{+}}^{ \beta }\tilde{G}\left(\mathfrak{e},\mathfrak{v}\right)\oplus {\mathcal{I}}_{{\mathfrak{v}}^{-}}^{\beta }\tilde{G}\left(\mathfrak{g},\mathfrak{u}\right)\oplus {\mathcal{I}}_{{\mathfrak{u}}^{+}}^{\beta }\tilde{G}\left(\mathfrak{g},\mathfrak{v}\right)\oplus {\mathcal{I}}_{{\mathfrak{v}}^{-}}^{\beta }\tilde{G}\left(\mathfrak{g},\mathfrak{u}\right)\right]\times {\int }_0^1{\tau }^{\alpha -1}\mathrm{Ԓ}\left(\tau \right)+\mathrm{Ԓ}\left(1-\tau \right)d\tau \\ {\supseteq }_{\mathbb{F}}\alpha \beta \left[\tilde{G}\left(\mathfrak{e},\mathfrak{u}\right)\oplus \tilde{G}\left(\mathfrak{g},\mathfrak{u}\right)\oplus \tilde{G}\left(\mathfrak{e},\mathfrak{v}\right)\oplus \tilde{G}\left(\mathfrak{g},\mathfrak{v}\right)\right]\times {\int }_0^1{\kappa }^{\beta -1}\left[\mathrm{Ԓ}\left(\kappa \right)+\mathrm{Ԓ}\left(1-\kappa \right)\right]d\kappa {\int }_0^1{\tau }^{\alpha -1}\left[\mathrm{Ԓ}\left(\tau \right)+\mathrm{Ԓ}\left(1-\tau \right)\right]d\tau .\end{array}$$

(28)

If $\tilde{G}\left(x,y\right)$ is a coordinated $\mathsf{U}\mathsf{D}$-$\mathbf{Ԓ}$-concave $\mathsf{F}\mathsf{N}\mathsf{V}\mathsf{M},$ then

$$\begin{array}{c}\frac{1}{{\mathrm{Ԓ}}^2\left(\frac{1}{2}\right)}\tilde{G}\left(\frac{\mathfrak{e}+\mathfrak{g}}{2}, \frac{\mathfrak{u}+\mathfrak{v}}{2}\right)\\ {\subseteq }_{\mathbb{F}}\frac{\Gamma \left(\alpha +1\right)}{{2\mathrm{Ԓ}\left(\frac{1}{2}\right)\left(\mathfrak{g}-\mathfrak{e}\right)}^{\alpha }}\left[{\mathcal{I}}_{{\mathfrak{e}}^{+}}^{\alpha }\tilde{G}\left(\mathfrak{g},\frac{\mathfrak{u}+\mathfrak{v}}{2}\right)\oplus {\mathcal{I}}_{{\mathfrak{g}}^{-}}^{\alpha }\tilde{G}\left(\mathfrak{e},\frac{\mathfrak{u}+\mathfrak{v}}{2}\right)\right]\oplus \frac{\Gamma \left(\beta +1\right)}{{2\mathrm{Ԓ}\left(\frac{1}{2}\right)\left(\mathfrak{v}-\mathfrak{u}\right)}^{\beta }}\left[{\mathcal{I}}_{{\mathfrak{u}}^{+}}^{\beta }\tilde{G}\left(\frac{\mathfrak{e}+\mathfrak{g}}{2},\mathfrak{v}\right)\oplus {\mathcal{I}}_{{\mathfrak{v}}^{-}}^{\beta }\tilde{G}\left(\frac{\mathfrak{e}+\mathfrak{g}}{2},\mathfrak{u}\right)\right]\\ {\subseteq }_{\mathbb{F}}\frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{{\left(\mathfrak{g}-\mathfrak{e}\right)}^{\alpha }{\left(\mathfrak{v}-\mathfrak{u}\right)}^{\beta }}\left[{\mathcal{I}}_{{\mathfrak{e}}^{+},{\mathfrak{u}}^{+}}^{\alpha , \beta }\tilde{G}\left(\mathfrak{g},\mathfrak{v}\right)\oplus {\mathcal{I}}_{{\mathfrak{e}}^{+},{\mathfrak{v}}^{-}}^{\alpha , \beta }\tilde{G}\left(\mathfrak{g},\mathfrak{u}\right)\oplus {\mathcal{I}}_{{\mathfrak{g}}^{-},{\mathfrak{u}}^{+}}^{\alpha , \beta }\tilde{G}\left(\mathfrak{e},\mathfrak{v}\right)\oplus {\mathcal{I}}_{{\mathfrak{g}}^{-},{\mathfrak{v}}^{-}}^{\alpha , \beta }\tilde{G}\left(\mathfrak{e},\mathfrak{u}\right)\right]\\ {\subseteq }_{\mathbb{F}}\frac{\beta \Gamma \left(\alpha +1\right)}{{\left(\mathfrak{g}-\mathfrak{e}\right)}^{\alpha }}\left[{\mathcal{I}}_{{\mathfrak{e}}^{+}}^{\alpha }\tilde{G}\left(\mathfrak{g},\mathfrak{u}\right)\oplus {\mathcal{I}}_{{\mathfrak{e}}^{+}}^{\alpha }\tilde{G}\left(\mathfrak{g},\mathfrak{v}\right)\oplus {\mathcal{I}}_{{\mathfrak{g}}^{-}}^{\alpha }\tilde{G}\left(\mathfrak{e},\mathfrak{u}\right)\oplus {\mathcal{I}}_{{\mathfrak{g}}^{-}}^{\alpha }\tilde{G}\left(\mathfrak{e},\mathfrak{v}\right)\right]\times {\int }_0^1{\kappa }^{\beta -1}\left[\mathrm{Ԓ}\left(\kappa \right)+\mathrm{Ԓ}\left(1-\kappa \right)\right]d\kappa \\ \oplus \frac{\alpha \Gamma \left(\beta +1\right)}{{\left(\mathfrak{v}-\mathfrak{u}\right)}^{\beta }}\left[{\mathcal{I}}_{{\mathfrak{u}}^{+}}^{ \beta }\tilde{G}\left(\mathfrak{e},\mathfrak{v}\right)\oplus {\mathcal{I}}_{{\mathfrak{v}}^{-}}^{\beta }\tilde{G}\left(\mathfrak{g},\mathfrak{u}\right)\oplus {\mathcal{I}}_{{\mathfrak{u}}^{+}}^{\beta }\tilde{G}\left(\mathfrak{g},\mathfrak{v}\right)\oplus {\mathcal{I}}_{{\mathfrak{v}}^{-}}^{\beta }\tilde{G}\left(\mathfrak{g},\mathfrak{u}\right)\right]\times {\int }_0^1{\tau }^{\alpha -1}\left[\mathrm{Ԓ}\left(\tau \right)+\mathrm{Ԓ}\left(1-\tau \right)\right]d\tau \\ {\subseteq }_{\mathbb{F}}\alpha \beta \left[\tilde{G}\left(\mathfrak{e},\mathfrak{u}\right)\oplus \tilde{G}\left(\mathfrak{g},\mathfrak{u}\right)\oplus \tilde{G}\left(\mathfrak{e},\mathfrak{v}\right)\oplus \tilde{G}\left(\mathfrak{g},\mathfrak{v}\right)\right]\times {\int }_0^1{\kappa }^{\beta -1}\left[\mathrm{Ԓ}\left(\kappa \right)+\mathrm{Ԓ}\left(1-\kappa \right)\right]d\kappa {\int }_0^1{\tau }^{\alpha -1}\left[\mathrm{Ԓ}\left(\tau \right)+\mathrm{Ԓ}\left(1-\tau \right)\right]d\tau .\end{array}$$

(29)

Proof. 

Let $\tilde{G}:\left[\mathfrak{e},\mathfrak{g}\right]\to {\mathbb{F}}_0$ be a coordinated $\mathsf{U}\mathsf{D}$-$\mathbf{Ԓ}$-convex $\mathsf{F}\mathsf{N}\mathsf{V}\mathsf{M}$. Then, from our hypothesis, we have

$$\frac{1}{{\mathrm{Ԓ}}^2\left(\frac{1}{2}\right)}\tilde{G}\left(\frac{\mathfrak{e}+\mathfrak{g}}{2},\frac{\mathfrak{u}+\mathfrak{v}}{2}\right){\supseteq }_{\mathbb{F}}\tilde{G}\left(\tau \mathfrak{e}+\left(1-\tau \right)\mathfrak{g},\tau \mathfrak{u}+\left(1-\tau \right)\mathfrak{v}\right)\oplus \tilde{G}\left(\left(1-\tau \right)\mathfrak{e}+\tau \mathfrak{g},\left(1-\tau \right)\mathfrak{u}+\tau \mathfrak{v}\right).$$

By using Theorem 6, for every $\gamma \in \left[0, 1\right]$, we have

$$\begin{array}{c}\frac{1}{{\mathrm{Ԓ}}^2\left(\frac{1}{2}\right)}{G}_{\ast }\left(\left(\frac{\mathfrak{e}+\mathfrak{g}}{2},\frac{\mathfrak{u}+\mathfrak{v}}{2}\right), \gamma \right) \\ \le G_{\ast }\left(\left(\tau \mathfrak{e}+\left(1-\tau \right)\mathfrak{g},\tau \mathfrak{u}+\left(1-\tau \right)\mathfrak{v}\right), \gamma \right)+G_{\ast }\left(\left(\left(1-\tau \right)\mathfrak{e}+\tau \mathfrak{g},\left(1-\tau \right)\mathfrak{u}+\tau \mathfrak{v}\right), \gamma \right), \\ \frac{1}{{\mathrm{Ԓ}}^2\left(\frac{1}{2}\right)}{G}^{\ast }\left(\left(\frac{\mathfrak{e}+\mathfrak{g}}{2},\frac{\mathfrak{u}+\mathfrak{v}}{2}\right), \gamma \right) \\ \ge G^{\ast }\left(\left(\tau \mathfrak{e}+\left(1-\tau \right)\mathfrak{g},\tau \mathfrak{u}+\left(1-\tau \right)\mathfrak{v}\right), \gamma \right)+G^{\ast }\left(\left(\left(1-\tau \right)\mathfrak{e}+\tau \mathfrak{g},\left(1-\tau \right)\mathfrak{u}+\tau \mathfrak{v}\right),\gamma \right).\end{array}$$

By using Lemma 1, we have

$$\begin{array}{c}\frac{1}{\mathrm{Ԓ}\left(\frac{1}{2}\right)}{G}_{\ast }\left(\left(x,\frac{\mathfrak{u}+\mathfrak{v}}{2}\right), \gamma \right)\le G_{\ast }\left(\left(x, \tau \mathfrak{u}+\left(1-\tau \right)\mathfrak{v}\right), \gamma \right)+G_{\ast }\left(\left(x, \left(1-\tau \right)\mathfrak{u}+\tau \mathfrak{v}\right), \gamma \right), \\ \frac{1}{\mathrm{Ԓ}\left(\frac{1}{2}\right)}{G}^{\ast }\left(\left(x,\frac{\mathfrak{u}+\mathfrak{v}}{2}\right), \gamma \right)\ge G^{\ast }\left(\left(x, \tau \mathfrak{u}+\left(1-\tau \right)\mathfrak{v}\right), \gamma \right)+G^{\ast }\left(\left(x, \left(1-\tau \right)\mathfrak{u}+\tau \mathfrak{v}\right),\gamma \right),\end{array}$$

(30)

$$\begin{array}{c}\frac{1}{\mathrm{Ԓ}\left(\frac{1}{2}\right)}{G}_{\ast }\left(\left(\frac{\mathfrak{e}+\mathfrak{g}}{2},y\right), \gamma \right)\le G_{\ast }\left(\left(\tau \mathfrak{e}+\left(1-\tau \right)\mathfrak{g},y\right), \gamma \right)+G_{\ast }\left(\left(\left(1-\tau \right)\mathfrak{e}+\tau \mathfrak{g},y\right), \gamma \right), \\ \frac{1}{\mathrm{Ԓ}\left(\frac{1}{2}\right)}{G}^{\ast }\left(\left(\frac{\mathfrak{e}+\mathfrak{g}}{2},y\right), \gamma \right)\ge G^{\ast }\left(\left(\tau \mathfrak{e}+\left(1-\tau \right)\mathfrak{g},y\right), \gamma \right)+G^{\ast }\left(\left(\left(1-\tau \right)\mathfrak{e}+\tau \mathfrak{g},y\right),\gamma \right).\end{array}$$

(31)

From (30) and (31), we have

$$\begin{array}{c} \frac{1}{\mathrm{Ԓ}\left(\frac{1}{2}\right)}\left[G_{\ast }\left(\left(x,\frac{\mathfrak{u}+\mathfrak{v}}{2}\right), \gamma \right),G^{\ast }\left(\left(x,\frac{\mathfrak{u}+\mathfrak{v}}{2}\right), \gamma \right)\right]\\ {\supseteq }_I\left[G_{\ast }\left(\left(x, \tau \mathfrak{u}+\left(1-\tau \right)\mathfrak{v}\right), \gamma \right),G^{\ast }\left(\left(x, \tau \mathfrak{u}+\left(1-\tau \right)\mathfrak{v}\right), \gamma \right)\right]\\ +\left[G_{\ast }\left(\left(x, \left(1-\tau \right)\mathfrak{u}+\tau \mathfrak{v}\right), \gamma \right),G^{\ast }\left(\left(x, \left(1-\tau \right)\mathfrak{u}+\tau \mathfrak{v}\right),\gamma \right)\right],\end{array}$$

and

$$\begin{array}{c}\frac{1}{\mathrm{Ԓ}\left(\frac{1}{2}\right)}\left[G_{\ast }\left(\left(\frac{\mathfrak{e}+\mathfrak{g}}{2},y\right), \gamma \right),G^{\ast }\left(\left(\frac{\mathfrak{e}+\mathfrak{g}}{2},y\right), \gamma \right)\right]\\ {\supseteq }_I\left[G_{\ast }\left(\left(\tau \mathfrak{e}+\left(1-\tau \right)\mathfrak{g},y\right), \gamma \right),G^{\ast }\left(\left(\tau \mathfrak{e}+\left(1-\tau \right)\mathfrak{g},y\right), \gamma \right)\right]\\ +\left[G_{\ast }\left(\left(\tau \mathfrak{e}+\left(1-\tau \right)\mathfrak{g},y\right), \gamma \right),G^{\ast }\left(\left(\tau \mathfrak{e}+\left(1-\tau \right)\mathfrak{g},y\right),\gamma \right)\right],\end{array}$$

It follows that

$$\frac{1}{\mathrm{Ԓ}\left(\frac{1}{2}\right)}{G}_{\gamma }\left(x,\frac{\mathfrak{u}+\mathfrak{v}}{2}\right){\supseteq }_I{G}_{\gamma }\left(x, \tau \mathfrak{u}+\left(1-\tau \right)\mathfrak{v}\right)+G_{\gamma }\left(x, \left(1-\tau \right)\mathfrak{u}+\tau \mathfrak{v}\right),$$

(32)

and

$$\frac{1}{\mathrm{Ԓ}\left(\frac{1}{2}\right)}{G}_{\gamma }\left(\frac{\mathfrak{e}+\mathfrak{g}}{2},y\right){\supseteq }_I{G}_{\gamma }\left(\tau \mathfrak{e}+\left(1-\tau \right)\mathfrak{g},y\right)+G_{\gamma }\left(\tau \mathfrak{e}+\left(1-\tau \right)\mathfrak{g},y\right).$$

(33)

Since $G_{\gamma }\left(x,.\right)$ and $G_{\gamma }\left(.,y\right)$ are both coordinated $\mathsf{U}\mathsf{D}$-$\mathbf{Ԓ}$-convex-$\mathsf{I}\mathsf{V}\mathsf{M}$s, then from inequality (15), for every $\gamma \in \left[0, 1\right]$, and from inequalities (32) and (43), we have

$$\begin{array}{cc}\frac{1}{\beta \mathrm{Ԓ}\left(\frac{1}{2}\right)}{G_{\gamma }}_x\left(\frac{\mathfrak{u}+\mathfrak{v}}{2}\right)& {\supseteq }_I\frac{\Gamma \left(\beta \right)}{{\left(\mathfrak{v}-\mathfrak{u}\right)}^{\beta }}\left[{\mathcal{I}}_{{\mathfrak{u}}^{+}}^{\beta }{G_{\gamma }}_x\left(\mathfrak{v}\right)+{\mathcal{I}}_{{\mathfrak{v}}^{-}}^{\beta } {G_{\gamma }}_x\left(\mathfrak{u}\right)\right]\\ & {\supseteq }_I\left[{G_{\gamma }}_x\left(\mathfrak{u}\right)+{G_{\gamma }}_x\left(\mathfrak{v}\right)\right]{\int }_0^1{\kappa }^{\beta -1}\left[\mathrm{Ԓ}\left(\kappa \right)+\mathrm{Ԓ}\left(1-\kappa \right)\right]d\kappa \end{array}$$

(34)

and

$$\begin{array}{cc}\frac{1}{\alpha \mathrm{Ԓ}\left(\frac{1}{2}\right)}{G_{\gamma }}_y\left(\frac{\mathfrak{e}+\mathfrak{g}}{2}\right)& {\supseteq }_I\frac{\Gamma \left(\alpha \right)}{{\left(\mathfrak{g}-\mathfrak{e}\right)}^{\alpha }}\left[{\mathcal{I}}_{{\mathfrak{e}}^{+}}^{\alpha }{G_{\gamma }}_y\left(\mathfrak{g}\right)+{\mathcal{I}}_{{\mathfrak{g}}^{-}}^{\alpha }{G_{\gamma }}_y\left(\mathfrak{e}\right)\right]\\ & {\supseteq }_I\left[{G_{\gamma }}_y\left(\mathfrak{e}\right)+{G_{\gamma }}_y\left(\mathfrak{g}\right)\right]{\int }_0^1{\tau }^{\alpha -1}\mathrm{Ԓ}\left(\tau \right)+\mathrm{Ԓ}\left(1-\tau \right)d\tau \end{array}$$

(35)

Since ${G_{\gamma }}_x\left(w\right)=G_{\gamma }\left(x,w\right)$, (34) can be written as

$$\begin{array}{cc}\frac{1}{\beta \mathrm{Ԓ}\left(\frac{1}{2}\right)}{G}_{\gamma }\left(x,\frac{\mathfrak{u}+\mathfrak{v}}{2}\right)& {\supseteq }_I\frac{\Gamma \left(\beta \right)}{{\left(\mathfrak{v}-\mathfrak{u}\right)}^{\beta }}\left[{\mathcal{I}}_{{\mathfrak{u}}^{+}}^{\alpha }{G}_{\gamma }\left(x,\mathfrak{v}\right)+{\mathcal{I}}_{{\mathfrak{v}}^{-}}^{\alpha }{G}_{\gamma }\left(x,\mathfrak{u}\right)\right]\\ & {\supseteq }_I\left[G_{\gamma }\left(x,\mathfrak{u}\right)+G_{\gamma }\left(x,\mathfrak{v}\right)\right]{\int }_0^1{\kappa }^{\beta -1}\left[\mathrm{Ԓ}\left(\kappa \right)+\mathrm{Ԓ}\left(1-\kappa \right)\right]d\kappa .\end{array}$$

(36)

That is

$$\begin{array}{cc}\frac{1}{\beta \mathrm{Ԓ}\left(\frac{1}{2}\right)}{G}_{\gamma }\left(x,\frac{\mathfrak{u}+\mathfrak{v}}{2}\right)& {\supseteq }_I\frac{1}{{\left(\mathfrak{v}-\mathfrak{u}\right)}^{\beta }}\left[ {\int }_{\mathfrak{u}}^{\mathfrak{v}}{\left(\mathfrak{v}-\kappa \right)}^{\beta -1}{G}_{\gamma }\left(x,\kappa \right)d\kappa +{\int }_{\mathfrak{u}}^{\mathfrak{v}}{\left(\kappa -\mathfrak{u}\right)}^{\beta -1}{G}_{\gamma }\left(x,\kappa \right)d\kappa \right]\\ & {\supseteq }_I\left[G_{\gamma }\left(x,\mathfrak{u}\right)+G_{\gamma }\left(x,\mathfrak{v}\right)\right]{\int }_0^1{\kappa }^{\beta -1}\left[\mathrm{Ԓ}\left(\kappa \right)+\mathrm{Ԓ}\left(1-\kappa \right)\right]d\kappa .\end{array}$$

Multiplying the double inequality of (36) by $\frac{{\left(\mathfrak{g}-x\right)}^{\alpha -1}}{{\left(\mathfrak{g}-\mathfrak{e}\right)}^{\alpha }}$ and integrating with respect to $x$ over $\left[\mathfrak{e},\mathfrak{g}\right],$ we have

$$\begin{array}{c}\frac{1}{{\beta \left(\mathfrak{g}-\mathfrak{e}\right)}^{\alpha }\mathrm{Ԓ}\left(\frac{1}{2}\right)}{\int }_{\mathfrak{e}}^{\mathfrak{g}}{G}_{\gamma }\left(x,\frac{\mathfrak{u}+\mathfrak{v}}{2}\right){\left(\mathfrak{g}-x\right)}^{\alpha -1}dx\\ {\supseteq }_I\frac{1}{{{\left(\mathfrak{g}-\mathfrak{e}\right)}^{\alpha }\left(\mathfrak{v}-\mathfrak{u}\right)}^{\beta }}{\int }_{\mathfrak{e}}^{\mathfrak{g}}{\int }_{\mathfrak{u}}^{\mathfrak{v}}{\left(\mathfrak{g}-x\right)}^{\alpha -1}{\left(\mathfrak{v}-\kappa \right)}^{\beta -1}{G}_{\gamma }\left(x,\kappa \right)d\kappa dx+{\int }_{\mathfrak{e}}^{\mathfrak{g}}{\int }_{\mathfrak{u}}^{\mathfrak{v}}{\left(\mathfrak{g}-x\right)}^{\alpha -1}{\left(\kappa -\mathfrak{u}\right)}^{\beta -1}{G}_{\gamma }\left(x,\kappa \right)d\kappa dx\\ {\supseteq }_I\frac{1}{{\left(\mathfrak{g}-\mathfrak{e}\right)}^{\alpha }}\left[{\int }_{\mathfrak{e}}^{\mathfrak{g}}{\left(\mathfrak{g}-x\right)}^{\alpha -1}{G}_{\gamma }\left(x,\mathfrak{u}\right)dx+{\int }_{\mathfrak{e}}^{\mathfrak{g}}{\left(\mathfrak{g}-x\right)}^{\alpha -1}{G}_{\gamma }\left(x,\mathfrak{v}\right)dx\right]{\int }_0^1{\kappa }^{\beta -1}\left[\mathrm{Ԓ}\left(\kappa \right)+\mathrm{Ԓ}\left(1-\kappa \right)\right]d\kappa .\end{array}$$

(37)

Again, multiplying the double inequality of (36) by $\frac{{\left(x-\mathfrak{e}\right)}^{\alpha -1}}{{\left(\mathfrak{g}-\mathfrak{e}\right)}^{\alpha }}$ and integrating with respect to $x$ over $\left[\mathfrak{e},\mathfrak{g}\right],$ we have

$$\begin{array}{c}\frac{1}{{\beta \left(\mathfrak{g}-\mathfrak{e}\right)}^{\alpha }\mathrm{Ԓ}\left(\frac{1}{2}\right)}{\int }_{\mathfrak{e}}^{\mathfrak{g}}{G}_{\gamma }\left(x,\frac{\mathfrak{u}+\mathfrak{v}}{2}\right){\left(x-\mathfrak{e}\right)}^{\alpha -1}dx\\ {\supseteq }_I\frac{1}{{\left(\mathfrak{g}-\mathfrak{e}\right)}^{\alpha }{\left(\mathfrak{v}-\mathfrak{u}\right)}^{\beta }}{\int }_{\mathfrak{e}}^{\mathfrak{g}}{\int }_{\mathfrak{u}}^{\mathfrak{v}}{\left(x-\mathfrak{e}\right)}^{\alpha -1}{\left(\mathfrak{v}-\kappa \right)}^{\beta -1}{G}_{\gamma }\left(x,\kappa \right)d\kappa dx\\ +\frac{1}{{\left(\mathfrak{g}-\mathfrak{e}\right)}^{\alpha }{\left(\mathfrak{v}-\mathfrak{u}\right)}^{\beta }}{\int }_{\mathfrak{e}}^{\mathfrak{g}}{\int }_{\mathfrak{u}}^{\mathfrak{v}}{\left(x-\mathfrak{e}\right)}^{\alpha -1}{\left(\kappa -\mathfrak{u}\right)}^{\beta -1}{G}_{\gamma }\left(x,\kappa \right)d\kappa dx\\ {\supseteq }_I\frac{1}{{\left(\mathfrak{g}-\mathfrak{e}\right)}^{\alpha }}\left[{\int }_{\mathfrak{e}}^{\mathfrak{g}}{\left(x-\mathfrak{e}\right)}^{\alpha -1}{G}_{\gamma }\left(x,\mathfrak{u}\right)dx+{\int }_{\mathfrak{e}}^{\mathfrak{g}}{\left(x-\mathfrak{e}\right)}^{\alpha -1}{G}_{\gamma }\left(x,\mathfrak{v}\right)dx\right]{\int }_0^1{\kappa }^{\beta -1}\left[\mathrm{Ԓ}\left(\kappa \right)+\mathrm{Ԓ}\left(1-\kappa \right)\right]d\kappa .\end{array}$$

(38)

From (37), we have

$$\begin{array}{c}\frac{\Gamma \left(\alpha +1\right)}{{2\mathrm{Ԓ}\left(\frac{1}{2}\right)\left(\mathfrak{g}-\mathfrak{e}\right)}^{\alpha }}\left[{\mathcal{I}}_{{\mathfrak{e}}^{+}}^{\alpha }{G}_{\gamma }\left(\mathfrak{g},\frac{\mathfrak{u}+\mathfrak{v}}{2}\right)\right]\\ {\supseteq }_I\frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{{\left(\mathfrak{g}-\mathfrak{e}\right)}^{\alpha }{\left(\mathfrak{v}-\mathfrak{u}\right)}^{\beta }}\left[{\mathcal{I}}_{{\mathfrak{e}}^{+},{\mathfrak{u}}^{+}}^{\alpha , \beta }{G}_{\gamma }\left(\mathfrak{g},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathfrak{g}}^{-},{\mathfrak{u}}^{+}}^{\alpha , \beta }{G}_{\gamma }\left(\mathfrak{g},\mathfrak{u}\right)\right]\\ {\supseteq }_I\frac{\beta \Gamma \left(\alpha +1\right)}{{\left(\mathfrak{g}-\mathfrak{e}\right)}^{\alpha }}\left[{\mathcal{I}}_{{\mathfrak{e}}^{+}}^{\alpha }{G}_{\gamma }\left(\mathfrak{g},\mathfrak{u}\right)+{\mathcal{I}}_{{\mathfrak{e}}^{+}}^{\alpha }{G}_{\gamma }\left(\mathfrak{g},\mathfrak{v}\right)\right]{\int }_0^1{\kappa }^{\beta -1}\left[\mathrm{Ԓ}\left(\kappa \right)+\mathrm{Ԓ}\left(1-\kappa \right)\right]d\kappa .\end{array}$$

(39)

From (38), we have

$$\begin{array}{c}\frac{\Gamma \left(\alpha +1\right)}{{2\mathrm{Ԓ}\left(\frac{1}{2}\right)\left(\mathfrak{g}-\mathfrak{e}\right)}^{\alpha }}\left[{\mathcal{I}}_{{\mathfrak{g}}^{-}}^{\alpha }{G}_{\gamma }\left(\mathfrak{e},\frac{\mathfrak{u}+\mathfrak{v}}{2}\right)\right]\\ {\supseteq }_I\frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{{\left(\mathfrak{g}-\mathfrak{e}\right)}^{\alpha }{\left(\mathfrak{v}-\mathfrak{u}\right)}^{\beta }}\left[{\mathcal{I}}_{{\mathfrak{g}}^{-},{\mathfrak{u}}^{+}}^{\alpha , \beta }{G}_{\gamma }\left(\mathfrak{e},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathfrak{g}}^{-},{\mathfrak{v}}^{-}}^{\alpha , \beta }{G}_{\gamma }\left(\mathfrak{e},\mathfrak{u}\right)\right]\\ {\supseteq }_I\frac{\beta \Gamma \left(\alpha +1\right)}{{\left(\mathfrak{g}-\mathfrak{e}\right)}^{\alpha }}\left[{\mathcal{I}}_{{\mathfrak{g}}^{-}}^{\alpha }{G}_{\gamma }\left(\mathfrak{e},\mathfrak{u}\right)+{\mathcal{I}}_{{\mathfrak{g}}^{-}}^{\alpha }{G}_{\gamma }\left(\mathfrak{e},\mathfrak{v}\right)\right]{\int }_0^1{\kappa }^{\beta -1}\left[\mathrm{Ԓ}\left(\kappa \right)+\mathrm{Ԓ}\left(1-\kappa \right)\right]d\kappa .\end{array}$$

(40)

Since, from $\gamma$-cuts, we obtain the collection of $\mathsf{I}\mathsf{V}\mathsf{M}$s $G_{\gamma }:∆\to {\mathbb{R}}_I^{+}$, we have

$$\begin{array}{c}\frac{\Gamma \left(\alpha +1\right)}{{2\mathrm{Ԓ}\left(\frac{1}{2}\right)\left(\mathfrak{g}-\mathfrak{e}\right)}^{\alpha }}\left[{\mathcal{I}}_{{\mathfrak{e}}^{+}}^{\alpha }\tilde{G}\left(\mathfrak{g},\frac{\mathfrak{u}+\mathfrak{v}}{2}\right)\right]\\ {\supseteq }_{\mathbb{F}}\frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{{\left(\mathfrak{g}-\mathfrak{e}\right)}^{\alpha }{\left(\mathfrak{v}-\mathfrak{u}\right)}^{\beta }}\left[{\mathcal{I}}_{{\mathfrak{e}}^{+},{\mathfrak{u}}^{+}}^{\alpha , \beta }\tilde{G}\left(\mathfrak{g},\mathfrak{v}\right)\oplus {\mathcal{I}}_{{\mathfrak{g}}^{-},{\mathfrak{u}}^{+}}^{\alpha , \beta }\tilde{G}\left(\mathfrak{g},\mathfrak{u}\right)\right]\\ {\supseteq }_{\mathbb{F}}\frac{\beta \Gamma \left(\alpha +1\right)}{{\left(\mathfrak{g}-\mathfrak{e}\right)}^{\alpha }}\left[{\mathcal{I}}_{{\mathfrak{e}}^{+}}^{\alpha }\tilde{G}\left(\mathfrak{g},\mathfrak{u}\right)\oplus {\mathcal{I}}_{{\mathfrak{e}}^{+}}^{\alpha }\tilde{G}\left(\mathfrak{g},\mathfrak{v}\right)\right]{\int }_0^1{\kappa }^{\beta -1}\left[\mathrm{Ԓ}\left(\kappa \right)+\mathrm{Ԓ}\left(1-\kappa \right)\right]d\kappa .\end{array}$$

(41)

And

$$\begin{array}{c}\frac{\Gamma \left(\alpha +1\right)}{{2\mathrm{Ԓ}\left(\frac{1}{2}\right)\left(\mathfrak{g}-\mathfrak{e}\right)}^{\alpha }}\left[{\mathcal{I}}_{{\mathfrak{g}}^{-}}^{\alpha }\tilde{G}\left(\mathfrak{e},\frac{\mathfrak{u}+\mathfrak{v}}{2}\right)\right]\\ {\supseteq }_{\mathbb{F}}\frac{\beta \Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{{\left(\mathfrak{g}-\mathfrak{e}\right)}^{\alpha }{\left(\mathfrak{v}-\mathfrak{u}\right)}^{\beta }}\left[{\mathcal{I}}_{{\mathfrak{g}}^{-},{\mathfrak{u}}^{+}}^{\alpha , \beta }\tilde{G}\left(\mathfrak{e},\mathfrak{v}\right)\oplus {\mathcal{I}}_{{\mathfrak{g}}^{-},{\mathfrak{v}}^{-}}^{\alpha , \beta }\tilde{G}\left(\mathfrak{e},\mathfrak{u}\right)\right]\\ {\supseteq }_{\mathbb{F}}\frac{\beta \Gamma \left(\alpha +1\right)}{{\left(\mathfrak{g}-\mathfrak{e}\right)}^{\alpha }}\left[{\mathcal{I}}_{{\mathfrak{g}}^{-}}^{\alpha }\tilde{G}\left(\mathfrak{e},\mathfrak{u}\right)\oplus {\mathcal{I}}_{{\mathfrak{g}}^{-}}^{\alpha }\tilde{G}\left(\mathfrak{e},\mathfrak{v}\right)\right]{\int }_0^1{\kappa }^{\beta -1}\left[\mathrm{Ԓ}\left(\kappa \right)+\mathrm{Ԓ}\left(1-\kappa \right)\right]d\kappa .\end{array}$$

(42)

Similarly, since ${\tilde{G}}_y\left(z\right)=\tilde{G}\left(z,y\right)$, from (35), (41), and (42), we have

$$\begin{array}{c}\frac{\Gamma \left(\beta +1\right)}{{2\mathrm{Ԓ}\left(\frac{1}{2}\right)\left(\mathfrak{v}-\mathfrak{u}\right)}^{\beta }}\left[{\mathcal{I}}_{{\mathfrak{u}}^{+}}^{\beta }\tilde{G}\left(\frac{\mathfrak{e}+\mathfrak{g}}{2},\mathfrak{v}\right)\right]\\ {\supseteq }_{\mathbb{F}}\frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{{\left(\mathfrak{g}-\mathfrak{e}\right)}^{\alpha }{\left(\mathfrak{v}-\mathfrak{u}\right)}^{\beta }}\left[{\mathcal{I}}_{{\mathfrak{e}}^{+},{\mathfrak{u}}^{+}}^{\alpha , \beta }\tilde{G}\left(\mathfrak{g},\mathfrak{v}\right)\oplus {\mathcal{I}}_{{\mathfrak{g}}^{-},{\mathfrak{u}}^{+}}^{\alpha , \beta }\tilde{G}\left(\mathfrak{e},\mathfrak{v}\right)\right]\\ {\supseteq }_{\mathbb{F}}\frac{\alpha \Gamma \left(\beta +1\right)}{{\left(\mathfrak{v}-\mathfrak{u}\right)}^{\beta }}\left[{\mathcal{I}}_{{\mathfrak{u}}^{+} }^{\beta }\tilde{G}\left(\mathfrak{e},\mathfrak{v}\right)\oplus {\mathcal{I}}_{{\mathfrak{u}}^{+}}^{\beta } \tilde{G}\left(\mathfrak{g},\mathfrak{v}\right)\right].\end{array}$$

(43)

And

$$\begin{array}{c}\frac{\Gamma \left(\beta +1\right)}{{2\mathrm{Ԓ}\left(\frac{1}{2}\right)\left(\mathfrak{v}-\mathfrak{u}\right)}^{\alpha }}\left[{\mathcal{I}}_{{\mathfrak{v}}^{-}}^{\beta }\tilde{G}\left(\frac{\mathfrak{e}+\mathfrak{g}}{2},\mathfrak{u}\right)\right]\\ {\supseteq }_{\mathbb{F}}\frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{{\left(\mathfrak{g}-\mathfrak{e}\right)}^{\alpha }{\left(\mathfrak{v}-\mathfrak{u}\right)}^{\beta }}\left[{\mathcal{I}}_{{\mathfrak{e}}^{+},{\mathfrak{v}}^{-}}^{\alpha , \beta }\tilde{G}\left(\mathfrak{g},\mathfrak{u}\right)\oplus {\mathcal{I}}_{{\mathfrak{g}}^{-},{\mathfrak{v}}^{-}}^{\alpha , \beta }\tilde{G}\left(\mathfrak{e},\mathfrak{u}\right)\right]\\ {\supseteq }_{\mathbb{F}}\frac{\alpha \Gamma \left(\beta +1\right)}{{\left(\mathfrak{v}-\mathfrak{u}\right)}^{\beta }}\left[{\mathcal{I}}_{{\mathfrak{v}}^{-}}^{\beta }\tilde{G}\left(\mathfrak{e},\mathfrak{u}\right)\oplus {\mathcal{I}}_{{\mathfrak{v}}^{-}}^{\beta }\tilde{G}\left(\mathfrak{g},\mathfrak{u}\right)\right].\end{array}$$

(44)

The second, third, and fourth inequalities of (28) will be the consequence of adding the inequalities (41)–(44).

Now, for any $\gamma \in \left[0, 1\right]$, we have inequality (15)’s left side:

$$\frac{1}{{\mathrm{Ԓ}}^2\left(\frac{1}{2}\right)}{G}_{\gamma }\left(\frac{\mathfrak{e}+\mathfrak{g}}{2},\frac{\mathfrak{u}+\mathfrak{v}}{2}\right){\supseteq }_I\frac{\Gamma \left(\beta +1\right)}{{\mathrm{Ԓ}\left(\frac{1}{2}\right)\left(\mathfrak{v}-\mathfrak{u}\right)}^{\beta }}\left[{\mathcal{I}}_{{\mathfrak{u}}^{+}}^{\beta }{G}_{\gamma }\left(\frac{\mathfrak{e}+\mathfrak{g}}{2},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathfrak{v}}^{-}}^{\beta }{G}_{\gamma }\left(\frac{\mathfrak{e}+\mathfrak{g}}{2},\mathfrak{u}\right)\right]$$

(45)

And

$$\frac{1}{{\mathrm{Ԓ}}^2\left(\frac{1}{2}\right)}{G}_{\gamma }\left(\frac{\mathfrak{e}+\mathfrak{g}}{2},\frac{\mathfrak{u}+\mathfrak{v}}{2}\right){\supseteq }_I\frac{\Gamma \left(\alpha +1\right)}{{\mathrm{Ԓ}\left(\frac{1}{2}\right)\left(\mathfrak{g}-\mathfrak{e}\right)}^{\alpha }}\left[{\mathcal{I}}_{{\mathfrak{e}}^{+}}^{\alpha }{G}_{\gamma }\left(\mathfrak{g},\frac{\mathfrak{u}+\mathfrak{v}}{2}\right)+{\mathcal{I}}_{{\mathfrak{g}}^{-}}^{\alpha }{G}_{\gamma }\left(\mathfrak{e},\frac{\mathfrak{u}+\mathfrak{v}}{2}\right)\right]$$

(46)

The following inequality is created by adding the two inequalities (45) and (46):

$$\begin{array}{c}\frac{1}{{\mathrm{Ԓ}}^2\left(\frac{1}{2}\right)}{G}_{\gamma }\left(\frac{\mathfrak{e}+\mathfrak{g}}{2},\frac{\mathfrak{u}+\mathfrak{v}}{2}\right){\supseteq }_I\frac{\Gamma \left(\alpha +1\right)}{{\mathrm{Ԓ}\left(\frac{1}{2}\right)\left(\mathfrak{g}-\mathfrak{e}\right)}^{\alpha }}\left[{\mathcal{I}}_{{\mathfrak{e}}^{+}}^{\alpha }{G}_{\gamma }\left(\mathfrak{g},\frac{\mathfrak{u}+\mathfrak{v}}{2}\right)+{\mathcal{I}}_{{\mathfrak{g}}^{-}}^{\alpha }{G}_{\gamma }\left(\mathfrak{e},\frac{\mathfrak{u}+\mathfrak{v}}{2}\right)\right]\\ +\frac{\Gamma \left(\beta +1\right)}{{\mathrm{Ԓ}\left(\frac{1}{2}\right)\left(\mathfrak{v}-\mathfrak{u}\right)}^{\beta }}\left[{\mathcal{I}}_{{\mathfrak{u}}^{+}}^{\beta }{G}_{\gamma }\left(\frac{\mathfrak{e}+\mathfrak{g}}{2},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathfrak{v}}^{-}}^{\beta }{G}_{\gamma }\left(\frac{\mathfrak{e}+\mathfrak{g}}{2},\mathfrak{u}\right)\right].\end{array}$$

Similarly, since we obtain the set of $\mathsf{I}\mathsf{V}\mathsf{M}$s $G_{\gamma }:∆\to {\mathbb{R}}_I^{+}$ for $\gamma \in \left[0, 1\right]$, the inequality can be expressed as follows:

$$\begin{array}{c}\frac{1}{{\mathrm{Ԓ}}^2\left(\frac{1}{2}\right)}\tilde{G}\left(\frac{\mathfrak{e}+\mathfrak{g}}{2},\frac{\mathfrak{u}+\mathfrak{v}}{2}\right)\\ {\supseteq }_{\mathbb{F}}\frac{\Gamma \left(\alpha +1\right)}{{\mathrm{Ԓ}\left(\frac{1}{2}\right)\left(\mathfrak{g}-\mathfrak{e}\right)}^{\alpha }}\left[{\mathcal{I}}_{{\mathfrak{e}}^{+}}^{\alpha }\tilde{G}\left(\mathfrak{g},\frac{\mathfrak{u}+\mathfrak{v}}{2}\right)\oplus {\mathcal{I}}_{{\mathfrak{g}}^{-}}^{\alpha }\tilde{G}\left(\mathfrak{e},\frac{\mathfrak{u}+\mathfrak{v}}{2}\right)\right]\oplus \frac{\Gamma \left(\beta +1\right)}{{\mathrm{Ԓ}\left(\frac{1}{2}\right)\left(\mathfrak{v}-\mathfrak{u}\right)}^{\beta }}\left[{\mathcal{I}}_{{\mathfrak{u}}^{+}}^{\beta }\tilde{G}\left(\frac{\mathfrak{e}+\mathfrak{g}}{2},\mathfrak{v}\right)\oplus {\mathcal{I}}_{{\mathfrak{v}}^{-}}^{\beta }\tilde{G}\left(\frac{\mathfrak{e}+\mathfrak{g}}{2},\mathfrak{u}\right)\right].\end{array}$$

(47)

The first inequality of (28) is this one.

Now, for any $\gamma \in \left[0, 1\right]$, we have inequality (15)’s right side:

$$\frac{\Gamma \left(\beta \right)}{{\left(\mathfrak{v}-\mathfrak{u}\right)}^{\beta }}\left[{\mathcal{I}}_{{\mathfrak{u}}^{+}}^{\beta }{G}_{\gamma }\left(\mathfrak{e},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathfrak{v}}^{-}}^{\beta } G_{\gamma }\left(\mathfrak{e},\mathfrak{u}\right)\right]{\supseteq }_I\left[G_{\gamma }\left(\mathfrak{e},\mathfrak{u}\right)+G_{\gamma }\left(\mathfrak{e},\mathfrak{v}\right)\right]\times {\int }_0^1{\kappa }^{\beta -1}\left[\mathrm{Ԓ}\left(\kappa \right)+\mathrm{Ԓ}\left(1-\kappa \right)\right]d\kappa$$

(48)

$$\frac{\Gamma \left(\beta \right)}{{\left(\mathfrak{v}-\mathfrak{u}\right)}^{\beta }}\left[{\mathcal{I}}_{{\mathfrak{u}}^{+}}^{\beta }{G}_{\gamma }\left(\mathfrak{g},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathfrak{v}}^{-}}^{\beta } G_{\gamma }\left(\mathfrak{g},\mathfrak{u}\right)\right]{\supseteq }_I\left[G_{\gamma }\left(\mathfrak{g},\mathfrak{u}\right)+G_{\gamma }\left(\mathfrak{g},\mathfrak{v}\right)\right]\times {\int }_0^1{\kappa }^{\beta -1}\left[\mathrm{Ԓ}\left(\kappa \right)+\mathrm{Ԓ}\left(1-\kappa \right)\right]d\kappa$$

(49)

$$\frac{\Gamma \left(\alpha \right)}{{\left(\mathfrak{g}-\mathfrak{e}\right)}^{\alpha }}\left[{\mathcal{I}}_{{\mathfrak{e}}^{+}}^{\alpha }{G}_{\gamma }\left(\mathfrak{g},\mathfrak{u}\right)+{\mathcal{I}}_{{\mathfrak{g}}^{-}}^{\alpha } G_{\gamma }\left(\mathfrak{e},\mathfrak{u}\right)\right]{\supseteq }_I\left[G_{\gamma }\left(\mathfrak{e},\mathfrak{u}\right)+G_{\gamma }\left(\mathfrak{g},\mathfrak{u}\right)\right]\times {\int }_0^1{\tau }^{\alpha -1}\mathrm{Ԓ}\left(\tau \right)+\mathrm{Ԓ}\left(1-\tau \right)d\tau$$

(50)

$$\frac{\Gamma \left(\alpha \right)}{{\left(\mathfrak{g}-\mathfrak{e}\right)}^{\alpha }}\left[{\mathcal{I}}_{{\mathfrak{e}}^{+}}^{\alpha }{G}_{\gamma }\left(\mathfrak{g},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathfrak{g}}^{-}}^{\alpha } G_{\gamma }\left(\mathfrak{e},\mathfrak{v}\right)\right]{\supseteq }_I\left[G_{\gamma }\left(\mathfrak{e},\mathfrak{v}\right)+G_{\gamma }\left(\mathfrak{g},\mathfrak{v}\right)\right]\times {\int }_0^1{\tau }^{\alpha -1}\mathrm{Ԓ}\left(\tau \right)+\mathrm{Ԓ}\left(1-\tau \right)d\tau$$

(51)

Summing inequalities (48)–(51) and then taking the multiplication of the result with $\alpha \beta$, we have

$$\begin{array}{c}\frac{\beta \Gamma \left(\alpha +1\right)}{{\left(\mathfrak{g}-\mathfrak{e}\right)}^{\alpha }}\left[{\mathcal{I}}_{{\mathfrak{e}}^{+}}^{\alpha }{G}_{\gamma }\left(\mathfrak{g},\mathfrak{u}\right)+{\mathcal{I}}_{{\mathfrak{g}}^{-}}^{\alpha }{G}_{\gamma }\left(\mathfrak{e},\mathfrak{u}\right)+{\mathcal{I}}_{{\mathfrak{e}}^{+}}^{\alpha }{G}_{\gamma }\left(\mathfrak{g},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathfrak{g}}^{-}}^{\alpha }{G}_{\gamma }\left(\mathfrak{e},\mathfrak{v}\right)\right]\\ +\frac{\alpha \Gamma \left(\beta +1\right)}{{\left(\mathfrak{v}-\mathfrak{u}\right)}^{\beta }}\left[{\mathcal{I}}_{{\mathfrak{u}}^{+}}^{\beta }{G}_{\gamma }\left(\mathfrak{e},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathfrak{v}}^{-}}^{\beta }{G}_{\gamma }\left(\mathfrak{e},\mathfrak{u}\right)+{\mathcal{I}}_{{\mathfrak{u}}^{+}}^{\beta }{G}_{\gamma }\left(\mathfrak{g},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathfrak{v}}^{-}}^{\beta }{G}_{\gamma }\left(\mathfrak{g},\mathfrak{u}\right)\right]\\ {\supseteq }_I\left[G_{\gamma }\left(\mathfrak{e},\mathfrak{u}\right)+G_{\gamma }\left(\mathfrak{e},\mathfrak{v}\right)+G_{\gamma }\left(\mathfrak{g},\mathfrak{u}\right)+G_{\gamma }\left(\mathfrak{g},\mathfrak{v}\right)\right]\times {\int }_0^1{\kappa }^{\beta -1}\left[\mathrm{Ԓ}\left(\kappa \right)+\mathrm{Ԓ}\left(1-\kappa \right)\right]\\ d\kappa {\int }_0^1{\tau }^{\alpha -1}\mathrm{Ԓ}\left(\tau \right)+\mathrm{Ԓ}\left(1-\tau \right)d\tau .\end{array}$$

Since we receive the collection of $\mathsf{I}\mathsf{V}\mathsf{M}$s $G_{\gamma }:∆\to {\mathbb{R}}_I^{+}$ from $\gamma$-cuts, we have

$$\begin{array}{c}\frac{\beta \Gamma \left(\alpha +1\right)}{{\left(\mathfrak{g}-\mathfrak{e}\right)}^{\alpha }}\left[{\mathcal{I}}_{{\mathfrak{e}}^{+}}^{\alpha }\tilde{G}\left(\mathfrak{g},\mathfrak{u}\right)\oplus {\mathcal{I}}_{{\mathfrak{g}}^{-}}^{\alpha }\tilde{G}\left(\mathfrak{e},\mathfrak{u}\right)\oplus {\mathcal{I}}_{{\mathfrak{e}}^{+}}^{\alpha }\tilde{G}\left(\mathfrak{g},\mathfrak{v}\right)\oplus {\mathcal{I}}_{{\mathfrak{g}}^{-}}^{\alpha }\tilde{G}\left(\mathfrak{e},\mathfrak{v}\right)\right]\\ \oplus \frac{\alpha \Gamma \left(\beta +1\right)}{{\left(\mathfrak{v}-\mathfrak{u}\right)}^{\beta }}\left[{\mathcal{I}}_{{\mathfrak{u}}^{+}}^{\beta }\tilde{G}\left(\mathfrak{e},\mathfrak{v}\right)\oplus {\mathcal{I}}_{{\mathfrak{v}}^{-}}^{\beta }\tilde{G}\left(\mathfrak{e},\mathfrak{u}\right)\oplus {\mathcal{I}}_{{\mathfrak{u}}^{+}}^{\beta }\tilde{G}\left(\mathfrak{g},\mathfrak{v}\right)\oplus {\mathcal{I}}_{{\mathfrak{v}}^{-}}^{\beta }\tilde{G}\left(\mathfrak{g},\mathfrak{u}\right)\right]\\ {\supseteq }_{\mathbb{F}}\left[\tilde{G}\left(\mathfrak{e},\mathfrak{u}\right)\oplus \tilde{G}\left(\mathfrak{e},\mathfrak{v}\right)\oplus \tilde{G}\left(\mathfrak{g},\mathfrak{u}\right)\oplus \tilde{G}\left(\mathfrak{g},\mathfrak{v}\right)\right]\times {\int }_0^1{\kappa }^{\beta -1}\left[\mathrm{Ԓ}\left(\kappa \right)+\mathrm{Ԓ}\left(1-\kappa \right)\right]\\ d\kappa {\int }_0^1{\tau }^{\alpha -1}\mathrm{Ԓ}\left(\tau \right)+\mathrm{Ԓ}\left(1-\tau \right)d\tau .\end{array}$$

(52)

This is the final inequality of (28), and a conclusion has been established. □

Example 3.

We assume the $\mathsf{F}\mathsf{N}\mathsf{V}\mathsf{M}$s $\tilde{G}:\left[0, 2\right]\times \left[0, 2\right]\to {\mathbb{F}}_0$ are defined by

$$G\left(x,y\right)\left(\sigma \right)=\left\{\begin{array}{c}\frac{\sigma -\left(2-\sqrt{x}\right)\left(2-\sqrt{y}\right)}{4-\left(2-\sqrt{x}\right)\left(2-\sqrt{y}\right)}, \sigma \in \left[\left(2-\sqrt{x}\right)\left(2-\sqrt{y}\right), 4\right]\\ \frac{\left(2+\sqrt{x}\right)\left(2+\sqrt{y}\right)-\sigma }{\left(2+\sqrt{x}\right)\left(2+\sqrt{y}\right)-4}, \sigma \in \left(4, \left(2+\sqrt{x}\right)\left(2+\sqrt{y}\right)\right]\\ 0, otherwise,\end{array}\right.$$

(53)

and then, for each $\gamma \in \left[0, 1\right],$ we have $G_{\gamma }\left(x,y\right)=\left[\left(1-\gamma \right)\left(2-\sqrt{x}\right)\left(2-\sqrt{y}\right)+4\gamma ,\left(1-\gamma \right)\left(2+\sqrt{x}\right)\left(2+\sqrt{y}\right)+4\gamma \right]$. Since the endpoint functions $G_{\ast }\left(\left(x,y\right),\gamma \right)$ and $G^{\ast }\left(\left(x,y\right),\gamma \right)$ are coordinate $\mathrm{Ԓ}$-convex and $\mathrm{Ԓ}$-concave functions for each $\gamma \in \left[0, 1\right],$ $\tilde{G}\left(x,y\right)$ is a $\mathsf{U}\mathsf{D}$-$\mathrm{Ԓ}$-coordinate convex $\mathsf{F}\mathsf{N}\mathsf{V}\mathsf{M}$.

$$\begin{array}{c}{G}_{\gamma }\left(\frac{\mathfrak{e}+\mathfrak{g}}{2}, \frac{\mathfrak{u}+\mathfrak{v}}{2}\right)=\left[\left(1-\gamma \right)+4\gamma ,9\left(1-\gamma \right)+4\gamma \right],\\ \frac{\Gamma \left(\alpha +1\right)}{{4\left(\mathfrak{g}-\mathfrak{e}\right)}^{\alpha }}\left[{\mathcal{I}}_{{\mathfrak{e}}^{+}}^{\alpha }\tilde{G}\left(\mathfrak{g},\frac{\mathfrak{u}+\mathfrak{v}}{2}\right)\oplus {\mathcal{I}}_{{\mathfrak{g}}^{-}}^{\alpha }\tilde{G}\left(\mathfrak{e},\frac{\mathfrak{u}+\mathfrak{v}}{2}\right)\right]\oplus \frac{\Gamma \left(\beta +1\right)}{{4\left(\mathfrak{v}-\mathfrak{u}\right)}^{\beta }}\left[{\mathcal{I}}_{{\mathfrak{u}}^{+}}^{\beta }\tilde{G}\left(\frac{\mathfrak{e}+\mathfrak{g}}{2},\mathfrak{v}\right)\oplus {\mathcal{I}}_{{\mathfrak{v}}^{-}}^{\beta }\tilde{G}\left(\frac{\mathfrak{e}+\mathfrak{g}}{2},\mathfrak{u}\right)\right]\\ =\left[\left(1-\gamma \right)\left(2-\frac{\sqrt{2}}{4}-\frac{\sqrt{2}}{8}\pi \right)+4\gamma ,\left(1-\gamma \right)\left(2+\frac{\sqrt{2}}{4}+\frac{\sqrt{2}}{8}\pi \right)+4\gamma \right]\\ \frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{{4\left(\mathfrak{g}-\mathfrak{e}\right)}^{\alpha }{\left(\mathfrak{v}-\mathfrak{u}\right)}^{\beta }}\left[{\mathcal{I}}_{{\mathfrak{e}}^{+},{\mathfrak{u}}^{+}}^{\alpha , \beta }{G}_{\gamma }\left(\mathfrak{g},\mathfrak{v}\right)\oplus {\mathcal{I}}_{{\mathfrak{e}}^{+},{\mathfrak{v}}^{-}}^{\alpha , \beta }{G}_{\gamma }\left(\mathfrak{g},\mathfrak{u}\right)\oplus {\mathcal{I}}_{{\mathfrak{g}}^{-},{\mathfrak{u}}^{+}}^{\alpha , \beta }{G}_{\gamma }\left(\mathfrak{e},\mathfrak{v}\right)\oplus {\mathcal{I}}_{{\mathfrak{g}}^{-},{\mathfrak{v}}^{-}}^{\alpha , \beta }{G}_{\gamma }\left(\mathfrak{e},\mathfrak{u}\right)\right]\\ =\left[\left(1-\gamma \right)\left(\frac{33}{8}-\sqrt{2}-\frac{\sqrt{2}}{2}\pi +\frac{\pi }{8}+\frac{{\pi }^2}{32}\right)+4\gamma ,\left(1-\gamma \right)\left(\frac{33}{8}+\sqrt{2}+\frac{\sqrt{2}}{2}\pi +\frac{\pi }{8}+\frac{{\pi }^2}{32}\right)+4\gamma \right]\\ \frac{\Gamma \left(\alpha +1\right)}{{8\left(\mathfrak{g}-\mathfrak{e}\right)}^{\alpha }}\left[{\mathcal{I}}_{{\mathfrak{e}}^{+}}^{\alpha }\tilde{G}\left(\mathfrak{g},\mathfrak{u}\right)\oplus {\mathcal{I}}_{{\mathfrak{e}}^{+}}^{\alpha }\tilde{G}\left(\mathfrak{g},\mathfrak{v}\right)\oplus {\mathcal{I}}_{{\mathfrak{g}}^{-}}^{\alpha }\tilde{G}\left(\mathfrak{e},\mathfrak{u}\right)\oplus {\mathcal{I}}_{{\mathfrak{g}}^{-}}^{\alpha }\tilde{G}\left(\mathfrak{e},\mathfrak{v}\right)\right]\\ \oplus \frac{\Gamma \left(\beta +1\right)}{{8\left(\mathfrak{v}-\mathfrak{u}\right)}^{\beta }}\left[{\mathcal{I}}_{{\mathfrak{u}}^{+}}^{\beta }\tilde{G}\left(\mathfrak{e},\mathfrak{v}\right)\oplus {\mathcal{I}}_{{\mathfrak{u}}^{+}}^{\beta }\tilde{G}\left(\mathfrak{g},\mathfrak{v}\right)\oplus {\mathcal{I}}_{{\mathfrak{v}}^{-}}^{\beta }\tilde{G}\left(\mathfrak{e},\mathfrak{u}\right)\oplus {\mathcal{I}}_{{\mathfrak{v}}^{-}}^{\beta }\tilde{G}\left(\mathfrak{g},\mathfrak{u}\right)\right]\\ =\left[\frac{34\sqrt{2}+\left(\sqrt{2}-4\right)\pi -24}{8\sqrt{2}}\left(1-\gamma \right)+4\gamma ,\frac{34\sqrt{2}+\left(\sqrt{2}+4\right)\pi +24}{8\sqrt{2}}\left(1-\gamma \right)+4\gamma \right]\\ \frac{G_{\gamma }\left(\mathfrak{u},\mathfrak{g}\right)+G_{\gamma }\left(\sigma ,\mathfrak{g}\right)+G_{\gamma }\left(\mathfrak{u},\mathfrak{v}\right)+G_{\gamma }\left(\sigma ,\mathfrak{v}\right)}{4}=\left[\left(1-\gamma \right)\left(\frac{9}{2}-2\sqrt{2}\right)+4\gamma ,\left(1-\gamma \right)\left(\frac{9}{2}+2\sqrt{2}\right)+4\gamma \right].\end{array}$$

That is,

$$\begin{array}{c}\left[\left(1-\gamma \right)+4\gamma ,9\left(1-\gamma \right)+4\gamma \right]{\supseteq }_I\left[\left(1-\gamma \right)\left(2-\frac{\sqrt{2}}{4}-\frac{\sqrt{2}}{8}\pi \right)+4\gamma ,\left(1-\gamma \right)\left(2+\frac{\sqrt{2}}{4}+\frac{\sqrt{2}}{8}\pi \right)+4\gamma \right]\\ {\supseteq }_I\left[\left(1-\gamma \right)\left(\frac{33}{8}-\sqrt{2}-\frac{\sqrt{2}}{2}\pi +\frac{\pi }{8}+\frac{{\pi }^2}{32}\right)+4\gamma ,\left(1-\gamma \right)\left(\frac{33}{8}+\sqrt{2}+\frac{\sqrt{2}}{2}\pi +\frac{\pi }{8}+\frac{{\pi }^2}{32}\right)+4\gamma \right]\\ {\supseteq }_I\left[\frac{34\sqrt{2}+\left(\sqrt{2}-4\right)\pi -24}{8\sqrt{2}}\left(1-\gamma \right)+4\gamma ,\frac{34\sqrt{2}+\left(\sqrt{2}+4\right)\pi +24}{8\sqrt{2}}\left(1-\gamma \right)+4\gamma \right]\\ {\supseteq }_I\frac{34\sqrt{2}+\left(\sqrt{2}-4\right)\pi -24}{8\sqrt{2}}\left(1-\gamma \right)+4\gamma .\end{array}$$

Hence, Theorem 8 has been verified.

Remark 2.

If one assumes that $\alpha =1$, $\beta =1$, and $\mathrm{Ԓ}\left(\tau \right)=\tau , \mathrm{Ԓ}\left(\kappa \right)=\kappa$, then, from (28), as a result, there will be an inequality (see [70]):

$$\begin{array}{c}\tilde{G}\left(\frac{\mathfrak{e}+\mathfrak{g}}{2}, \frac{\mathfrak{u}+\mathfrak{v}}{2}\right)\\ {\supseteq }_{\mathbb{F}}\frac{1}{2}\left[\frac{1}{\mathfrak{g}-\mathfrak{e}}{\int }_{\mathfrak{e}}^{\mathfrak{g}}\tilde{G}\left(x,\frac{\mathfrak{u}+\mathfrak{v}}{2}\right)dx\oplus \frac{1}{\mathfrak{v}-\mathfrak{u}}{\int }_{\mathfrak{u}}^{\mathfrak{v}}\tilde{G}\left(\frac{\mathfrak{e}+\mathfrak{g}}{2},y\right)dy\right]{\supseteq }_{\mathbb{F}}\frac{1}{\left(\mathfrak{g}-\mathfrak{e}\right)\left(\mathfrak{v}-\mathfrak{u}\right)}{\int }_{\mathfrak{e}}^{\mathfrak{g}}{\int }_{\mathfrak{u}}^{\mathfrak{v}}\tilde{G}\left(x,y\right)dydx\\ {\supseteq }_{\mathbb{F}}\frac{1}{4\left(\mathfrak{g}-\mathfrak{e}\right)}\left[{\int }_{\mathfrak{e}}^{\mathfrak{g}}\tilde{G}\left(x,\mathfrak{u}\right)dx\oplus {\int }_{\mathfrak{e}}^{\mathfrak{g}}\tilde{G}\left(x,\mathfrak{v}\right)dx\right]\oplus \frac{1}{4\left(\mathfrak{v}-\mathfrak{u}\right)}\left[{\int }_{\mathfrak{u}}^{\mathfrak{v}}\tilde{G}\left(\mathfrak{e},y\right)dy\oplus {\int }_{\mathfrak{u}}^{\mathfrak{v}}\tilde{G}\left(\mathfrak{g},y\right)dy\right]\\ {\supseteq }_{\mathbb{F}}\frac{\tilde{G}\left(\mathfrak{e},\mathfrak{u}\right)\oplus \tilde{G}\left(\mathfrak{g},\mathfrak{u}\right)\oplus \tilde{G}\left(\mathfrak{e},\mathfrak{v}\right)\oplus \tilde{G}\left(\mathfrak{g},\mathfrak{v}\right)}{4}.\end{array}$$

(54)

If one assumes that $\alpha =1$, $\beta =1$, $\mathrm{Ԓ}\left(\tau \right)=\tau , \mathrm{Ԓ}\left(\kappa \right)=\kappa ,$ and $\tilde{G}$ is a coordinated left-$\mathsf{U}\mathsf{D}$-$\mathrm{Ԓ}$-convex, then, from (28), as a result, there will be an inequality (see [59):

$$\begin{array}{c}\tilde{G}\left(\frac{\mathfrak{e}+\mathfrak{g}}{2}, \frac{\mathfrak{u}+\mathfrak{v}}{2}\right)\\ {\le }_{\mathbb{F}}\frac{1}{2}\left[\frac{1}{\mathfrak{g}-\mathfrak{e}}{\int }_{\mathfrak{e}}^{\mathfrak{g}}\tilde{G}\left(x,\frac{\mathfrak{u}+\mathfrak{v}}{2}\right)dx\oplus \frac{1}{\mathfrak{v}-\mathfrak{u}}{\int }_{\mathfrak{u}}^{\mathfrak{v}}\tilde{G}\left(\frac{\mathfrak{e}+\mathfrak{g}}{2},y\right)dy\right]{\le }_{\mathbb{F}}\frac{1}{\left(\mathfrak{g}-\mathfrak{e}\right)\left(\mathfrak{v}-\mathfrak{u}\right)}{\int }_{\mathfrak{e}}^{\mathfrak{g}}{\int }_{\mathfrak{u}}^{\mathfrak{v}}\tilde{G}\left(x,y\right)dydx\\ {\le }_{\mathbb{F}}\frac{1}{4\left(\mathfrak{g}-\mathfrak{e}\right)}\left[{\int }_{\mathfrak{e}}^{\mathfrak{g}}\tilde{G}\left(x,\mathfrak{u}\right)dx\oplus {\int }_{\mathfrak{e}}^{\mathfrak{g}}\tilde{G}\left(x,\mathfrak{v}\right)dx\right]\oplus \frac{1}{4\left(\mathfrak{v}-\mathfrak{u}\right)}\left[{\int }_{\mathfrak{u}}^{\mathfrak{v}}\tilde{G}\left(\mathfrak{e},y\right)dy\oplus {\int }_{\mathfrak{u}}^{\mathfrak{v}}\tilde{G}\left(\mathfrak{g},y\right)dy\right]\\ {\le }_{\mathbb{F}}\frac{\tilde{G}\left(\mathfrak{e},\mathfrak{u}\right)\oplus \tilde{G}\left(\mathfrak{g},\mathfrak{u}\right)\oplus \tilde{G}\left(\mathfrak{e},\mathfrak{v}\right)\oplus \tilde{G}\left(\mathfrak{g},\mathfrak{v}\right)}{4}.\end{array}$$

(55)

If $\mathrm{Ԓ}\left(\tau \right)=\tau , \mathrm{Ԓ}\left(\kappa \right)=\kappa ,$ and $G_{\ast }\left(\left(x,y\right), \gamma \right)\ne G^{\ast }\left(\left(x,y\right), \gamma \right)$ with $\gamma =1$, then, from (28), we succeed in bringing about the upcoming inequality (see [55]):

$$\begin{array}{c}G\left(\frac{\mathfrak{e}+\mathfrak{g}}{2}, \frac{\mathfrak{u}+\mathfrak{v}}{2}\right)\\ \supseteq \frac{\Gamma \left(\alpha +1\right)}{{4\left(\mathfrak{g}-\mathfrak{e}\right)}^{\alpha }}\left[{\mathcal{I}}_{{\mathfrak{e}}^{+}}^{\alpha }G\left(\mathfrak{g},\frac{\mathfrak{u}+\mathfrak{v}}{2}\right)+{\mathcal{I}}_{{\mathfrak{g}}^{-}}^{\alpha }G\left(\mathfrak{e},\frac{\mathfrak{u}+\mathfrak{v}}{2}\right)\right]+\frac{\Gamma \left(\beta +1\right)}{{4\left(\mathfrak{v}-\mathfrak{u}\right)}^{\beta }}\left[{\mathcal{I}}_{{\mathfrak{u}}^{+}}^{\beta }G\left(\frac{\mathfrak{e}+\mathfrak{g}}{2},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathfrak{v}}^{-}}^{\beta }G\left(\frac{\mathfrak{e}+\mathfrak{g}}{2},\mathfrak{u}\right)\right]\\ \supseteq \frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{{4\left(\mathfrak{g}-\mathfrak{e}\right)}^{\alpha }{\left(\mathfrak{v}-\mathfrak{u}\right)}^{\beta }} \left[{\mathcal{I}}_{{\mathfrak{e}}^{+},{\mathfrak{u}}^{+}}^{\alpha , \beta }G\left(\mathfrak{g},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathfrak{e}}^{+},{\mathfrak{v}}^{-}}^{\alpha , \beta }G\left(\mathfrak{g},\mathfrak{u}\right)+{\mathcal{I}}_{{\mathfrak{g}}^{-},{\mathfrak{u}}^{+}}^{\alpha , \beta }G\left(\mathfrak{e},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathfrak{g}}^{-},{\mathfrak{v}}^{-}}^{\alpha , \beta }G\left(\mathfrak{e},\mathfrak{u}\right)\right]\\ \supseteq \frac{\Gamma \left(\alpha +1\right)}{{8\left(\mathfrak{g}-\mathfrak{e}\right)}^{\alpha }}\left[{\mathcal{I}}_{{\mathfrak{e}}^{+} }^{\alpha }G\left(\mathfrak{g},\mathfrak{u}\right)+{\mathcal{I}}_{{\mathfrak{e}}^{+}}^{\alpha }G\left(\mathfrak{g},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathfrak{g}}^{-}}^{\alpha }G\left(\mathfrak{e},\mathfrak{u}\right)+{\mathcal{I}}_{{\mathfrak{g}}^{-}}^{\alpha }G\left(\mathfrak{e},\mathfrak{v}\right)\right]\\ +\frac{\Gamma \left(\beta +1\right)}{{8\left(\mathfrak{v}-\mathfrak{u}\right)}^{\beta }}\left[{\mathcal{I}}_{{\mathfrak{u}}^{+} }^{ \beta }G\left(\mathfrak{e},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathfrak{v}}^{-}}^{\beta }G\left(\mathfrak{e},\mathfrak{u}\right)+{\mathcal{I}}_{{\mathfrak{u}}^{+}}^{\beta }G\left(\mathfrak{g},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathfrak{v}}^{-}}^{\beta }G\left(\mathfrak{g},\mathfrak{u}\right)\right]\\ \supseteq \frac{G\left(\mathfrak{e},\mathfrak{u}\right)+G\left(\mathfrak{g},\mathfrak{u}\right)+G\left(\mathfrak{e},\mathfrak{v}\right)+G\left(\mathfrak{g},\mathfrak{v}\right)}{4}.\end{array}$$

(56)

If $\mathrm{Ԓ}\left(\tau \right)=\tau , \mathrm{Ԓ}\left(\kappa \right)=\kappa$ and $G_{\ast }\left(\left(x,y\right), \gamma \right)\ne G^{\ast }\left(\left(x,y\right), \gamma \right)$ with $\gamma =1$, then, from (28), we succeed in bringing about the upcoming inequality (see [68]):

$$\begin{array}{c}G\left(\frac{\mathfrak{e}+\mathfrak{g}}{2}, \frac{\mathfrak{u}+\mathfrak{v}}{2}\right)\\ \supseteq \frac{1}{2}\left[\frac{1}{\mathfrak{g}-\mathfrak{e}}{\int }_{\mathfrak{e}}^{\mathfrak{g}}G\left(x,\frac{\mathfrak{u}+\mathfrak{v}}{2}\right)dx+\frac{1}{\mathfrak{v}-\mathfrak{u}}{\int }_{\mathfrak{u}}^{\mathfrak{v}}G\left(\frac{\mathfrak{e}+\mathfrak{g}}{2},y\right)dy\right]\subseteq \frac{1}{\left(\mathfrak{g}-\mathfrak{e}\right)\left(\mathfrak{v}-\mathfrak{u}\right)} {\int }_{\mathfrak{e}}^{\mathfrak{g}}{\int }_{\mathfrak{u}}^{\mathfrak{v}}G\left(x,y\right)dydx\\ \supseteq \frac{1}{4\left(\mathfrak{g}-\mathfrak{e}\right)}\left[{\int }_{\mathfrak{e}}^{\mathfrak{g}}G\left(x,\mathfrak{u}\right)dx+{\int }_{\mathfrak{e}}^{\mathfrak{g}}G\left(x,\mathfrak{v}\right)dx\right]+\frac{1}{4\left(\mathfrak{v}-\mathfrak{u}\right)}\left[{\int }_{\mathfrak{u}}^{\mathfrak{v}}G\left(\mathfrak{e},y\right)dy+{\int }_{\mathfrak{u}}^{\mathfrak{v}}G\left(\mathfrak{g},y\right)dy\right]\\ \supseteq \frac{G\left(\mathfrak{e},\mathfrak{u}\right)+G\left(\mathfrak{g},\mathfrak{u}\right)+G\left(\mathfrak{e},\mathfrak{v}\right)+G\left(\mathfrak{g},\mathfrak{v}\right)}{4}.\end{array}$$

(57)

If $\tilde{G}$ is a coordinated right-$\mathsf{U}\mathsf{D}$-$\mathrm{Ԓ}$-convex function with $\mathrm{Ԓ}\left(\tau \right)=\tau , \mathrm{Ԓ}\left(\kappa \right)=\kappa$ and $G_{\ast }\left(\left(x,y\right), \gamma \right)=G^{\ast }\left(\left(x,y\right), \gamma \right)$ with $\gamma =1$, then, from (28), we succeed in bringing about the upcoming inequality (see [71]):

$$\begin{array}{c}G\left(\frac{\mathfrak{e}+\mathfrak{g}}{2}, \frac{\mathfrak{u}+\mathfrak{v}}{2}\right)\\ \le \frac{\Gamma \left(\alpha +1\right)}{{4\left(\mathfrak{g}-\mathfrak{e}\right)}^{\alpha }}\left[{\mathcal{I}}_{{\mathfrak{e}}^{+}}^{\alpha } G\left(\mathfrak{g},\frac{\mathfrak{u}+\mathfrak{v}}{2}\right)+{\mathcal{I}}_{{\mathfrak{g}}^{-}}^{\alpha }G\left(\mathfrak{e},\frac{\mathfrak{u}+\mathfrak{v}}{2}\right)\right]+\frac{\Gamma \left(\beta +1\right)}{{4\left(\mathfrak{v}-\mathfrak{u}\right)}^{\beta }}\left[{\mathcal{I}}_{{\mathfrak{u}}^{+}}^{\beta } G\left(\frac{\mathfrak{e}+\mathfrak{g}}{2},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathfrak{v}}^{-}}^{\beta } G\left(\frac{\mathfrak{e}+\mathfrak{g}}{2},\mathfrak{u}\right)\right]\\ \le \frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{{4\left(\mathfrak{g}-\mathfrak{e}\right)}^{\alpha }{\left(\mathfrak{v}-\mathfrak{u}\right)}^{\beta }} \left[{\mathcal{I}}_{{\mathfrak{e}}^{+},{\mathfrak{u}}^{+} }^{\alpha , \beta }G\left(\mathfrak{g},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathfrak{e}}^{+},{\mathfrak{v}}^{-}}^{\alpha , \beta } G\left(\mathfrak{g},\mathfrak{u}\right)+{\mathcal{I}}_{{\mathfrak{g}}^{-},{\mathfrak{u}}^{+} }^{\alpha , \beta }G\left(\mathfrak{e},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathfrak{g}}^{-},{\mathfrak{v}}^{-}}^{\alpha , \beta } G\left(\mathfrak{e},\mathfrak{u}\right)\right]\\ \le \frac{\Gamma \left(\alpha +1\right)}{{8\left(\mathfrak{g}-\mathfrak{e}\right)}^{\alpha }}\left[{\mathcal{I}}_{{\mathfrak{e}}^{+} }^{\alpha }G\left(\mathfrak{g},\mathfrak{u}\right)G{\mathcal{I}}_{{\mathfrak{e}}^{+}}^{\alpha } G\left(\mathfrak{g},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathfrak{g}}^{-} }^{\alpha }G\left(\mathfrak{e},\mathfrak{u}\right)+{\mathcal{I}}_{{\mathfrak{g}}^{-}}^{\alpha } G\left(\mathfrak{e},\mathfrak{v}\right)\right].\\ +\frac{\Gamma \left(\beta +1\right)}{{8\left(\mathfrak{v}-\mathfrak{u}\right)}^{\beta }}\left[{\mathcal{I}}_{{\mathfrak{u}}^{+} }^{ \beta }G\left(\mathfrak{e},\mathfrak{v}\right)\widetilde{+}{\mathcal{I}}_{{\mathfrak{v}}^{-}}^{\beta } G\left(\mathfrak{e},\mathfrak{u}\right)+{\mathcal{I}}_{{\mathfrak{u}}^{+} }^{\beta }G\left(\mathfrak{g},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathfrak{v}}^{-}}^{\beta } G\left(\mathfrak{g},\mathfrak{u}\right)\right]\\ \le \frac{G\left(\mathfrak{e},\mathfrak{u}\right)+G\left(\mathfrak{g},\mathfrak{u}\right)+G\left(\mathfrak{e},\mathfrak{v}\right)+G\left(\mathfrak{g},\mathfrak{v}\right)}{4}.\end{array}$$

(58)

In the next section, we are going to find very interesting outcomes that will be obtained over a product of two coordinate $\mathsf{U}\mathsf{D}$-$\mathrm{Ԓ}$-convex $\mathsf{F}\mathsf{N}\mathsf{V}\mathsf{M}$s. These inequalities are known as Pachpatte inequalities.

Theorem 9.

Let $\tilde{G}, \tilde{\mathcal{J}}:∆\to {\mathbb{F}}_0^{+}$ be two coordinated $\mathsf{U}\mathsf{D}$-$\mathrm{Ԓ}$-convex $\mathsf{F}\mathsf{N}\mathsf{V}\mathsf{M}$s on $∆$, and let ${\mathrm{Ԓ}}_1,{\mathrm{Ԓ}}_2 :\left[0, 1\right]\to {\mathbb{R}}^{+}$. Then, from $\gamma$-cuts, we set up the sequence of $\mathsf{I}\mathsf{V}\mathsf{M}$s $G_{\gamma },{\mathcal{J}}_{\gamma }:∆\to {\mathbb{R}}_I^{+},$ which is given by $G_{\gamma }\left(x,y\right)=\left[G_{\ast }\left(\left(x,y\right),\gamma \right), G^{\ast }\left(\left(x,y\right),\gamma \right)\right]$ and ${\mathcal{J}}_{\gamma }\left(x,y\right)=\left[{\mathcal{J}}_{\ast }\left(\left(x,y\right),\gamma \right), {\mathcal{J}}^{\ast }\left(\left(x,y\right),\gamma \right)\right]$ for all $\left(x,y\right)\in ∆$ and for all $\gamma \in \left[0, 1\right]$. If ${\tilde{G}\otimes \tilde{\mathcal{J}}\in \mathcal{F}\mathfrak{O}}_{∆}$, then the following inequalities hold:

$$\begin{array}{c}\frac{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{{\left(\mathfrak{g}-\mathfrak{e}\right)}^{\alpha }{\left(\mathfrak{v}-\mathfrak{u}\right)}^{\beta }}\left[{\mathcal{I}}_{{\mathfrak{e}}^{+},{\mathfrak{u}}^{+}}^{\alpha , \beta }\tilde{G}\left(\mathfrak{g},\mathfrak{v}\right)\otimes \tilde{\mathcal{J}}\left(\mathfrak{g},\mathfrak{v}\right)\oplus {\mathcal{I}}_{{\mathfrak{e}}^{+},{\mathfrak{v}}^{-}}^{\alpha , \beta }\tilde{G}\left(\mathfrak{g},\mathfrak{u}\right)\otimes \tilde{\mathcal{J}}\left(\mathfrak{g},\mathfrak{u}\right)\right]\\ \oplus \frac{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{{\left(\mathfrak{g}-\mathfrak{e}\right)}^{\alpha }{\left(\mathfrak{v}-\mathfrak{u}\right)}^{\beta }}\left[{\mathcal{I}}_{{\mathfrak{g}}^{-},{\mathfrak{u}}^{+}}^{\alpha , \beta }\tilde{G}\left(\mathfrak{e},\mathfrak{v}\right)\otimes \tilde{\mathcal{J}}\left(\mathfrak{e},\mathfrak{v}\right)\oplus {\mathcal{I}}_{{\mathfrak{g}}^{-},{\mathfrak{v}}^{-}}^{\alpha , \beta }\tilde{G}\left(\mathfrak{e},\mathfrak{u}\right)\otimes \tilde{\mathcal{J}}\left(\mathfrak{e},\mathfrak{u}\right)\right]\\ {\supseteq }_{\mathbb{F}}\tilde{\mathcal{M}}\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right){\int }_0^1{\tau }^{\alpha -1}{\kappa }^{\beta -1}[{\mathrm{Ԓ}}_1\left(1-\tau \right){\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_1\left(1-\kappa \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)+{\mathrm{Ԓ}}_1\left(1-\tau \right)\\ {\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right){\mathrm{Ԓ}}_2\left(\kappa \right)+{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_1\left(1-\kappa \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)+\\ {\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right){\mathrm{Ԓ}}_2\left(\kappa \right)]d\tau d\kappa \\ \oplus \tilde{P}\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right){\int }_0^1{\tau }^{\alpha -1}{\kappa }^{\beta -1}[{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_1\left(1-\kappa \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)+{\mathrm{Ԓ}}_1\left(1-\tau \right)\\ {\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_1\left(1-\kappa \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)+{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right){\mathrm{Ԓ}}_2\left(\kappa \right)+{\mathrm{Ԓ}}_1\left(1-\tau \right)\\ {\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right){\mathrm{Ԓ}}_2\left(\kappa \right)]d\tau d\kappa \\ \oplus \tilde{\mathcal{N}}\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right){\int }_0^1{\tau }^{\alpha -1}{\kappa }^{\beta -1}[{\mathrm{Ԓ}}_1\left(1-\tau \right){\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)+{\mathrm{Ԓ}}_1\left(1-\tau \right){\mathrm{Ԓ}}_2\left(1-\tau \right)\\ {\mathrm{Ԓ}}_1\left(1-\kappa \right){\mathrm{Ԓ}}_2\left(\kappa \right)+{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_1\left(1-\kappa \right){\mathrm{Ԓ}}_2\left(\kappa \right)+{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)]d\tau d\kappa \\ \oplus \tilde{Q}\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right){\int }_0^1{\tau }^{\alpha -1}{}^{\beta -1}[{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)+{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_1\left(1-\kappa \right)\\ {\mathrm{Ԓ}}_2\left(\kappa \right)+{\mathrm{Ԓ}}_1\left(1-\tau \right){\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)+{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)]d\tau d\kappa .\end{array}$$

(59)

If $\tilde{G}$ and $\tilde{\mathcal{J}}$ are both coordinated $\mathsf{U}\mathsf{D}$-$\mathrm{Ԓ}$-concave $\mathsf{F}\mathsf{N}\mathsf{V}\mathsf{M}$s on $∆$, then the inequality above can be expressed as follows:

$$\begin{array}{c}\frac{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{{\left(\mathfrak{g}-\mathfrak{e}\right)}^{\alpha }{\left(\mathfrak{v}-\mathfrak{u}\right)}^{\beta }}\left[{\mathcal{I}}_{{\mathfrak{e}}^{+},{\mathfrak{u}}^{+}}^{\alpha , \beta }\tilde{G}\left(\mathfrak{g},\mathfrak{v}\right)\otimes \tilde{\mathcal{J}}\left(\mathfrak{g},\mathfrak{v}\right)\oplus {\mathcal{I}}_{{\mathfrak{e}}^{+},{\mathfrak{v}}^{-}}^{\alpha , \beta }\tilde{G}\left(\mathfrak{g},\mathfrak{u}\right)\otimes \tilde{\mathcal{J}}\left(\mathfrak{g},\mathfrak{u}\right)\right]\\ \oplus \frac{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{{\left(\mathfrak{g}-\mathfrak{e}\right)}^{\alpha }{\left(\mathfrak{v}-\mathfrak{u}\right)}^{\beta }}\left[{\mathcal{I}}_{{\mathfrak{g}}^{-},{\mathfrak{u}}^{+}}^{\alpha , \beta }\tilde{G}\left(\mathfrak{e},\mathfrak{v}\right)\otimes \tilde{\mathcal{J}}\left(\mathfrak{e},\mathfrak{v}\right)\oplus {\mathcal{I}}_{{\mathfrak{g}}^{-},{\mathfrak{v}}^{-}}^{\alpha , \beta }\tilde{G}\left(\mathfrak{e},\mathfrak{u}\right)\otimes \tilde{\mathcal{J}}\left(\mathfrak{e},\mathfrak{u}\right)\right]\\ {\subseteq }_{\mathbb{F}}\tilde{\mathcal{M}}\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right){\int }_0^1{\tau }^{\alpha -1}{\kappa }^{\beta -1}[{\mathrm{Ԓ}}_1\left(1-\tau \right){\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_1\left(1-\kappa \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)+{\mathrm{Ԓ}}_1\left(1-\tau \right)\\ {\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right){\mathrm{Ԓ}}_2\left(\kappa \right)+{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_1\left(1-\kappa \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)+\\ {\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right){\mathrm{Ԓ}}_2\left(\kappa \right)]d\tau d\kappa \\ \oplus \tilde{P}\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right){\int }_0^1{\tau }^{\alpha -1}{\kappa }^{\beta -1}[{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_1\left(1-\kappa \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)+\\ {\mathrm{Ԓ}}_1\left(1-\tau \right){\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_1\left(1-\kappa \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)+{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right){\mathrm{Ԓ}}_2\left(\kappa \right)+{\mathrm{Ԓ}}_1\left(1-\tau \right)\\ {\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right){\mathrm{Ԓ}}_2\left(\kappa \right)]d\tau d\kappa \\ \oplus \tilde{\mathcal{N}}\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right){\int }_0^1{\tau }^{\alpha -1}{\kappa }^{\beta -1}[{\mathrm{Ԓ}}_1\left(1-\tau \right){\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)+{\mathrm{Ԓ}}_1\left(1-\tau \right){\mathrm{Ԓ}}_2\left(1-\tau \right)\\ {\mathrm{Ԓ}}_1\left(1-\kappa \right){\mathrm{Ԓ}}_2\left(\kappa \right)+{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_1\left(1-\kappa \right){\mathrm{Ԓ}}_2\left(\kappa \right)+{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)]d\tau d\kappa \\ \oplus \tilde{Q}\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right){\int }_0^1{\tau }^{\alpha -1}{\kappa }^{\beta -1}[{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)+{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_1\left(1-\kappa \right)\\ {\mathrm{Ԓ}}_2\left(\kappa \right)+{\mathrm{Ԓ}}_1\left(1-\tau \right){\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)+{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)]d\tau d\kappa \end{array}$$

(60)

where

$$\begin{array}{c}\tilde{\mathcal{M}}\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right)=\tilde{G}\left(\mathfrak{e},\mathfrak{u}\right)\otimes \tilde{\mathcal{J}}\left(\mathfrak{e},\mathfrak{u}\right)\oplus \tilde{G}\left(\mathfrak{g},\mathfrak{u}\right)\otimes \tilde{\mathcal{J}}\left(\mathfrak{g},\mathfrak{u}\right)\oplus \tilde{G}\left(\mathfrak{e},\mathfrak{v}\right)\otimes \tilde{\mathcal{J}}\left(\mathfrak{e},\mathfrak{v}\right)\oplus \tilde{G}\left(\mathfrak{g},\mathfrak{v}\right)\otimes \tilde{\mathcal{J}}\left(\mathfrak{g},\mathfrak{v}\right),\\ \tilde{P}\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right)=\tilde{G}\left(\mathfrak{e},\mathfrak{u}\right)\otimes \tilde{\mathcal{J}}\left(\mathfrak{g},\mathfrak{u}\right)\oplus \tilde{G}\left(\mathfrak{g},\mathfrak{u}\right)\otimes \tilde{\mathcal{J}}\left(\mathfrak{e},\mathfrak{u}\right)\oplus \tilde{G}\left(\mathfrak{e},\mathfrak{v}\right)\otimes \tilde{\mathcal{J}}\left(\mathfrak{g},\mathfrak{v}\right)\oplus \tilde{G}\left(\mathfrak{g},\mathfrak{v}\right)\otimes \tilde{\mathcal{J}}\left(\mathfrak{e},\mathfrak{v}\right),\\ \tilde{\mathcal{N}}\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right)=\tilde{G}\left(\mathfrak{e},\mathfrak{u}\right)\otimes \tilde{\mathcal{J}}\left(\mathfrak{e},\mathfrak{v}\right)\oplus \tilde{G}\left(\mathfrak{g},\mathfrak{u}\right)\otimes \tilde{\mathcal{J}}\left(\mathfrak{g},\mathfrak{v}\right)\oplus \tilde{G}\left(\mathfrak{e},\mathfrak{v}\right)\otimes \tilde{\mathcal{J}}\left(\mathfrak{e},\mathfrak{u}\right)\oplus \tilde{G}\left(\mathfrak{g},\mathfrak{v}\right)\otimes \tilde{\mathcal{J}}\left(\mathfrak{g},\mathfrak{u}\right),\\ \tilde{Q}\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right)=\tilde{G}\left(\mathfrak{e},\mathfrak{u}\right)\otimes \tilde{\mathcal{J}}\left(\mathfrak{g},\mathfrak{v}\right)\oplus \tilde{G}\left(\mathfrak{g},\mathfrak{u}\right)\otimes \tilde{\mathcal{J}}\left(\mathfrak{e},\mathfrak{v}\right)\oplus \tilde{G}\left(\mathfrak{e},\mathfrak{v}\right)\otimes \tilde{\mathcal{J}}\left(\mathfrak{g},\mathfrak{u}\right)\oplus \tilde{G}\left(\mathfrak{g},\mathfrak{v}\right)\otimes \tilde{\mathcal{J}}\left(\mathfrak{e},\mathfrak{u}\right),\end{array}$$

and for each $\gamma \in \left[0, 1\right],$ $\tilde{\mathcal{M}}\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right)$, $\tilde{P}\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right)$, $\tilde{\mathcal{N}}\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right),$ and $\tilde{Q}\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right)$ are defined as follows:

$$\begin{array}{c}{\mathcal{M}}_{\gamma }\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right)=\left[{\mathcal{M}}_{\ast }\left(\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right), \gamma \right), {\mathcal{M}}^{\ast }\left(\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right), \gamma \right)\right],\\ P_{\gamma }\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right)=\left[P_{\ast }\left(\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right), \gamma \right), P^{\ast }\left(\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right), \gamma \right)\right],\\ {\mathcal{N}}_{\gamma }\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right)=\left[{\mathcal{N}}_{\ast }\left(\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right), \gamma \right), {\mathcal{N}}^{\ast }\left(\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right), \gamma \right)\right],\\ Q_{\gamma }\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right)=\left[Q_{\ast }\left(\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right), \gamma \right), Q^{\ast }\left(\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right), \gamma \right)\right].\end{array}$$

Proof. 

Let $\tilde{G}$ and $\tilde{\mathcal{J}}$ be two coordinated $\mathsf{U}\mathsf{D}$-${\mathrm{Ԓ}}_1$ and ${\mathrm{Ԓ}}_2$-convex $\mathsf{F}\mathsf{N}\mathsf{V}\mathsf{M}$s on $\left[\mathfrak{e},\mathfrak{g}\right]\times \left[\mathfrak{u},\mathfrak{v}\right]$, respectively. Then,

$$\begin{array}{c}\tilde{G}\left(\tau \mathfrak{e}+\left(1-\tau \right)\mathfrak{g},\kappa \mathfrak{u}+\left(1-\kappa \right)\mathfrak{v}\right)\\ {\supseteq }_{\mathbb{F}}{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right)\tilde{G}\left(\mathfrak{e},\mathfrak{u}\right)\oplus {\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_1\left(1-\kappa \right)\tilde{G}\left(\mathfrak{e},\mathfrak{v}\right)\oplus {\mathrm{Ԓ}}_1\left(1-\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right)\tilde{G}\left(\mathfrak{g},\mathfrak{u}\right)\oplus {\mathrm{Ԓ}}_1\left(1-\tau \right)\\ {\mathrm{Ԓ}}_1\left(1-\kappa \right)\tilde{G}\left(\mathfrak{g},\mathfrak{v}\right),\\ \tilde{G}\left(\tau \mathfrak{e}+\left(1-\tau \right)\mathfrak{g},\left(1-\kappa \right)\mathfrak{u}+\kappa \mathfrak{v}\right)\\ {\supseteq }_{\mathbb{F}}{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_1\left(1-\kappa \right)\tilde{G}\left(\mathfrak{e},\mathfrak{u}\right)\oplus {\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right)\tilde{G}\left(\mathfrak{e},\mathfrak{v}\right)\oplus {\mathrm{Ԓ}}_1\left(1-\tau \right){\mathrm{Ԓ}}_1\left(1-\kappa \right)\tilde{G}\left(\mathfrak{g},\mathfrak{u}\right)\oplus \\ {\mathrm{Ԓ}}_1\left(1-\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right)\tilde{G}\left(\mathfrak{g},\mathfrak{v}\right),\end{array}$$

$$\begin{array}{c}\tilde{G}\left(\left(1-\tau \right)\mathfrak{e}+\tau \mathfrak{g},\kappa \mathfrak{u}+\left(1-\kappa \right)\mathfrak{v}\right)\\ {\supseteq }_{\mathbb{F}}{\mathrm{Ԓ}}_1\left(1-\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right)\tilde{G}\left(\mathfrak{e},\mathfrak{u}\right)\oplus {\mathrm{Ԓ}}_1\left(1-\tau \right){\mathrm{Ԓ}}_1\left(1-\kappa \right)\tilde{G}\left(\mathfrak{e},\mathfrak{v}\right)\oplus {\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right)\tilde{G}\left(\mathfrak{g},\mathfrak{u}\right)\oplus \\ {\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_1\left(1-\kappa \right)\tilde{G}\left(\mathfrak{g},\mathfrak{v}\right),\\ \tilde{G}\left(\left(1-\tau \right)\mathfrak{e}+\tau \mathfrak{g},\left(1-\kappa \right)\mathfrak{u}+\kappa \mathfrak{v}\right)\\ {\supseteq }_{\mathbb{F}}{\mathrm{Ԓ}}_1\left(1-\tau \right){\mathrm{Ԓ}}_1\left(1-\kappa \right)\tilde{G}\left(\mathfrak{e},\mathfrak{u}\right)\oplus {\mathrm{Ԓ}}_1\left(1-\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right)\tilde{G}\left(\mathfrak{e},\mathfrak{v}\right)\oplus {\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_1\left(1-\kappa \right)\tilde{G}\left(\mathfrak{g},\mathfrak{u}\right)\oplus \\ {\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right)\tilde{G}\left(\mathfrak{g},\mathfrak{v}\right),\end{array}$$

and

$$\begin{array}{c}\tilde{\mathcal{J}}\left(\tau \mathfrak{e}+\left(1-\tau \right)\mathfrak{g},\kappa \mathfrak{u}+\left(1-\kappa \right)\mathfrak{v}\right)\\ {\supseteq }_{\mathbb{F}}{\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)\tilde{\mathcal{J}}\left(\mathfrak{e},\mathfrak{u}\right)\oplus {\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)\tilde{\mathcal{J}}\left(\mathfrak{e},\mathfrak{v}\right)\oplus {\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)\tilde{\mathcal{J}}\left(\mathfrak{g},\mathfrak{u}\right)\oplus {\mathrm{Ԓ}}_2\left(1-\tau \right)\\ {\mathrm{Ԓ}}_2\left(1-\kappa \right)\tilde{\mathcal{J}}\left(\mathfrak{g},\mathfrak{v}\right),\\ \tilde{\mathcal{J}}\left(\tau \mathfrak{e}+\left(1-\tau \right)\mathfrak{g},\left(1-\kappa \right)\mathfrak{u}+\kappa \mathfrak{v}\right)\\ {\supseteq }_{\mathbb{F}}{\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)\tilde{\mathcal{J}}\left(\mathfrak{e},\mathfrak{u}\right)\oplus {\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)\tilde{\mathcal{J}}\left(\mathfrak{e},\mathfrak{v}\right)\oplus {\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)\tilde{\mathcal{J}}\left(\mathfrak{g},\mathfrak{u}\right)\oplus \\ {\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)\tilde{\mathcal{J}}\left(\mathfrak{g},\mathfrak{v}\right),\end{array}$$

$$\begin{array}{c}\tilde{\mathcal{J}}\left(\left(1-\tau \right)\mathfrak{e}+\tau \mathfrak{g},\kappa \mathfrak{u}+\left(1-\kappa \right)\mathfrak{v}\right)\\ {\supseteq }_{\mathbb{F}}{\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)\tilde{\mathcal{J}}\left(\mathfrak{e},\mathfrak{u}\right)\oplus {\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)\tilde{\mathcal{J}}\left(\mathfrak{e},\mathfrak{v}\right)\oplus {\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)\tilde{\mathcal{J}}\left(\mathfrak{g},\mathfrak{u}\right)\oplus \\ {\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)\tilde{\mathcal{J}}\left(\mathfrak{g},\mathfrak{v}\right),\\ \tilde{\mathcal{J}}\left(\left(1-\tau \right)\mathfrak{e}+\tau \mathfrak{g},\left(1-\kappa \right)\mathfrak{u}+\kappa \mathfrak{v}\right)\\ {\supseteq }_{\mathbb{F}}{\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)\tilde{\mathcal{J}}\left(\mathfrak{e},\mathfrak{u}\right)\oplus {\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)\tilde{\mathcal{J}}\left(\mathfrak{e},\mathfrak{v}\right)\oplus {\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)\tilde{\mathcal{J}}\left(\mathfrak{g},\mathfrak{u}\right)\oplus \\ {\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)\tilde{\mathcal{J}}\left(\mathfrak{g},\mathfrak{v}\right),\end{array}$$

Since $\tilde{G}$ and $\tilde{\mathcal{J}}$ are both coordinated $\mathsf{U}\mathsf{D}$-${\mathrm{Ԓ}}_1$- and ${\mathrm{Ԓ}}_2$-convex $\mathsf{F}\mathsf{N}\mathsf{V}\mathsf{M}$s on $\left[\mathfrak{e},\mathfrak{g}\right]\times \left[\mathfrak{u},\mathfrak{v}\right]$, respectively, then, for any $\gamma \in \left[0, 1\right]$, we have

$$\begin{array}{c}{G}_{\gamma }\left(\tau \mathfrak{e}+\left(1-\tau \right)\mathfrak{g},\kappa \mathfrak{u}+\left(1-\kappa \right)\mathfrak{v}\right)\times {\mathcal{J}}_{\gamma }\left(\tau \mathfrak{e}+\left(1-\tau \right)\mathfrak{g},\kappa \mathfrak{u}+\left(1-\kappa \right)\mathfrak{v}\right)\\ +G_{\gamma }\left(\tau \mathfrak{e}+\left(1-\tau \right)\mathfrak{g},\left(1-\kappa \right)\mathfrak{u}+\kappa \mathfrak{v}\right)\times {\mathcal{J}}_{\gamma }\left(\tau \mathfrak{e}+\left(1-\tau \right)\mathfrak{g},\left(1-\kappa \right)\mathfrak{u}+\kappa \mathfrak{v}\right)\\ +G_{\gamma }\left(\left(1-\tau \right)\mathfrak{e}+\tau \mathfrak{g},\kappa \mathfrak{u}+\left(1-\kappa \right)\mathfrak{v}\right)\times {\mathcal{J}}_{\gamma }\left(\left(1-\tau \right)\mathfrak{e}+\tau \mathfrak{g},\kappa \mathfrak{u}+\left(1-\kappa \right)\mathfrak{v}\right)\\ +G_{\gamma }\left(\left(1-\tau \right)\mathfrak{e}+\tau \mathfrak{g},\left(1-\kappa \right)\mathfrak{u}+\kappa \mathfrak{v}\right)\times {\mathcal{J}}_{\gamma }\left(\left(1-\tau \right)\mathfrak{e}+\tau \mathfrak{g},\left(1-\kappa \right)\mathfrak{u}+\kappa \mathfrak{v}\right)\\ {\supseteq }_I{\mathcal{M}}_{\gamma }\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right)[{\mathrm{Ԓ}}_1\left(1-\tau \right){\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_1\left(1-\kappa \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)+{\mathrm{Ԓ}}_1\left(1-\tau \right){\mathrm{Ԓ}}_2\left(1-\tau \right)\\ {\mathrm{Ԓ}}_1\left(\kappa \right){\mathrm{Ԓ}}_2\left(\kappa \right)+{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_1\left(1-\kappa \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)+{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right){\mathrm{Ԓ}}_2\left(\kappa \right)]\\ +P_{\gamma }\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right)[{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_1\left(1-\kappa \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)+{\mathrm{Ԓ}}_1\left(1-\tau \right){\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_1\left(1-\kappa \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)\\ +{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right){\mathrm{Ԓ}}_2\left(\kappa \right)+{\mathrm{Ԓ}}_1\left(1-\tau \right){\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right){\mathrm{Ԓ}}_2\left(\kappa \right)]\\ +{\mathcal{N}}_{\gamma }\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right)[{\mathrm{Ԓ}}_1\left(1-\tau \right){\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)+{\mathrm{Ԓ}}_1\left(1-\tau \right){\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_1\left(1-\kappa \right)\\ {\mathrm{Ԓ}}_2\left(\kappa \right)+{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_1\left(1-\kappa \right){\mathrm{Ԓ}}_2\left(\kappa \right)+{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)]\\ +Q_{\gamma }\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right)[{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)+{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_1\left(1-\kappa \right){\mathrm{Ԓ}}_2\left(\kappa \right)+\\ {\mathrm{Ԓ}}_1\left(1-\tau \right){\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)+{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)].\end{array}$$

Taking the multiplication of the above fuzzy inclusion with ${\tau }^{\alpha -1}{\kappa }^{\beta -1}$ and then taking the double integration of the result over $\left[0, 1\right]\times \left[0, 1\right]$ with respect to ($\tau , \kappa$) gives

$$\begin{array}{c}{\int }_0^1{\int }_0^1{\tau }^{\alpha -1}{\kappa }^{\beta -1}{G}_{\gamma }\left(\tau \mathfrak{e}+\left(1-\tau \right)\mathfrak{g},\kappa \mathfrak{u}+\left(1-\kappa \right)\mathfrak{v}\right)\times {\mathcal{J}}_{\gamma }\left(\tau \mathfrak{e}+\left(1-\tau \right)\mathfrak{g},\kappa \mathfrak{u}+\left(1-\kappa \right)\mathfrak{v}\right)d\tau d\kappa \\ +{\int }_0^1{\int }_0^1{\tau }^{\alpha -1}{\kappa }^{\beta -1}{G}_{\gamma }\left(\tau \mathfrak{e}+\left(1-\tau \right)\mathfrak{g},\left(1-\kappa \right)\mathfrak{u}+\kappa \mathfrak{v}\right)\times {\mathcal{J}}_{\gamma }\left(\tau \mathfrak{e}+\left(1-\tau \right)\mathfrak{g},\left(1-\kappa \right)\mathfrak{u}+\kappa \mathfrak{v}\right)d\tau d\kappa \\ +{\int }_0^1{\int }_0^1{\tau }^{\alpha -1}{\kappa }^{\beta -1}{G}_{\gamma }\left(\left(1-\tau \right)\mathfrak{e}+\tau \mathfrak{g},\kappa \mathfrak{u}+\left(1-\kappa \right)\mathfrak{v}\right)\times {\mathcal{J}}_{\gamma }\left(\left(1-\tau \right)\mathfrak{e}+\tau \mathfrak{g},\kappa \mathfrak{u}+\left(1-\kappa \right)\mathfrak{v}\right)d\tau d\kappa \\ +{\int }_0^1{\int }_0^1{\tau }^{\alpha -1}{\kappa }^{\beta -1}{G}_{\gamma }\left(\left(1-\tau \right)\mathfrak{e}+\tau \mathfrak{g},\left(1-\kappa \right)\mathfrak{u}+\kappa \mathfrak{v}\right)\times {\mathcal{J}}_{\gamma }\left(\left(1-\tau \right)\mathfrak{e}+\tau \mathfrak{g},\left(1-\kappa \right)\mathfrak{u}+\kappa \mathfrak{v}\right)d\tau d\kappa \\ \begin{array}{cc}{\supseteq }_I{\mathcal{M}}_{\gamma }\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right)& {\int }_0^1{\int }_0^1{\tau }^{\alpha -1}{\kappa }^{\beta -1}[{\mathrm{Ԓ}}_1\left(1-\tau \right){\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_1\left(1-\kappa \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)\\ & +{\mathrm{Ԓ}}_1\left(1-\tau \right){\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right){\mathrm{Ԓ}}_2\left(\kappa \right)+{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_1\left(1-\kappa \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)\\ & +{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right){\mathrm{Ԓ}}_2\left(\kappa \right)]d\tau d\kappa \end{array}\\ \begin{array}{cc}+P_{\gamma }\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right)& {\int }_0^1{\int }_0^1{\tau }^{\alpha -1}{\kappa }^{\beta -1}[{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_1\left(1-\kappa \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)\\ & +{\mathrm{Ԓ}}_1\left(1-\tau \right){\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_1\left(1-\kappa \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)+{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right){\mathrm{Ԓ}}_2\left(\kappa \right)\\ & +{\mathrm{Ԓ}}_1\left(1-\tau \right){\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right){\mathrm{Ԓ}}_2\left(\kappa \right)]d\tau d\kappa \end{array}\\ \begin{array}{cc}+{\mathcal{N}}_{\gamma }\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right)& {\int }_0^1{\int }_0^1{\tau }^{\alpha -1}{\kappa }^{\beta -1}[{\mathrm{Ԓ}}_1\left(1-\tau \right){\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)\\ & +{\mathrm{Ԓ}}_1\left(1-\tau \right){\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_1\left(1-\kappa \right){\mathrm{Ԓ}}_2\left(\kappa \right)+{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_1\left(1-\kappa \right){\mathrm{Ԓ}}_2\left(\kappa \right)\\ & +{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)]d\tau d\kappa \end{array}\\ \begin{array}{cc}+Q_{\gamma }\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right)& {\int }_0^1{\int }_0^1{\tau }^{\alpha -1}{\kappa }^{\beta -1}[{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)\\ & +{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_1\left(1-\kappa \right){\mathrm{Ԓ}}_2\left(\kappa \right)+{\mathrm{Ԓ}}_1\left(1-\tau \right){\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)\\ & +{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)]d\tau d\kappa \end{array}\end{array}$$

(61)

From the right-hand side of (61), we have

$$\begin{array}{c}{\int }_0^1{\int }_0^1{\tau }^{\alpha -1}{\kappa }^{\beta -1}{G}_{\gamma }\left(\tau \mathfrak{e}+\left(1-\tau \right)\mathfrak{g},\kappa \mathfrak{u}+\left(1-\kappa \right)\mathfrak{v}\right)\times {\mathcal{J}}_{\gamma }\left(\tau \mathfrak{e}+\left(1-\tau \right)\mathfrak{g},\kappa \mathfrak{u}+\left(1-\kappa \right)\mathfrak{v}\right)d\tau d\kappa \\ +{\int }_0^1{\int }_0^1{\tau }^{\alpha -1}{\kappa }^{\beta -1}{G}_{\gamma }\left(\tau \mathfrak{e}+\left(1-\tau \right)\mathfrak{g},\left(1-\kappa \right)\mathfrak{u}+\kappa \mathfrak{v}\right)\times {\mathcal{J}}_{\gamma }\left(\tau \mathfrak{e}+\left(1-\tau \right)\mathfrak{g},\left(1-\kappa \right)\mathfrak{u}+\kappa \mathfrak{v}\right)d\tau d\kappa \\ +{\int }_0^1{\int }_0^1{\tau }^{\alpha -1}{\kappa }^{\beta -1}{G}_{\gamma }\left(\left(1-\tau \right)\mathfrak{e}+\tau \mathfrak{g},\kappa \mathfrak{u}+\left(1-\kappa \right)\mathfrak{v}\right)\times {\mathcal{J}}_{\gamma }\left(\left(1-\tau \right)\mathfrak{e}+\tau \mathfrak{g},\kappa \mathfrak{u}+\left(1-\kappa \right)\mathfrak{v}\right)d\tau d\kappa \\ +{\int }_0^1{\int }_0^1{\tau }^{\alpha -1}{\kappa }^{\beta -1}{G}_{\gamma }\left(\left(1-\tau \right)\mathfrak{e}+\tau \mathfrak{g},\left(1-\kappa \right)\mathfrak{u}+\kappa \mathfrak{v}\right)\times {\mathcal{J}}_{\gamma }\left(\left(1-\tau \right)\mathfrak{e}+\tau \mathfrak{g},\left(1-\kappa \right)\mathfrak{u}+\kappa \mathfrak{v}\right)d\tau d\kappa \\ =\frac{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{{\left(\mathfrak{g}-\mathfrak{e}\right)}^{\alpha }{\left(\mathfrak{v}-\mathfrak{u}\right)}^{\beta }}\left[{\mathcal{I}}_{{\mathfrak{e}}^{+},{\mathfrak{u}}^{+}}^{\alpha , \beta }{G}_{\gamma }\left(\mathfrak{g},\mathfrak{v}\right)\times {\mathcal{J}}_{\gamma }\left(\mathfrak{g},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathfrak{e}}^{+},{\mathfrak{v}}^{-}}^{\alpha , \beta }{G}_{\gamma }\left(\mathfrak{g},\mathfrak{u}\right)\times {\mathcal{J}}_{\gamma }\left(\mathfrak{g},\mathfrak{u}\right)\right]\end{array}$$

(62)

Combining (61) and (62), for each $\gamma \in \left[0, 1\right],$ we have

$$\begin{array}{c}\frac{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{{\left(\mathfrak{g}-\mathfrak{e}\right)}^{\alpha }{\left(\mathfrak{v}-\mathfrak{u}\right)}^{\beta }}\left[{\mathcal{I}}_{{\mathfrak{e}}^{+},{\mathfrak{u}}^{+}}^{\alpha , \beta }{G}_{\gamma }\left(\mathfrak{g},\mathfrak{v}\right)\times {\mathcal{J}}_{\gamma }\left(\mathfrak{g},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathfrak{e}}^{+},{\mathfrak{v}}^{-}}^{\alpha , \beta }{G}_{\gamma }\left(\mathfrak{g},\mathfrak{u}\right)\times {\mathcal{J}}_{\gamma }\left(\mathfrak{g},\mathfrak{u}\right)\right]\\ \begin{array}{cc}{\supseteq }_I{\mathcal{M}}_{\gamma }\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right)& {\int }_0^1{\int }_0^1{\tau }^{\alpha -1}{\kappa }^{\beta -1}[{\mathrm{Ԓ}}_1\left(1-\tau \right){\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_1\left(1-\kappa \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)\\ & +{\mathrm{Ԓ}}_1\left(1-\tau \right){\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right){\mathrm{Ԓ}}_2\left(\kappa \right)+{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_1\left(1-\kappa \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)\\ & +{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right){\mathrm{Ԓ}}_2\left(\kappa \right)]d\tau d\kappa \end{array}\\ \begin{array}{cc}+P_{\gamma }\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right)& {\int }_0^1{\int }_0^1{\tau }^{\alpha -1}{\kappa }^{\beta -1}[{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_1\left(1-\kappa \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)\\ & +{\mathrm{Ԓ}}_1\left(1-\tau \right){\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_1\left(1-\kappa \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)+{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right){\mathrm{Ԓ}}_2\left(\kappa \right)\\ & +{\mathrm{Ԓ}}_1\left(1-\tau \right){\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right){\mathrm{Ԓ}}_2\left(\kappa \right)]d\tau d\kappa \end{array}\\ \begin{array}{cc}+{\mathcal{N}}_{\gamma }\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right)& {\int }_0^1{\int }_0^1{\tau }^{\alpha -1}{\kappa }^{\beta -1}[{\mathrm{Ԓ}}_1\left(1-\tau \right){\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)\\ & +{\mathrm{Ԓ}}_1\left(1-\tau \right){\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_1\left(1-\kappa \right){\mathrm{Ԓ}}_2\left(\kappa \right)+{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_1\left(1-\kappa \right){\mathrm{Ԓ}}_2\left(\kappa \right)\\ & +{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)]d\tau d\kappa \end{array}\\ \begin{array}{cc}+Q_{\gamma }\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right)& {\int }_0^1{\int }_0^1{\tau }^{\alpha -1}{\kappa }^{\beta -1}[{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)\\ & +{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_1\left(1-\kappa \right){\mathrm{Ԓ}}_2\left(\kappa \right)+{\mathrm{Ԓ}}_1\left(1-\tau \right){\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)\\ & +{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)]d\tau d\kappa .\end{array}\end{array}$$

Moreover, we have

$$\begin{array}{c}\frac{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{{\left(\mathfrak{g}-\mathfrak{e}\right)}^{\alpha }{\left(\mathfrak{v}-\mathfrak{u}\right)}^{\beta }}\left[{\mathcal{I}}_{{\mathfrak{e}}^{+},{\mathfrak{u}}^{+}}^{\alpha , \beta }\tilde{G}\left(\mathfrak{g},\mathfrak{v}\right)\otimes \tilde{\mathcal{J}}\left(\mathfrak{g},\mathfrak{v}\right)\oplus {\mathcal{I}}_{{\mathfrak{e}}^{+},{\mathfrak{v}}^{-}}^{\alpha , \beta }\tilde{G}\left(\mathfrak{g},\mathfrak{u}\right)\otimes \tilde{\mathcal{J}}\left(\mathfrak{g},\mathfrak{u}\right)\right]\\ \oplus \frac{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{{\left(\mathfrak{g}-\mathfrak{e}\right)}^{\alpha }{\left(\mathfrak{v}-\mathfrak{u}\right)}^{\beta }}\left[{\mathcal{I}}_{{\mathfrak{g}}^{-},{\mathfrak{u}}^{+}}^{\alpha , \beta }\tilde{G}\left(\mathfrak{e},\mathfrak{v}\right)\otimes \tilde{\mathcal{J}}\left(\mathfrak{e},\mathfrak{v}\right)\oplus {\mathcal{I}}_{{\mathfrak{g}}^{-},{\mathfrak{v}}^{-}}^{\alpha , \beta }\tilde{G}\left(\mathfrak{e},\mathfrak{u}\right)\otimes \tilde{\mathcal{J}}\left(\mathfrak{e},\mathfrak{u}\right)\right]\\ {\supseteq }_{\mathbb{F}}\tilde{\mathcal{M}}\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right){\int }_0^1{\tau }^{\alpha -1}{\kappa }^{\beta -1}[{\mathrm{Ԓ}}_1\left(1-\tau \right){\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_1\left(1-\kappa \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)+{\mathrm{Ԓ}}_1\left(1-\tau \right)\\ {\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right){\mathrm{Ԓ}}_2\left(\kappa \right)+{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_1\left(1-\kappa \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)+\\ {\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right){\mathrm{Ԓ}}_2\left(\kappa \right)]d\tau d\kappa \\ \oplus \tilde{P}\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right){\int }_0^1{\tau }^{\alpha -1}{\kappa }^{\beta -1}[{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_1\left(1-\kappa \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)+{\mathrm{Ԓ}}_1\left(1-\tau \right)\\ {\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_1\left(1-\kappa \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)+{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right){\mathrm{Ԓ}}_2\left(\kappa \right)+{\mathrm{Ԓ}}_1\left(1-\tau \right)\\ {\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right){\mathrm{Ԓ}}_2\left(\kappa \right)]d\tau d\kappa \\ \oplus \tilde{\mathcal{N}}\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right){\int }_0^1{\tau }^{\alpha -1}{\kappa }^{\beta -1}[{\mathrm{Ԓ}}_1\left(1-\tau \right){\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)+{\mathrm{Ԓ}}_1\left(1-\tau \right){\mathrm{Ԓ}}_2\left(1-\tau \right)\\ {\mathrm{Ԓ}}_1\left(1-\kappa \right){\mathrm{Ԓ}}_2\left(\kappa \right)+{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_1\left(1-\kappa \right){\mathrm{Ԓ}}_2\left(\kappa \right)+{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)]d\tau d\kappa \\ \oplus \tilde{Q}\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right){\int }_0^1{\tau }^{\alpha -1}{\kappa }^{\beta -1}[{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)+{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_1\left(1-\kappa \right)\\ {\mathrm{Ԓ}}_2\left(\kappa \right)+{\mathrm{Ԓ}}_1\left(1-\tau \right){\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)+{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)]d\tau d\kappa .\end{array}$$

Hence, we obtain the required result. □

Remark 3.

If one assumes that$\mathrm{Ԓ}\left(\tau \right)=\tau , \mathrm{Ԓ}\left(\kappa \right)=\kappa$, $\alpha =1,$ and $\beta =1$, then, from (59), as a result, there will be an inequality (see [70]):

$$\begin{array}{c}\frac{1}{\left(\mathfrak{g}-\mathfrak{e}\right)\left(\mathfrak{v}-\mathfrak{u}\right)}{\int }_{\mathfrak{e}}^{\mathfrak{g}}{\int }_{\mathfrak{u}}^{\mathfrak{v}}\tilde{G}\left(x,y\right)\otimes \tilde{\mathcal{J}}\left(x,y\right)dydx\\ {\supseteq }_{\mathbb{F}}\frac{1}{9}\tilde{\mathcal{M}}\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right)\oplus \frac{1}{18}\left[\tilde{P}\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right)\oplus \tilde{\mathcal{N}}\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right)\right]\oplus \frac{1}{36}\tilde{Q}\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right).\end{array}$$

(63)

If $\tilde{G}$ is a coordinated left-$\mathsf{U}\mathsf{D}$-$\mathrm{Ԓ}$-convex function with $\mathrm{Ԓ}\left(\tau \right)=\tau , \mathrm{Ԓ}\left(\kappa \right)=\kappa$ and one assumes that $\alpha =1$ and $\beta =1$, then, from (59), as a result, there will be an inequality (see [59]):

$$\begin{array}{c}\frac{1}{\left(\mathfrak{g}-\mathfrak{e}\right)\left(\mathfrak{v}-\mathfrak{u}\right)}{\int }_{\mathfrak{e}}^{\mathfrak{g}}{\int }_{\mathfrak{u}}^{\mathfrak{v}}\tilde{G}\left(x,y\right)\otimes \tilde{\mathcal{J}}\left(x,y\right)dydx\\ {\le }_{\mathbb{F}}\frac{1}{9}\tilde{\mathcal{M}}\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right)\oplus \frac{1}{18}\left[\tilde{P}\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right)\oplus \tilde{\mathcal{N}}\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right)\right]\oplus \frac{1}{36}\tilde{Q}\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right).\end{array}$$

(64)

If $G_{\ast }\left(\left(x,y\right), \gamma \right)\ne G^{\ast }\left(\left(x,y\right), \gamma \right)$ with $\gamma =1$ and $\mathrm{Ԓ}\left(\tau \right)=\tau , \mathrm{Ԓ}\left(\kappa \right)=\kappa$, then, from (59), we succeed in bringing about the upcoming inequality (see [55]):

$$\begin{array}{c}\frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{{4\left(\mathfrak{g}-\mathfrak{e}\right)}^{\alpha }{\left(\mathfrak{v}-\mathfrak{u}\right)}^{\beta }}\left[\begin{array}{c}{\mathcal{I}}_{{\mathfrak{e}}^{+},{\mathfrak{u}}^{+}}^{\alpha , \beta }G\left(\mathfrak{g},\mathfrak{v}\right)\times \mathcal{J}\left(\mathfrak{g},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathfrak{e}}^{+},{\mathfrak{v}}^{-}}^{\alpha , \beta }G\left(\mathfrak{g},\mathfrak{u}\right)\times \mathcal{J}\left(\mathfrak{g},\mathfrak{u}\right)\end{array}\right]\\ +\frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{{4\left(\mathfrak{g}-\mathfrak{e}\right)}^{\alpha }{\left(\mathfrak{v}-\mathfrak{u}\right)}^{\beta }}\left[\begin{array}{c}{\mathcal{I}}_{{\mathfrak{g}}^{-},{\mathfrak{u}}^{+}}^{\alpha , \beta }G\left(\mathfrak{e},\mathfrak{v}\right)\times \mathcal{J}\left(\mathfrak{e},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathfrak{g}}^{-},{\mathfrak{v}}^{-}}^{\alpha , \beta }G\left(\mathfrak{e},\mathfrak{u}\right)\times \mathcal{J}\left(\mathfrak{e},\mathfrak{u}\right)\end{array}\right]\\ \supseteq \left(\frac{1}{2}-\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\right)\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)\mathcal{M}\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right)+\frac{\alpha }{(\alpha +1)(\alpha +2)}\left(\frac{1}{2}-\frac{\beta }{(\beta +1)(\beta +2)}\right)P\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right)\\ +\left(\frac{1}{2}-\frac{\alpha }{(\alpha +1)(\alpha +2)}\right)\frac{\beta }{(\beta +1)(\beta +2)}\mathcal{N}\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right)+\frac{\beta }{(\beta +1)(\beta +2)}\frac{\alpha }{(\alpha +1)(\alpha +2)}Q\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right).\end{array}$$

(65)

If $\mathrm{Ԓ}\left(\tau \right)=\tau , \mathrm{Ԓ}\left(\kappa \right)=\kappa ,$ and $G_{\ast }\left(\left(x,y\right), \gamma \right)\ne G^{\ast }\left(\left(x,y\right), \gamma \right)$ with $\gamma =1$, then, from (59), we succeed in bringing about the upcoming inequality (see [68]):

$$\begin{array}{c}\frac{1}{\left(\mathfrak{g}-\mathfrak{e}\right)\left(\mathfrak{v}-\mathfrak{u}\right)}{\int }_{\mathfrak{e}}^{\mathfrak{g}}{\int }_{\mathfrak{u}}^{\mathfrak{v}}G\left(x,y\right)\times \mathcal{J}\left(x,y\right)dydx\\ \supseteq \frac{1}{9}\mathcal{M}\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right)+\frac{1}{18}\left[P\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right)+\mathcal{N}\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right)\right]+\frac{1}{36}Q\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right).\end{array}$$

(66)

If $G_{\ast }\left(\left(x,y\right), \gamma \right)=G^{\ast }\left(\left(x,y\right), \gamma \right)$ and ${\mathcal{J}}_{\ast }\left(\left(x,y\right), \gamma \right)={\mathcal{J}}^{\ast }\left(\left(x,y\right), \gamma \right)$ with $\gamma =1$ and $\mathrm{Ԓ}\left(\tau \right)=\tau , \mathrm{Ԓ}\left(\kappa \right)=\kappa$, then, from (59), we succeed in bringing about the upcoming inequality (see [69]):

$$\begin{array}{c}\frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{{4\left(\mathfrak{g}-\mathfrak{e}\right)}^{\alpha }{\left(\mathfrak{v}-\mathfrak{u}\right)}^{\beta }}\left[\begin{array}{c}{\mathcal{I}}_{{\mathfrak{e}}^{+},{\mathfrak{u}}^{+}}^{\alpha , \beta }G\left(\mathfrak{g},\mathfrak{v}\right)\times \mathcal{J}\left(\mathfrak{g},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathfrak{e}}^{+},{\mathfrak{v}}^{-}}^{\alpha , \beta }G\left(\mathfrak{g},\mathfrak{u}\right)\times \mathcal{J}\left(\mathfrak{g},\mathfrak{u}\right)\end{array}\right]\\ +\frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{{4\left(\mathfrak{g}-\mathfrak{e}\right)}^{\alpha }{\left(\mathfrak{v}-\mathfrak{u}\right)}^{\beta }} \left[\begin{array}{c}+{\mathcal{I}}_{{\mathfrak{g}}^{-},{\mathfrak{u}}^{+}}^{\alpha , \beta }G\left(\mathfrak{e},\mathfrak{v}\right)\times \mathcal{J}\left(\mathfrak{e},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathfrak{g}}^{-},{\mathfrak{v}}^{-}}^{\alpha , \beta }G\left(\mathfrak{e},\mathfrak{u}\right)\times \mathcal{J}\left(\mathfrak{e},\mathfrak{u}\right)\end{array}\right]\\ \le \left(\frac{1}{2}-\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\right)\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)\mathcal{M}\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right)+\frac{\alpha }{(\alpha +1)(\alpha +2)}\left(\frac{1}{2}-\frac{\beta }{(\beta +1)(\beta +2)}\right)P\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right)\\ +\left(\frac{1}{2}-\frac{\alpha }{(\alpha +1)(\alpha +2)}\right)\frac{\beta }{(\beta +1)(\beta +2)}\mathcal{N}\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right)+\frac{\beta }{(\beta +1)(\beta +2)}\frac{\alpha }{(\alpha +1)(\alpha +2)}Q\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right).\end{array}$$

(67)

Theorem 10.

Let $\tilde{G}, \tilde{\mathcal{J}}:∆\to {\mathbb{F}}_0^{+}$ be a coordinated $\mathsf{U}\mathsf{D}$-$\mathrm{Ԓ}$-convex $\mathsf{F}\mathsf{N}\mathsf{V}\mathsf{M}$ on $∆$, and let $\mathrm{Ԓ} :\left[0, 1\right]\to {\mathbb{R}}^{+}$. Then, from $\gamma$-cuts, we set up the sequence of $\mathsf{I}\mathsf{V}\mathsf{M}$s $G_{\gamma },{\mathcal{J}}_{\gamma }:∆\to {\mathbb{R}}_I^{+}, which$ is given by $G_{\gamma }\left(x,y\right)=\left[G_{\ast }\left(\left(x,y\right),\gamma \right), G^{\ast }\left(\left(x,y\right),\gamma \right)\right]$ and ${\mathcal{J}}_{\gamma }\left(x,y\right)=\left[{\mathcal{J}}_{\ast }\left(\left(x,y\right),\gamma \right), {\mathcal{J}}^{\ast }\left(\left(x,y\right),\gamma \right)\right]$ for all $\left(x,y\right)\in ∆$ and for all $\gamma \in \left[0, 1\right]$. If ${\tilde{G}\otimes \tilde{\mathcal{J}}\in \mathcal{F}\mathfrak{O}}_{∆}$, then the following inequalities holds:

$$\begin{array}{c}\frac{1}{2\alpha \beta {{\mathrm{Ԓ}}_1}^2\left(\frac{1}{2}\right){{\mathrm{Ԓ}}_2}^2\left(\frac{1}{2}\right)}\tilde{G}\left(\frac{\mathfrak{e}+\mathfrak{g}}{2}, \frac{\mathfrak{u}+\mathfrak{v}}{2}\right)\otimes \tilde{\mathcal{J}}\left(\frac{\mathfrak{e}+\mathfrak{g}}{2}, \frac{\mathfrak{u}+\mathfrak{v}}{2}\right)\\ {\supseteq }_{\mathbb{F}}\frac{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{{2\left(\mathfrak{g}-\mathfrak{e}\right)}^{\alpha }{\left(\mathfrak{v}-\mathfrak{u}\right)}^{\beta }}\left[{\mathcal{I}}_{{\mathfrak{e}}^{+},{\mathfrak{u}}^{+}}^{\alpha , \beta }\tilde{G}\left(\mathfrak{g},\mathfrak{v}\right)\otimes \tilde{\mathcal{J}}\left(\mathfrak{g},\mathfrak{v}\right)\oplus {\mathcal{I}}_{{\mathfrak{e}}^{+},{\mathfrak{v}}^{-}}^{\alpha , \beta }\tilde{G}\left(\mathfrak{g},\mathfrak{u}\right)\otimes \tilde{\mathcal{J}}\left(\mathfrak{g},\mathfrak{u}\right)\right]\\ \oplus \frac{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{2{\left(\mathfrak{g}-\mathfrak{e}\right)}^{\alpha }{\left(\mathfrak{v}-\mathfrak{u}\right)}^{\beta }}\left[{\mathcal{I}}_{{\mathfrak{g}}^{-},{\mathfrak{u}}^{+}}^{\alpha , \beta }\tilde{G}\left(\mathfrak{e},\mathfrak{v}\right)\otimes \tilde{\mathcal{J}}\left(\mathfrak{e},\mathfrak{v}\right)\oplus {\mathcal{I}}_{{\mathfrak{g}}^{-},{\mathfrak{v}}^{-}}^{\alpha , \beta }\tilde{G}\left(\mathfrak{e},\mathfrak{u}\right)\otimes \tilde{\mathcal{J}}\left(\mathfrak{e},\mathfrak{u}\right)\right]\\ \oplus \tilde{\mathcal{M}}\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right){\int }_0^1{\tau }^{\alpha -1}{\kappa }^{\beta -1}\left[{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right)\right[{\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)+{\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)+{\mathrm{Ԓ}}_2\left(1-\tau \right)\\ {\mathrm{Ԓ}}_2\left(1-\kappa \right)]+{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_1\left(1-\kappa \right)\left[{\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)+{\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)+{\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)\right]]d\tau d\kappa \\ \oplus \tilde{P}\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right){\int }_0^1{\tau }^{\alpha -1}{\kappa }^{\beta -1}\left[{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right)\right[{\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)+{\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)+\\ {\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)]+{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_1\left(1-\kappa \right)[{\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)+{\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)+\\ {\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)]d\tau d\kappa \\ \oplus \tilde{\mathcal{N}}\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right){\int }_0^1{\tau }^{\alpha -1}{\kappa }^{\beta -1}\left[{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right)\right[{\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)+{\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)+{\mathrm{Ԓ}}_2\left(1-\tau \right)\\ {\mathrm{Ԓ}}_2\left(\kappa \right)]+{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_1\left(1-\kappa \right)\left[{\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)+{\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)+{\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)\right]]d\tau d\kappa \\ \oplus \tilde{Q}\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right){\int }_0^1{\tau }^{\alpha -1}{\kappa }^{\beta -1}[{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right)\left[{\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)+{\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)+{\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)\right]\\ +{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_1\left(1-\kappa \right)\left[{\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)+{\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)+{\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)\right]]d\tau d\kappa .\end{array}$$

(68)

If $\tilde{G}$ and $\tilde{\mathcal{J}}$ are both coordinate $\mathsf{U}\mathsf{D}$-$\mathrm{Ԓ}$-concave $\mathsf{F}\mathsf{N}\mathsf{V}\mathsf{M}$s on $∆$, then the inequality above can be expressed as follows:

$$\begin{array}{c}\frac{1}{2\alpha \beta {{\mathrm{Ԓ}}_1}^2\left(\frac{1}{2}\right){{\mathrm{Ԓ}}_2}^2\left(\frac{1}{2}\right)}\tilde{G}\left(\frac{\mathfrak{e}+\mathfrak{g}}{2}, \frac{\mathfrak{u}+\mathfrak{v}}{2}\right)\otimes \tilde{\mathcal{J}}\left(\frac{\mathfrak{e}+\mathfrak{g}}{2}, \frac{\mathfrak{u}+\mathfrak{v}}{2}\right)\\ {\subseteq }_{\mathbb{F}}\frac{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{{2\left(\mathfrak{g}-\mathfrak{e}\right)}^{\alpha }{\left(\mathfrak{v}-\mathfrak{u}\right)}^{\beta }}\left[{\mathcal{I}}_{{\mathfrak{e}}^{+},{\mathfrak{u}}^{+}}^{\alpha , \beta }\tilde{G}\left(\mathfrak{g},\mathfrak{v}\right)\otimes \tilde{\mathcal{J}}\left(\mathfrak{g},\mathfrak{v}\right)\oplus {\mathcal{I}}_{{\mathfrak{e}}^{+},{\mathfrak{v}}^{-}}^{\alpha , \beta }\tilde{G}\left(\mathfrak{g},\mathfrak{u}\right)\otimes \tilde{\mathcal{J}}\left(\mathfrak{g},\mathfrak{u}\right)\right]\\ \oplus \frac{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{2{\left(\mathfrak{g}-\mathfrak{e}\right)}^{\alpha }{\left(\mathfrak{v}-\mathfrak{u}\right)}^{\beta }}\left[{\mathcal{I}}_{{\mathfrak{g}}^{-},{\mathfrak{u}}^{+}}^{\alpha , \beta }\tilde{G}\left(\mathfrak{e},\mathfrak{v}\right)\otimes \tilde{\mathcal{J}}\left(\mathfrak{e},\mathfrak{v}\right)\oplus {\mathcal{I}}_{{\mathfrak{g}}^{-},{\mathfrak{v}}^{-}}^{\alpha , \beta }\tilde{G}\left(\mathfrak{e},\mathfrak{u}\right)\otimes \tilde{\mathcal{J}}\left(\mathfrak{e},\mathfrak{u}\right)\right]\\ \oplus \tilde{\mathcal{M}}\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right){\int }_0^1{\tau }^{\alpha -1}{\kappa }^{\beta -1}\left[{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right)\right[{\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)+{\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)+{\mathrm{Ԓ}}_2\left(1-\tau \right)\\ {\mathrm{Ԓ}}_2\left(1-\kappa \right)]+{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_1\left(1-\kappa \right)\left[{\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)+{\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)+{\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)\right]]d\tau d\kappa \\ \oplus \tilde{P}\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right){\int }_0^1{\tau }^{\alpha -1}{\kappa }^{\beta -1}\left[{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right)\right[{\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)+{\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)+\\ {\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)]+{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_1\left(1-\kappa \right)[{\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)+{\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)+\\ {\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)\left]\right]d\tau d\kappa \\ \oplus \tilde{\mathcal{N}}\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right){\int }_0^1{\tau }^{\alpha -1}{\kappa }^{\beta -1}[{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right)\left[{\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)+{\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)+{\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)\right]\\ +{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_1\left(1-\kappa \right)\left[{\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)+{\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)+{\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)\right]]d\tau d\kappa \\ \oplus \tilde{Q}\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right){\int }_0^1{\tau }^{\alpha -1}{\kappa }^{\beta -1}\left[{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right)\right[{\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)+{\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)+\\ {\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)]+{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_1\left(1-\kappa \right)\left[{\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)+{\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)+{\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)\right]]d\tau d\kappa .\end{array}$$

(69)

where $\tilde{\mathcal{M}}\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right)$, $\tilde{P}\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right)$, $\tilde{\mathcal{N}}\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right),$ and $\tilde{Q}\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right)$ are given in Theorem 9.

Proof. 

Since $\tilde{G},\tilde{\mathcal{J}} :∆\to {\mathbb{F}}_0$ are two $\mathsf{U}\mathsf{D}$-$\mathbf{Ԓ}$-convex $\mathsf{F}\mathsf{N}\mathsf{V}\mathsf{M}$s, then, from inequality (17) and for each $\gamma \in \left[0, 1\right],$ we have

$$\begin{array}{c}{G}_{\gamma }\left(\frac{\mathfrak{e}+\mathfrak{g}}{2},\frac{\mathfrak{u}+\mathfrak{v}}{2}\right)\times {\mathcal{J}}_{\gamma }\left(\frac{\mathfrak{e}+\mathfrak{g}}{2},\frac{\mathfrak{u}+\mathfrak{v}}{2}\right)\\ =G_{\gamma }\left(\frac{\tau \mathfrak{e}+\left(1-\tau \right)\mathfrak{g}}{2}+\frac{\left(1-\tau \right)\mathfrak{e}+\tau \mathfrak{g}}{2},\frac{\kappa \mathfrak{u}+\left(1-\kappa \right)\mathfrak{v}}{2}+\frac{\mathfrak{u}+\mathfrak{v}}{2}\right)\times {\mathcal{J}}_{\gamma }\left(\frac{\tau \mathfrak{e}+\left(1-\tau \right)\mathfrak{g}}{2}+\frac{\left(1-\tau \right)\mathfrak{e}+\tau \mathfrak{g}}{2},\frac{\kappa \mathfrak{u}+\left(1-\kappa \right)\mathfrak{v}}{2}+\frac{\left(1-\kappa \right)\mathfrak{u}+\kappa \mathfrak{v}}{2}\right)\\ {\supseteq }_I{{\mathrm{Ԓ}}_1}^2\left(\frac{1}{2}\right){{\mathrm{Ԓ}}_2}^2\left(\frac{1}{2}\right)\times \left[\begin{array}{c}{G}_{\gamma }\left(\tau \mathfrak{e}+\left(1-\tau \right)\mathfrak{g},\kappa \mathfrak{u}+\left(1-\kappa \right)\mathfrak{v}\right)+G_{\gamma }\left(\left(1-\tau \right)\mathfrak{e}+\tau \mathfrak{g},\kappa \mathfrak{u}+\left(1-\kappa \right)\mathfrak{v}\right)\\ +G_{\gamma }\left(\tau \mathfrak{e}+\left(1-\tau \right)\mathfrak{g},\left(1-\kappa \right)\mathfrak{u}+\kappa \mathfrak{v}\right)+G_{\gamma }\left(\left(1-\tau \right)\mathfrak{e}+\tau \mathfrak{g},\left(1-\kappa \right)\mathfrak{u}+\kappa \mathfrak{v}\right)\end{array}\right]\\ \times \left[\begin{array}{c}{\mathcal{J}}_{\gamma }\left(\tau \mathfrak{e}+\left(1-\tau \right)\mathfrak{g},\kappa \mathfrak{u}+\left(1-\kappa \right)\mathfrak{v}\right)+{\mathcal{J}}_{\gamma }\left(\left(1-\tau \right)\mathfrak{e}+\tau \mathfrak{g},\kappa \mathfrak{u}+\left(1-\kappa \right)\mathfrak{v}\right)\\ +{\mathcal{J}}_{\gamma }\left(\tau \mathfrak{e}+\left(1-\tau \right)\mathfrak{g},\left(1-\kappa \right)\mathfrak{u}+\kappa \mathfrak{v}\right)+{\mathcal{J}}_{\gamma }\left(\left(1-\tau \right)\mathfrak{e}+\tau \mathfrak{g},\left(1-\kappa \right)\mathfrak{u}+\kappa \mathfrak{v}\right)\end{array}\right]\\ {\supseteq }_I{{\mathrm{Ԓ}}_1}^2\left(\frac{1}{2}\right){{\mathrm{Ԓ}}_2}^2\left(\frac{1}{2}\right)\times \left[\begin{array}{c}{G}_{\gamma }\left(\tau \mathfrak{e}+\left(1-\tau \right)\mathfrak{g},\kappa \mathfrak{u}+\left(1-\kappa \right)\mathfrak{v}\right)\times {\mathcal{J}}_{\gamma }\left(\tau \mathfrak{e}+\left(1-\tau \right)\mathfrak{g},\kappa \mathfrak{u}+\left(1-\kappa \right)\mathfrak{v}\right)\\ +G_{\gamma }\left(\left(1-\tau \right)\mathfrak{e}+\tau \mathfrak{g},\kappa \mathfrak{u}+\left(1-\kappa \right)\mathfrak{v}\right)\times {\mathcal{J}}_{\gamma }\left(\left(1-\tau \right)\mathfrak{e}+\tau \mathfrak{g},\kappa \mathfrak{u}+\left(1-\kappa \right)\mathfrak{v}\right)\\ +G_{\gamma }\left(\tau \mathfrak{e}+\left(1-\tau \right)\mathfrak{g},\left(1-\kappa \right)\mathfrak{u}+\kappa \mathfrak{v}\right)\times {\mathcal{J}}_{\gamma }\left(\tau \mathfrak{e}+\left(1-\tau \right)\mathfrak{g},\left(1-\kappa \right)\mathfrak{u}+\kappa \mathfrak{v}\right)\\ +G_{\gamma }\left(\left(1-\tau \right)\mathfrak{e}+\tau \mathfrak{g},\left(1-\kappa \right)\mathfrak{u}+\kappa \mathfrak{v}\right)\times {\mathcal{J}}_{\gamma }\left(\left(1-\tau \right)\mathfrak{e}+\tau \mathfrak{g},\left(1-\kappa \right)\mathfrak{u}+\kappa \mathfrak{v}\right)\end{array}\right]\\ +{{\mathrm{Ԓ}}_1}^2\left(\frac{1}{2}\right){{\mathrm{Ԓ}}_2}^2\left(\frac{1}{2}\right)\times \\ \left[\begin{array}{c}{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right)\left[{\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)+{\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)+{\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)\right]\\ +{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_1\left(1-\kappa \right)\left[{\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)+{\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)+{\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)\right]\\ \begin{array}{c}+{\mathrm{Ԓ}}_1\left(1-\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right)\left[{\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)+{\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)+{\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)\right]\\ +{\mathrm{Ԓ}}_1\left(1-\tau \right){\mathrm{Ԓ}}_1\left(1-\kappa \right)\left[{\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)+{\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)+{\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)\right]\end{array}\end{array}\right]{\mathcal{M}}_{\gamma }\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right)\\ +{{\mathrm{Ԓ}}_1}^2\left(\frac{1}{2}\right){{\mathrm{Ԓ}}_2}^2\left(\frac{1}{2}\right)\times \\ \left[\begin{array}{c}{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right)\left[{\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)+{\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)+{\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)\right]\\ +{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_1\left(1-\kappa \right)\left[{\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)+{\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)+{\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)\right]\\ +{\mathrm{Ԓ}}_1\left(1-\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right)\left[{\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)+{\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)+{\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)\right]\\ +{\mathrm{Ԓ}}_1\left(1-\tau \right){\mathrm{Ԓ}}_1\left(1-\kappa \right)\left[{\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)+{\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)+{\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)\right]\end{array}\right]P_{\gamma }\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right)\\ +{{\mathrm{Ԓ}}_1}^2\left(\frac{1}{2}\right){{\mathrm{Ԓ}}_2}^2\left(\frac{1}{2}\right)\times \\ \left[\begin{array}{c}{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right)\left[{\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)+{\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)+{\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)\right]\\ +{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_1\left(1-\kappa \right)\left[{\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)+{\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)+{\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)\right]\\ +{\mathrm{Ԓ}}_1\left(1-\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right)\left[{\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)+{\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)+{\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)\right]\\ +{\mathrm{Ԓ}}_1\left(1-\tau \right){\mathrm{Ԓ}}_1\left(1-\kappa \right)\left[{\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)+{\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)+{\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)\right]\end{array}\right]{\mathcal{N}}_{\gamma }\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right)\\ +{{\mathrm{Ԓ}}_1}^2\left(\frac{1}{2}\right){{\mathrm{Ԓ}}_2}^2\left(\frac{1}{2}\right)\times \\ \left[\begin{array}{c}{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right)\left[{\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)+{\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)+{\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)\right]\\ +{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_1\left(1-\kappa \right)\left[{\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)+{\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)+{\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)\right]\\ \begin{array}{c}+{\mathrm{Ԓ}}_1\left(1-\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right)\left[{\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)+{\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)+{\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)\right]\\ +{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_1\left(1-\kappa \right)\left[{\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)+{\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)+{\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)\right]\end{array}\end{array}\right]Q_{\gamma }\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right).\end{array}$$

Taking the multiplication of the above fuzzy inclusion with ${\tau }^{\alpha -1}{\kappa }^{\beta -1}$ and then taking the double integration of the result over $\left[0, 1\right]\times \left[0, 1\right]$ with respect to ($\tau , \kappa$), we have

$$\begin{array}{c}{\int }_0^1{\int }_0^1{\tau }^{\alpha -1}{\kappa }^{\beta -1}{G}_{\gamma }\left(\frac{\mathfrak{e}+\mathfrak{g}}{2},\frac{\mathfrak{u}+\mathfrak{v}}{2}\right)\times {\mathcal{J}}_{\gamma }\left(\frac{\mathfrak{e}+\mathfrak{g}}{2},\frac{\mathfrak{u}+\mathfrak{v}}{2}\right)d\tau d\kappa \\ {\supseteq }_I{{\mathrm{Ԓ}}_1}^2\left(\frac{1}{2}\right){{\mathrm{Ԓ}}_2}^2\left(\frac{1}{2}\right)\\ \times {\int }_0^1{\int }_0^1{\tau }^{\alpha -1}{\kappa }^{\beta -1}\left[\begin{array}{c}{G}_{\gamma }\left(\tau \mathfrak{e}+\left(1-\tau \right)\mathfrak{g},\kappa \mathfrak{u}+\left(1-\kappa \right)\mathfrak{v}\right)\times {\mathcal{J}}_{\gamma }\left(\tau \mathfrak{e}+\left(1-\tau \right)\mathfrak{g},\kappa \mathfrak{u}+\left(1-\kappa \right)\mathfrak{v}\right)\\ +G_{\gamma }\left(\left(1-\tau \right)\mathfrak{e}+\tau \mathfrak{g},\kappa \mathfrak{u}+\left(1-\kappa \right)\mathfrak{v}\right)\times {\mathcal{J}}_{\gamma }\left(\left(1-\tau \right)\mathfrak{e}+\tau \mathfrak{g},\kappa \mathfrak{u}+\left(1-\kappa \right)\mathfrak{v}\right)\\ +G_{\gamma }\left(\tau \mathfrak{e}+\left(1-\tau \right)\mathfrak{g},\left(1-\kappa \right)\mathfrak{u}+\kappa \mathfrak{v}\right)\times {\mathcal{J}}_{\gamma }\left(\tau \mathfrak{e}+\left(1-\tau \right)\mathfrak{g},\left(1-\kappa \right)\mathfrak{u}+\kappa \mathfrak{v}\right)\\ +G_{\gamma }\left(\left(1-\tau \right)\mathfrak{e}+\tau \mathfrak{g},\left(1-\kappa \right)\mathfrak{u}+\kappa \mathfrak{v}\right)\times {\mathcal{J}}_{\gamma }\left(\left(1-\tau \right)\mathfrak{e}+\tau \mathfrak{g},\left(1-\kappa \right)\mathfrak{u}+\kappa \mathfrak{v}\right)\end{array}\right]d\tau d\kappa \\ +{{\mathrm{Ԓ}}_1}^2\left(\frac{1}{2}\right){{\mathrm{Ԓ}}_2}^2\left(\frac{1}{2}\right){\mathcal{M}}_{\gamma }\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right)\\ \times {\int }_0^1{\int }_0^1{\tau }^{\alpha -1}{\kappa }^{\beta -1}\left[\begin{array}{c}{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right)\left[{\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)+{\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)+{\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)\right]\\ +{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_1\left(1-\kappa \right)\left[{\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)+{\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)+{\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)\right]\\ \begin{array}{c}+{\mathrm{Ԓ}}_1\left(1-\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right)\left[{\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)+{\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)+{\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)\right]\\ +{\mathrm{Ԓ}}_1\left(1-\tau \right){\mathrm{Ԓ}}_1\left(1-\kappa \right)\left[{\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)+{\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)+{\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)\right]\end{array}\end{array}\right]d\tau d\kappa \\ +{{\mathrm{Ԓ}}_1}^2\left(\frac{1}{2}\right){{\mathrm{Ԓ}}_2}^2\left(\frac{1}{2}\right)P_{\gamma }\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right)\\ \times {\int }_0^1{\int }_0^1{\tau }^{\alpha -1}{\kappa }^{\beta -1}\left[\begin{array}{c}{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right)\left[{\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)+{\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)+{\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)\right]\\ +{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_1\left(1-\kappa \right)\left[{\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)+{\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)+{\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)\right]\\ +{\mathrm{Ԓ}}_1\left(1-\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right)\left[{\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)+{\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)+{\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)\right]\\ +{\mathrm{Ԓ}}_1\left(1-\tau \right){\mathrm{Ԓ}}_1\left(1-\kappa \right)\left[{\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)+{\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)+{\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)\right]\end{array}\right]d\tau d\kappa \\ +{{\mathrm{Ԓ}}_1}^2\left(\frac{1}{2}\right){{\mathrm{Ԓ}}_2}^2\left(\frac{1}{2}\right){\mathcal{N}}_{\gamma }\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right)\\ \times {\int }_0^1{\int }_0^1{\tau }^{\alpha -1}{\kappa }^{\beta -1}\left[\begin{array}{c}{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right)\left[{\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)+{\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)+{\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)\right]\\ +{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_1\left(1-\kappa \right)\left[{\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)+{\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)+{\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)\right]\\ +{\mathrm{Ԓ}}_1\left(1-\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right)\left[{\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)+{\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)+{\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)\right]\\ +{\mathrm{Ԓ}}_1\left(1-\tau \right){\mathrm{Ԓ}}_1\left(1-\kappa \right)\left[{\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)+{\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)+{\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)\right]\end{array}\right]d\tau d\kappa \\ +{{\mathrm{Ԓ}}_1}^2\left(\frac{1}{2}\right){{\mathrm{Ԓ}}_2}^2\left(\frac{1}{2}\right)Q_{\gamma }\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right)\\ \times {\int }_0^1{\int }_0^1{\tau }^{\alpha -1}{\kappa }^{\beta -1}\left[\begin{array}{c}{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right)\left[{\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)+{\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)+{\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)\right]\\ +{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_1\left(1-\kappa \right)\left[{\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)+{\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)+{\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)\right]\\ \begin{array}{c}+{\mathrm{Ԓ}}_1\left(1-\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right)\left[{\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)+{\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)+{\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)\right]\\ +{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_1\left(1-\kappa \right)\left[{\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)+{\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)+{\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)\right]\end{array}\end{array}\right]\end{array}$$

which implies that

$$\begin{array}{c}\frac{1}{\alpha \beta }{G}_{\gamma }\left(\frac{\mathfrak{e}+\mathfrak{g}}{2}, \frac{\mathfrak{u}+\mathfrak{v}}{2}\right)\times {\mathcal{J}}_{\gamma }\left(\frac{\mathfrak{e}+\mathfrak{g}}{2}, \frac{\mathfrak{u}+\mathfrak{v}}{2}\right)\\ {\supseteq }_{\mathbb{F}}\frac{\Gamma \left(\alpha \right)\Gamma \left(\beta \right){{\mathrm{Ԓ}}_1}^2\left(\frac{1}{2}\right){{\mathrm{Ԓ}}_2}^2\left(\frac{1}{2}\right)}{{\left(\mathfrak{g}-\mathfrak{e}\right)}^{\alpha }{\left(\mathfrak{v}-\mathfrak{u}\right)}^{\beta }}\left[{\mathcal{I}}_{{\mathfrak{e}}^{+},{\mathfrak{u}}^{+}}^{\alpha , \beta }{G}_{\gamma }\left(\mathfrak{g},\mathfrak{v}\right)\times {\mathcal{J}}_{\gamma }\left(\mathfrak{g},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathfrak{e}}^{+},{\mathfrak{v}}^{-}}^{\alpha , \beta }{G}_{\gamma }\left(\mathfrak{g},\mathfrak{u}\right)\times {\mathcal{J}}_{\gamma }\left(\mathfrak{g},\mathfrak{u}\right)\right]\\ +\frac{\Gamma \left(\alpha \right)\Gamma \left(\beta \right){{\mathrm{Ԓ}}_1}^2\left(\frac{1}{2}\right){{\mathrm{Ԓ}}_2}^2\left(\frac{1}{2}\right)}{{\left(\mathfrak{g}-\mathfrak{e}\right)}^{\alpha }{\left(\mathfrak{v}-\mathfrak{u}\right)}^{\beta }}\left[{\mathcal{I}}_{{\mathfrak{g}}^{-},{\mathfrak{u}}^{+}}^{\alpha , \beta }{G}_{\gamma }\left(\mathfrak{e},\mathfrak{v}\right)\times {\mathcal{J}}_{\gamma }\left(\mathfrak{e},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathfrak{g}}^{-},{\mathfrak{v}}^{-}}^{\alpha , \beta }{G}_{\gamma }\left(\mathfrak{e},\mathfrak{u}\right)\times {\mathcal{J}}_{\gamma }\left(\mathfrak{e},\mathfrak{u}\right)\right]\\ +2{{\mathrm{Ԓ}}_1}^2\left(\frac{1}{2}\right){{\mathrm{Ԓ}}_2}^2\left(\frac{1}{2}\right){\mathcal{M}}_{\gamma }\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right){\int }_0^1{\tau }^{\alpha -1}{\kappa }^{\beta -1}\left[{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right)\right[{\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)+{\mathrm{Ԓ}}_2\left(1-\tau \right)\\ {\mathrm{Ԓ}}_2\left(\kappa \right)+{\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)]+{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_1\left(1-\kappa \right)[{\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)+{\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)+\\ {\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)\left]\right]d\tau d\kappa \\ +2{{\mathrm{Ԓ}}_1}^2\left(\frac{1}{2}\right){{\mathrm{Ԓ}}_2}^2\left(\frac{1}{2}\right)P_{\gamma }\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right){\int }_0^1{\tau }^{\alpha -1}{\kappa }^{\beta -1}\left[{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right)\right[{\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)+\\ {\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)+{\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)]+{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_1\left(1-\kappa \right)[{\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)+{\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)+\\ {\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)\left]\right]d\tau d\kappa \\ +2{{\mathrm{Ԓ}}_1}^2\left(\frac{1}{2}\right){{\mathrm{Ԓ}}_2}^2\left(\frac{1}{2}\right){\mathcal{N}}_{\gamma }\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right){\int }_0^1{\tau }^{\alpha -1}{\kappa }^{\beta -1}\left[{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right)\right[{\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)+{\mathrm{Ԓ}}_2\left(1-\tau \right)\\ {\mathrm{Ԓ}}_2\left(1-\kappa \right)+{\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)]+{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_1\left(1-\kappa \right)[{\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)+{\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)+\\ {\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)\left]\right]d\tau d\kappa \\ +2{{\mathrm{Ԓ}}_1}^2\left(\frac{1}{2}\right){{\mathrm{Ԓ}}_2}^2\left(\frac{1}{2}\right)Q_{\gamma }\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right){\int }_0^1{\tau }^{\alpha -1}{\kappa }^{\beta -1}[{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_1\left(\kappa \right)\left[{\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)+{\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)+{\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)\right]\\ +{\mathrm{Ԓ}}_1\left(\tau \right){\mathrm{Ԓ}}_1\left(1-\kappa \right)\left[{\mathrm{Ԓ}}_2\left(1-\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)+{\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(\kappa \right)+{\mathrm{Ԓ}}_2\left(\tau \right){\mathrm{Ԓ}}_2\left(1-\kappa \right)\right]]d\tau d\kappa ,\end{array}$$

since $\gamma \in \left[0, 1\right]$, then, after simplification, we reach the required conclusion. □

Remark 4.

If one assumes that $\mathrm{Ԓ}\left(\tau \right)=\tau , \mathrm{Ԓ}\left(\kappa \right)=\kappa$, $\alpha =1,$ and $\beta =1$, then, from (68), as a result, there will be an inequality (see [69]):

$$\begin{array}{c}4\tilde{G}\left(\frac{\mathfrak{e}+\mathfrak{g}}{2},\frac{\mathfrak{u}+\mathfrak{v}}{2}\right)\otimes \tilde{\mathcal{J}}\left(\frac{\mathfrak{e}+\mathfrak{g}}{2},\frac{\mathfrak{u}+\mathfrak{v}}{2}\right)\\ {\supseteq }_{\mathbb{F}}\frac{1}{\left(\mathfrak{g}-\mathfrak{e}\right)\left(\mathfrak{v}-\mathfrak{u}\right)}{\int }_{\mathfrak{e}}^{\mathfrak{g}}{\int }_{\mathfrak{u}}^{\mathfrak{v}}\tilde{G}\left(x,y\right)\otimes \tilde{\mathcal{J}}\left(x,y\right)dydx\oplus \frac{5}{36}\tilde{\mathcal{M}}\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right)\\ \oplus \frac{7}{36}\left[\tilde{P}\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right)\widetilde{+}\tilde{\mathcal{N}}\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right)\right]\oplus \frac{2}{9}\tilde{Q}\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right).\end{array}$$

(70)

If $\tilde{G}$ is a coordinated left-$\mathsf{U}\mathsf{D}$-$\mathrm{Ԓ}$-convex function with $\mathrm{Ԓ}\left(\tau \right)=\tau , \mathrm{Ԓ}\left(\kappa \right)=\kappa$ and one assumes that $\alpha =1$ and $\beta =1$, then, from (68), as a result, there will be an inequality (see [59]):

$$\begin{array}{c}4\tilde{G}\left(\frac{\mathfrak{e}+\mathfrak{g}}{2},\frac{\mathfrak{u}+\mathfrak{v}}{2}\right)\otimes \tilde{\mathcal{J}}\left(\frac{\mathfrak{e}+\mathfrak{g}}{2},\frac{\mathfrak{u}+\mathfrak{v}}{2}\right)\\ {\le }_{\mathbb{F}}\frac{1}{\left(\mathfrak{g}-\mathfrak{e}\right)\left(\mathfrak{v}-\mathfrak{u}\right)}{\int }_{\mathfrak{e}}^{\mathfrak{g}}{\int }_{\mathfrak{u}}^{\mathfrak{v}}\tilde{G}\left(x,y\right)\otimes \tilde{\mathcal{J}}\left(x,y\right)dydx\oplus \frac{5}{36}\tilde{\mathcal{M}}\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right)\\ \oplus \frac{7}{36}\left[\tilde{P}\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right)\widetilde{+}\tilde{\mathcal{N}}\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right)\right]\oplus \frac{2}{9}\tilde{Q}\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right).\end{array}$$

(71)

If $G_{\ast }\left(\left(x,y\right), \gamma \right)\ne G^{\ast }\left(\left(x,y\right), \gamma \right)$ with $\mathrm{Ԓ}\left(\tau \right)=\tau , \mathrm{Ԓ}\left(\kappa \right)=\kappa$ and $\gamma =1$, then, from (68), we succeed in bringing about the upcoming inequality (see [55]):

$$\begin{array}{c}4 G\left(\frac{\mathfrak{e}+\mathfrak{g}}{2},\frac{\mathfrak{u}+\mathfrak{v}}{2}\right)\times \mathcal{J}\left(\frac{\mathfrak{e}+\mathfrak{g}}{2},\frac{\mathfrak{u}+\mathfrak{v}}{2}\right)\\ \supseteq \frac{1}{\left(\mathfrak{g}-\mathfrak{e}\right)\left(\mathfrak{v}-\mathfrak{u}\right)} {\int }_{\mathfrak{e}}^{\mathfrak{g}}{\int }_{\mathfrak{u}}^{\mathfrak{v}}G\left(x,y\right)\times \mathcal{J}\left(x,y\right)dydx+\frac{5}{36}\mathcal{M}\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right)\\ +\frac{7}{36}\left[P\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right)+\mathcal{N}\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right)\right]+\frac{2}{9}Q\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right).\end{array}$$

(72)

If $G_{\ast }\left(\left(x,y\right), \gamma \right)\ne G^{\ast }\left(\left(x,y\right), \gamma \right)$ with $\gamma =1$ and $\mathrm{Ԓ}\left(\tau \right)=\tau , \mathrm{Ԓ}\left(\kappa \right)=\kappa$, then, from (68), we succeed in bringing about the upcoming inequality (see [71]):

$$\begin{array}{c}4G\left(\frac{\mathfrak{e}+\mathfrak{g}}{2}, \frac{\mathfrak{u}+\mathfrak{v}}{2}\right)\times \mathcal{J}\left(\frac{\mathfrak{e}+\mathfrak{g}}{2}, \frac{\mathfrak{u}+\mathfrak{v}}{2}\right)\\ \supseteq \frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{{4\left(\mathfrak{g}-\mathfrak{e}\right)}^{\alpha }{\left(\mathfrak{v}-\mathfrak{u}\right)}^{\beta }} \left[\begin{array}{c}{\mathcal{I}}_{{\mathfrak{e}}^{+},{\mathfrak{u}}^{+} }^{\alpha , \beta }G\left(\mathfrak{g},\mathfrak{v}\right)\times \mathcal{J}\left(\mathfrak{g},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathfrak{e}}^{+},{\mathfrak{v}}^{-}}^{\alpha , \beta } G\left(\mathfrak{g},\mathfrak{u}\right)\times \mathcal{J}\left(\mathfrak{g},\mathfrak{u}\right)\\ +{\mathcal{I}}_{{\mathfrak{g}}^{-},{\mathfrak{u}}^{+} }^{\alpha , \beta }G\left(\mathfrak{e},\mathfrak{v}\right)\times \mathcal{J}\left(\mathfrak{e},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathfrak{g}}^{-},{\mathfrak{v}}^{-}}^{\alpha , \beta } G\left(\mathfrak{e},\mathfrak{u}\right)\times \mathcal{J}\left(\mathfrak{e},\mathfrak{u}\right)\end{array}\right]\\ +\left[\frac{\alpha }{2\left(\alpha +1\right)\left(\alpha +2\right)}+\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\left(\frac{1}{2}-\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\right)\right]\mathcal{M}\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right)\\ +\left[\frac{1}{2}\left(\frac{1}{2}-\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\right)+\frac{\alpha }{(\alpha +1)(\alpha +2)}\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right]P\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right)\\ +\left[\frac{1}{2}\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)+\frac{\alpha }{(\alpha +1)(\alpha +2)}\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right]\mathcal{N}\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right)\\ +\left[\frac{1}{4}-\frac{\alpha }{(\alpha +1)(\alpha +2)}\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right]Q\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right).\end{array}$$

(73)

If $G_{\ast }\left(\left(x,y\right), \gamma \right)=G^{\ast }\left(\left(x,y\right), \gamma \right)$ and ${\mathcal{J}}_{\ast }\left(\left(x,y\right), \gamma \right)={\mathcal{J}}^{\ast }\left(\left(x,y\right), \gamma \right)$ with $\gamma =1$ and $\mathrm{Ԓ}\left(\tau \right)=\tau , \mathrm{Ԓ}\left(\kappa \right)=\kappa$, then, from (68), we succeed in bringing about the upcoming inequality (see [69]):

$$\begin{array}{c}4G\left(\frac{\mathfrak{e}+\mathfrak{g}}{2}, \frac{\mathfrak{u}+\mathfrak{v}}{2}\right)\times \mathcal{J}\left(\frac{\mathfrak{e}+\mathfrak{g}}{2}, \frac{\mathfrak{u}+\mathfrak{v}}{2}\right)\\ \le \frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{{4\left(\mathfrak{g}-\mathfrak{e}\right)}^{\alpha }{\left(\mathfrak{v}-\mathfrak{u}\right)}^{\beta }}\left[\begin{array}{c}{\mathcal{I}}_{{\mathfrak{e}}^{+},{\mathfrak{u}}^{+}}^{\alpha , \beta }G\left(\mathfrak{g},\mathfrak{v}\right)\times \mathcal{J}\left(\mathfrak{g},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathfrak{e}}^{+},{\mathfrak{v}}^{-}}^{\alpha , \beta }G\left(\mathfrak{g},\mathfrak{u}\right)\times \mathcal{J}\left(\mathfrak{g},\mathfrak{u}\right)\\ +{\mathcal{I}}_{{\mathfrak{g}}^{-},{\mathfrak{u}}^{+}}^{\alpha , \beta }G\left(\mathfrak{e},\mathfrak{v}\right)\times \mathcal{J}\left(\mathfrak{e},\mathfrak{v}\right)+{\mathcal{I}}_{{\mathfrak{g}}^{-},{\mathfrak{v}}^{-}}^{\alpha , \beta }G\left(\mathfrak{e},\mathfrak{u}\right)\times \mathcal{J}\left(\mathfrak{e},\mathfrak{u}\right)\end{array}\right].\\ +\left[\frac{\alpha }{2\left(\alpha +1\right)\left(\alpha +2\right)}+\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\left(\frac{1}{2}-\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\right)\right]\mathcal{M}\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right)\\ +\left[\frac{1}{2}\left(\frac{1}{2}-\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\right)+\frac{\alpha }{(\alpha +1)(\alpha +2)}\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right]P\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right)\\ +\left[\frac{1}{2}\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)+\frac{\alpha }{(\alpha +1)(\alpha +2)}\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right]\mathcal{N}\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right)\\ +\left[\frac{1}{4}-\frac{\alpha }{(\alpha +1)(\alpha +2)}\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right]Q\left(\mathfrak{e},\mathfrak{g},\mathfrak{u},\mathfrak{v}\right).\end{array}$$

(74)

4. Conclusions

This study makes use of fuzzy-number-valued fractional integrals to handle certain fractional integral inclusions involving the Hermite–Hadamard integral inequality via a newly defined class of coordinated $\mathsf{U}\mathsf{D}$-$\mathbf{Ԓ}$-convex $\mathsf{F}\mathsf{N}\mathsf{V}\mathsf{M}s$. We also look into other set inclusion connections related to the fractional Pachpatte integral inequality. Additionally, a few examples are provided to support the accuracy of the conclusions drawn in the research. We highlight the links between the results obtained here and those previously published in order to demonstrate the generic properties of the fuzzy set inclusion relations offered. Based on published works [59,68] and the bibliographies cited in them, we can confidently conclude that fuzzy-number-valued analyses are commonly used in applied analyses, particularly in the field of optimality analysis. In the integration with the fuzzy-number-valued fractional integral operators, the fuzzy $\mathsf{U}\mathsf{D}$-inclusion relations are somewhat interesting and need more investigation.