Generalized version of Jensen and Hermite-Hadamard inequalities for interval-valued ( h 1 , h 2 ) -Godunova-Levin functions

: Interval analysis distinguishes between inclusion relation and order relation. Under the inclusion relation, convexity and nonconvexity contribute to di ﬀ erent kinds of inequalities. The construction and reﬁnement of classical inequalities have received a great deal of attention for many classes of convex as well as nonconvex functions. Convex theory, however, is commonly known to rely on Godunova-Levin functions because their properties enable us to determine inequality terms more precisely than those obtained from convex functions. The purpose of this study was to introduce a ( ⊆ ) relation to established Jensen-type and Hermite-Hadamard inequalities using ( h 1 , h 2 )-Godunova-Levin interval-valued functions. To strengthen the validity of our results, we provide several examples and obtain some new and previously unknown results.


Introduction
There has been much debate in the biography of interval analysis, but the key research outcomes, intervals, and interval-valued functions were first introduced by Moore [1], in 1950. The mathematical community has been paying close attention to this research field since its inception. Experts believe interval analysis is useful in global optimization and constraint solving algorithms. In scientific computation, interval analysis is useful, particularly for accuracy, round-off errors, and automatic validation. For the last five decades, there has been a lot of curiosity about it and it has been used in many areas, such as differential equations with intervals [2], aerodynamic load analysis [3], aeroelasticity [4], computer graphics [5], and so on. We recommend readers in addition to other interval analysis results and applications, see e.g., [6][7][8][9][10][11][12].
It is well known that the convexity of functions is important in mathematics, economics, probability theory, optimal control theory, and other scientific disciplines. According to various scholars, function convexity is based on inequality. Among elementary mathematics, the Hermite-Hadamard inequality is a popular subject since it offers the first geometrical interpretation of convex mappings. There has been extensive research on the Hermite-Hadamard inequality for various classes of convexity due to its importance. The following is the classical Hermite-Hadamard (H − H) inequality: where χ : S ⊆ R → R be a convex on interval S of real numbers and Υ, Ω ∈ S with Υ < Ω. A number of convexity classes have been considered in developing this inequality, see e.g., [13][14][15][16]. Since Varoşanec [17], introduced the notion of h-convex function in 2007, different authors have developed more refined Hermite-Hadamard inequalities related to h-convex functions, see e.g., [18][19][20][21]. This inequality was proved in 2018 by Awan et al. [22], using (h 1 , h 2 )-convex functions. (1.1) Later, An et al. [23], introduced the concept of (h 1 , h 2 )-convex interval-valued functions (in short I-V-Fs) and prove the above inequality in that generalization. Further Nwaeze et al. [24] developed the H − H inequality for n-polynomial for convex I-V-Fs; Ali et al. [25] and Kalsoom et al. [26] applied quantum calculus to refine this concept. Moreover, by Khan et al. [27][28][29][30][31][32] this concept has been generalized to convex fuzzy I-V-Fs as well. For some recent results related to these inequalities for interval-valued functions, see e.g., [33][34][35][36]. In 2019, Ohud Almutairi and his co-author proved the following inequality using the h-Godunova-Levin function [37].
Further, Costa et al. [38], present a fuzzy Jensen-type inequality for I-V-Fs while Hongxin Bai, et al. [39], develop a Jensen-type inequality for interval nonconvex (h 1 , h 2 ) functions. Motivated by Ohud Almutairi [37], An et al. [23], and Hongxin Bai, et al. [39], we introduce the notion of interval-valued (h 1 , h 2 )-Godunova-Levin functions and develop Jensen and H − H inequalities for this newly introduced class of functions. The article is divided into the following sections. The necessary mathematical background is provided in Section 2. Section 3 presents the problem description as well as our key findings. Section 4 provides conclusions.

Preliminaries
To begin, a short overview of terms, notations, and properties used in this paper is necessary [13]. Consider I ⊆ R, where I is closed as well as bounded. For any [Υ] ∈ I is defined by The interval [Υ] shows degeneration when Υ=Υ. We state [Υ] is positive when Υ > 0 or negative when Υ < 0. Assume that the bundle of all and positive intervals consists of the following R I , R + I , respectively. Consider any real number ν and [Υ], the interval ν[Υ] is given as: The metric space (R I , d) is often complete. An explanation of how operations are defined on R I give rise to a number of algebraic features that make it quasilinear space, see [41]. As follows, they can be categorized • (Associativity of addition) (Υ + Ω) + β = Υ + (Ω + β) ∀ Υ, Ω, β ∈ R I , • (Commutativity of addition) Υ + Ω = Ω + Υ ∀ Υ, Ω ∈ R I , • (Additivity element) In addition, inclusion ⊆ is one of the set property is defined as: A definition of Riemannian integrability of interval valued functions is given in [40] . .

Main results
As a closing to the current part of the preliminaries, we introduced a new concept of interval-valued (h 1 , h 2 )-Godunova-Levin convexity. . Then Proof. Let χ be interval valued (h 1 , h 2 )-Godunova-Levin convex function and suppose that x, y ∈ [Υ, Ω], ν ∈ (0, 1), then that is, It follows that we have and . This completes the proof. Then can be similar to a Proposition 3.1.
3.1. In this section, we can be established some variants of Hermite-Hadamard inequality by using the definition of (h 1 , h 2 )-Godunova-Levin I-V-Fs.
Proof. According to our hypothesis, As a result of integrating above inequality over (0, 1), we obtain It follows that we have This implies Now by Definition 3.1, As a result of integrating above inequality over (0, 1), we obtain It implies that Combinig Eqs (3.4) and (3.5), we get required result .
Thus we obtain Consequently, the above theorem is verified. .
Proof. Consider [Υ, Υ+Ω 2 ], we have As a result of integrating above inequality over (0, 1), we obtain Then above inequality become as This implies that Similarly for interval [ Υ+Ω 2 , Ω], we have Adding the inclusions (3.6) and (3.7), we get . χ 1 Thus we obtain As a result, the preceding theorem is confirmed.
It follows that As a result, the preceding theorem is confirmed.
Proof. By hypothesis, one has As a result of integrating above inequality over (0, 1), we obtain dx .

Multiply both sides by
, we get This completes the proof. Then M(Υ, Ω)  As a result, the preceding theorem is confirmed.

3.2.
In this section, we can be established Jensen-type inequality by using the definition of (h 1 , h 2 )-Godunova-Levin I-V-Fs.

Conclusions
In this paper, we introduce the (h 1 , h 2 )-Godunova-Levin concept for I-V-Fs. The purpose of the above concept was to study Jensen and Hermite-Hadamrd inequalities using I-V-Fs. The inequalities previously developed by An et al. [23], and Hongxin Bai, et al. [39] are generalized in our results. In addition, some useful examples are provided to support our main conclusions. Continuing this research direction, we will investigate Jensen and Hermite-Hadamrd type inequalities for I-V-Fs and fuzzy-valued functions over time scales. We think that this is an intriguing topic that can be explored in the future to find equivalent inequalities depending on the type of convexity. By utilizing these concepts, a new direction for convex optimization can be developed. By embracing this concept, we hope to support other authors in securing their roles in various fields of science.