Hermite–Hadamard and Jensen-Type Inequalities via Riemann Integral Operator for a Generalized Class of Godunova–Levin Functions

In last few years, the inequality theory gained the attention of many researchers working in analysis and other branches of mathematics [1–3]. Most of the real world problems may be viewed as integral equations. So, the generalization of integral inequalities is always appreciable and is more closer to applied problems[4, 5]. In the renowned celebrated book by Moore, the interactive analysis of numerical data starts the introduction to interval analysis in numerical analysis, see [6]. A tremendous number of applications have been developed over the past 50 years in areas including computer graphics [7], aeroelasticity [8], interval dierential equations [9], and neural network optimization [10]. Numerous various integral inequalities have been investigated recently by different authors in the context of interval-valued functions (see [11, 12]). It is well known that the convexity of functions plays an extremely important role inmathematics and other scienti¢c ¢elds such as economics, probability theory, and optimal control theory; moreover, several inequalities have been recorded in the literature (see [13–17]). In equality, the following inequality is called a classical Hermite–Hadamard inequality:


Introduction
In last few years, the inequality theory gained the attention of many researchers working in analysis and other branches of mathematics [1][2][3]. Most of the real world problems may be viewed as integral equations. So, the generalization of integral inequalities is always appreciable and is more closer to applied problems [4,5].
In the renowned celebrated book by Moore, the interactive analysis of numerical data starts the introduction to interval analysis in numerical analysis, see [6]. A tremendous number of applications have been developed over the past 50 years in areas including computer graphics [7], aeroelasticity [8], interval di erential equations [9], and neural network optimization [10]. Numerous various integral inequalities have been investigated recently by different authors in the context of interval-valued functions (see [11,12]).
It is well known that the convexity of functions plays an extremely important role in mathematics and other scienti c elds such as economics, probability theory, and optimal control theory; moreover, several inequalities have been recorded in the literature (see [13][14][15][16][17]). In equality, the following inequality is called a classical Hermite-Hadamard inequality: where φ: S ⊆ R ⟶ R is a convex on interval S and v, w ∈ S with v < w. In the context of di erent generalizations and extensions of this inequality (see [16,18]), the notion of hconvex was originally developed by Varoşanec in 2007 (see [19]). Several authors have developed more sophisticated Hermite-Hadamard inequalities that include h-convex functions (see [20,21]). Furthermore, Costa presented an inequality of the Jensen type for fuzzy interval-valued functions (see [22]). As well, Zhao et al. provide a new Hermite-Hadamard inequality for h-convex functions in the context of interval-valued functions (see [23]). e following inequality was proved in 2019 by Almutairi and Kiliman using the h-Godunova-Levin function (see [24]).
Motivated by Costa [22], Zhao et al. [23], Dragomir [25], and Almutairi and Kiliman [24], we introduce and explore the notion of h-Godunova-Levin interval-valued functions. Our new concept allowed us to develop fractional version of Hermite-Hadamard and Jensen-type inequalities via Riemann integral operator.

Preliminaries
In this study, we will review some fundamental definitions, properties, and notations. Let us say I is the collection of all intervals of R, [v] ∈ I is defined as follows: [ is said to be degenerated when v and v both are equal. We call [v] is positive when v > 0 or negative when v < 0. We denote the space of all intervals by R I of R and the set of all negative and positive intervals are represented as R − I and R + I , respectively. e inclusion "⊆" is defined as For any random real number μ and [v], the interval μ[v] is given as , algebraic operations are defined as For intervals [v, w], [v, w] the Hausdorff-Pompeiu distance is defined as It is well known that the entire (R I , d) is complete metric space.

Main Results
Now, we are ready to introduce the notion of interval-valued h-Godunova-Levin convex functions.
that is, It follows that we have and Proof. is can be similar to Proposition 1. In the next theorem, we Hermite-Hadamard-type inequality for h-Godunova-Levin interval-valued functions.
Proof. By supposition, we have Integrating the above inequality w.r.t "x" over (0, 1), we obtain It follows that we have Similarly, is implies that Journal of Mathematics As a result of applying the interval h-Godunova-Levin convex function, we have Integrate w.r.t "x" over (0, 1); we have Accordingly, Now, combining (24) and (26), we get the required result: □ Remark 2 (i) If h(x) � 1 , then eorem2has the following result for interval P functions: (ii) If h(x) � (1/x) , then eorem 2 has the following result for interval convex functions: (iii) If h(x) � (1/(x) s ) , then eorem 2 has the following result for interval s-convex function: (iv) If φ � φ , then eorem 2 has the following result of Ohud Almutairi and Adem Kiliman (see [24], eorem1).
As a result, is proves the above theorem. where (37) Integrate the above inequality over (0, 1) w.r.t x: (38) en, the above inequality becomes as Now, by using the property of integral, the above inequality becomes as Adding inequality (41) and (42), we obtain Now,

Journal of Mathematics
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