Hermite–Hadamard–type fractional–integral inequalities for ( p, h ) –convex fuzzy–interval– valued mappings

In this paper, fuzzy-interval-valued functions of the ( p, h ) -convex type, deﬁned recently by Khan et al. [ AIMS Math. 8 (2023) 7437–7470], are studied. Several Hermite-Hadamard-type inequalities in the said setting are obtained. A Hermite-Fejer-type inequality is also obtained, which generalizes several recently published results. Moreover, in order to supplement the obtained results, suitable numerical examples are given.


Introduction
Convex inequalities have been an active topic of mathematical research since the introduction of the first convex inequality, known as Jensen inequality. Many inequalities derived using convexity exist in the literature; for example, see the books [25,31]. Inequalities have various applications to different branches of mathematics, including numerical analysis, probability density functions, and optimization; see the papers [2,6,8,13,26,27,34].
The Hermite-Hadamard inequality [12], proved independently by Charles Hermite and Jacques Hadamard, is among those inequalities that have attracted the most attention in the mathematical community. This inequality has been generalized in various ways by many mathematicians. If F : I → R is a convex function on I and n, m ∈ I with n < m, then the said inequality is stated as Many researchers obtained this inequality as a consequence of the generalization using different kinds of convexity with fractional operators [3,7,21,23,28,32,[35][36][37][38][39]. Additional detail about the Hermite-Hadamard and convex inequalities can be found in the papers [4,5,15]. In the present paper, the fuzzy-interval-valued setting together with newly defined fuzzy convexity is utilized to derive various convex inequalities together with fractional integral operators. The recent results obtained by Khan et al. [22] are generalized in this paper. It is remarked here that the initial idea of fractional calculus was given by L'Hospital and Leibniz in 1695. This concept was extended by many mathematicians, including Riemann, Grünwald, Letnikov, Hadamard, and Weyl. These mathematicians made valuable contributions not only to fractional calculus but also to its various applications. Nowadays, fractional calculus is being used widely in describing various phenomena, such as the fractional conservation of mass and fractional Schrödinger equation in quantum theory; more detail about fractional calculus can be found in [14,29,41].

Definitions and preliminaries
Let K c and F c (R) be the collections of all closed and bounded intervals, and fuzzy intervals of R, respectively. Denote by K + c the set of all positive intervals. The collections of all Riemann-integrable real-valued functions, Riemann-integrable interval-valued functions (IVFs), and all Riemann-integrable fuzzy-interval-valued functions (FIVFs) over [u, v] are denoted by R [u,v] , IR [u,v] , and FR [u,v] , respectively. A brief overview of the interval-valued analysis and notions is given in this section; for additional detail, see [9,24,42].
Remark 2.1 (see [42]). The relation " l " is defined on K c as follows for all [r * , r * ], [m * , m * ] ∈ K c ; it is an order relation.
Remark 2.2 (see [16]). Let F c (R) be a set of fuzzy numbers. If ζ, w ∈ F c (R), then the relation " " is defined on F c (R) as follows this relation is known as partial order relation.
and for all φ ∈ [0, 1]. Then F is a fuzzy Riemann integrable over [u, v] if and only if F * (x, φ) and F * (x, φ) both are Riemann integrable over [u, v]. Moreover, if F is fuzzy Riemann integrable over [u, v], then The following definition given by Khan et al. [20] generalizes the previously defined convex types of functions.
Definition 2.5. Let K p be a p-convex set, let J ⊂ R be an interval containing (0, 1), and let h : J → R be a non-negative function. Then FIVF F : Remark 2.3. The following properties hold for the (p, h)-convex FIVF F: • If we take h ≡ I we get p-convex FIVF, that is • If we take p = 1 and h(α) = α s then from (p, h)-convex FIVF we archieve s-convex FIVF [19]; that is • If we take p ≡ 1 and h ≡ I then from (p, h)-convex FIVF we archieve convex FIVF (see [11,19]); that is Next, some fractional-type integrals are defined, which are used in the rest of the paper. The following definition defines Katugampola-fractional integrals, due to Udita Katugampola [17], which generalizes the Riemann-Liouville fractional integrals.

finite interval. Then, the left-and right-sided Katugampola fractional integrals of order
with a < x < b and p > 0, provided that the integrals exist, where is the gamma function [1]. Definition 2.7. Let p, α > 0 and L([ρ, ζ], E) be the collection of all Lebesgue measurable fuzzy-interval-valued mappings (FIVMs) on [ρ, ζ]. Then the fuzzy interval left and right generalized fractional integrals of F ∈ L([ρ, ζ], E) with order α > 0 are defined by respectively. The fuzzy interval left and right generalized fractional integral based on end-point mappings can be defined as where and Similarly, we can define right-generalized fractional integral F of x based on end-point mappings.

Main results
The first result presented is a variation of the Hermite-Hadamard type inequality in the fractional convex FIVF sense. Let a p = ζt p + (1 − ζ)s p and y p = (1 − ζ)t p + ζs p . Then, by the above inequality, one has Therefore, for every φ ∈ [0, 1] , one has Multiplying both sides by ζ α−1 and integrating the obtained result with respect to ζ over (0, 1) , one has Let ζt p + s p − ζs p = k p and ζs p + t p − ζt p = k p . Then one has Analogously, for F * (x, φ) one has That is, Thus, one has In a similar way as above, one gets Combining the left-and right-hand sides, one arrives at By setting h = I in Theorem 3.1, one gets Theorem 5 of Khan et al. [18].

From this we get
Thus, for every x ∈ [s, t] and for every φ ∈ [0, 1]. If F ∈ L([s, t], F c (R)), then By setting α = 1 2 , x p = 1−ζ 2 s p + 1+ζ 2 t p , and y p = 1+ζ 2 s p + 1−ζ 2 t p , one gets Therefore, for every φ ∈ [0, 1], one has F * a p +b p 2 Multiplying both sides by ζ α−1 and integrating the obtained result with respect to ζ, one gets Considering the following substitutions x p = 1−ζ 2 s p + 1+ζ 2 t p and y p = 1+ζ 2 s p + 1−ζ 2 t p , one has Identifying in terms of Katugampola integrals, one gets Combining the left-and right-hand sides, we obtain the desired inequality.

Corollary 3.3.
By setting h to be equal to the identity mapping, one obtains the following inequality Next, we present a generalized Hermite-Hadamard-Fejer inequality for the convex FIVF.
, as well as φ -levels define the family of IVMs for every x ∈ [s, t] and for every φ ∈ [0, 1]. If F ∈ L([a, b], F c (R)) and C : [a, b] → R, C 0, be a p-symmetric function with respect to Proof. Since F is a (p, h)-convex FIVF, one has .
Therefore, for every φ ∈ [0, 1] one has Multiplying both sides of the inequality with and integrating with respect to t from 0 to 1, one arrives at , φ dt By using the fact that C is p-symmetric with respect to a p +b p 2 1 p , one concludes that the left-hand side is equal to From the last inequality, one gets Finally, we obtain the required inequality   .