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

Zhang, Xiaoju; Shabbir, Khurram; Afzal, Waqar; Xiao, He; Lin, Dong

doi:https://doi.org/10.1155/2022/3830324

Journal of Mathematics

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Abstract Introduction Preliminaries Conclusions Data Availability Conflicts of Interest Authors’ Contributions References Copyright Related Articles

Research Article | Open Access

Volume 2022 | Article ID 3830324 | https://doi.org/10.1155/2022/3830324

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

Xiaoju Zhang,¹Khurram Shabbir,²Waqar Afzal,²He Xiao,¹and Dong Lin³

Academic Editor: Xiaolong Qin

Received14 Jun 2022

Accepted20 Jul 2022

Published31 Aug 2022

Abstract

The generalization of Godunova–Levin interval-valued functions has been drastically studied in last few decades, as it has a remarkable applications in both pure and applied mathematics. The goal of this study is to introduce the notion of h-Godunova–Levin interval-valued functions. We establish Hermite–Hadamard and Jensen-type inequalities via Riemann integral operator.

1. Introduction

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 differential 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 scientific fields 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:where is a convex on interval S and with . In the context of different generalizations and extensions of this inequality (see [16, 18]), the notion of h-convex 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]).

The following inequality was proved in 2019 by Almutairi and Kiliman using the h-Godunova-Levin function (see [24]).

Theorem 1. Let . If is h-Godunova–Levin convex function and. Then,

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.

2. Preliminaries

In this study, we will review some fundamental definitions, properties, and notations. Let us say I is the collection of all intervals of , is defined as follows:

, , where real interval is closed and bounded subset of . The interval is said to be degenerated when and both are equal. We call is positive when or negative when . We denote the space of all intervals by of and the set of all negative and positive intervals are represented as and , respectively. The inclusion “” is defined as

For any random real number and , the interval is given as

For and , algebraic operations are defined aswhere

For intervals the Hausdorff–Pompeiu distance is defined as

It is well known that the entire is complete metric space.

Definition 1 (see [26]). Let be such that , for eachandare Riemann integrable over interval. Then, we say that our function f is Riemann integrable over intervaland denotes

Definition 2 (see [27]). A function is known as convex function; if and , we have

Definition 3 (see [28]). A nonnegative function is known as Godunova–Levin function; ifand, we have

Definition 4 (see [19]). Let be a nonnegative function. We say is known as h-convex function, or ; if and , we haveIf the above inequality is inverted, it is referred to as h-concave or .

Definition 5 (see [24]). Let be a nonnegative function. We say is known as h-Godunova–Levin convex function, or; ifand, we haveIf the above inequality is inverted, it is referred to as h-Godunova–Levin concave or .

Remark 1. (i)If , then Definition 2.5 reduces to P-function [18](ii)If , then Definition 2.5 reduces to h-convex function [19](iii)If , then Definition 2.5 reduces to Godunova–Levin function [28](iv)If , then Definition 2.5 reduces to s-convex function [29](v)If , then Definition 2.5 reduces to s-Godunova–Levin function [30]

3. Main Results

Now, we are ready to introduce the notion of interval-valued h-Godunova–Levin convex functions.

Definition 6. Let be a nonnegative function. We say is the interval h-Godunova–Levin convex function, or; if, for all and , we haveIf the above inequality is inverted, it is referred to interval h-Godunova–Levin concave or . The space of all interval-valued h-Godunova–Levin convex and h-Godunova–Levin concave functions are denoted by and , respectively.

Proposition 1. Let be h-Godunova–Levin convex interval-valued function such that . Then, if if and only if , .

Proof. Let be h-Godunova–Levin convex interval-valued function, and suppose that , ; then,that is,It follows that we haveandThis shows that and . Conversely, suppose that if and , then, from above definition and set inclusion, we have This completes the proof.

Proposition 2. Let be h-Godunova–Levin concave interval-valued function such that. Then, if if and only if , .

Proof. This can be similar to Proposition 1.
In the next theorem, we Hermite–Hadamard-type inequality for h-Godunova–Levin interval-valued functions.

Theorem 2. Let and ; if and , then we have

Proof. By supposition, we haveIntegrating the above inequality w.r.t “x” over , we obtainIt follows that we haveSimilarly,This implies thatAs a result of applying the interval -Godunova–Levin convex function, we haveIntegrate w.r.t “x” over ; we haveAccordingly,Now, combining (24) and (26), we get the required result:

Remark 2. (i)If , then Theorem2has the following result for interval functions:(ii)If , then Theorem 2has the following result for interval convex functions:(iii)If , then Theorem 2has the following result for interval s-convex function:(iv)If , then Theorem 2has the following result of Ohud Almutairi and Adem Kiliman (see [24], Theorem1).

Example 1. Suppose that is defined as for , and is defined as , whereAs a result,This proves the above theorem.

Corollary 1. Let and if and ; then, we have

Theorem 3. Let and ; if and , then we havewhere

Proof. We break the interval into and ; then, for interval ,Integrate the above inequality over w.r.t :Then, the above inequality becomes asNow, by using the property of integral, the above inequality becomes asSimilarly, for interval , we haveAdding inequality (41) and (42), we obtainNow,

Example 2. Let be defined as , and for , is defined as , whereConsider and put values:Thus, we obtainThis proves the above theorem.

Corollary 2. Let and ; if and , then we have

Theorem 4. Let be a continuous function. If , , and , then we havewhere

Proof. We assume that and ; then, we haveThen,Integrating both sides over , w.r.t “x,” we haveIt follows thatTheorem is proved.

Example 3. Suppose that defined as for , , and is defined as .
Then,It follows thatConsequently, the above theorem is verified.

Corollary 3. Let be a continuous function. If , , and , then we have

Theorem 5. Let be a continuous function. If and , then we have

Proof. By hypothesis, one hasThen,Integrating above inequality over , w.r.t x, we obtainMultiply both sides by ; from the above equation, we obtainThis completes the proof.

Example 4. Suppose that is defined asfor,, andis defined as.
Then,It follows thatThis proves the above theorem.

Corollary 4. Let be a continuous function. If , and , then we have

Theorem 6. (Jensen-type inequality). Let with . If is super multiplicative nonnegative function and if , then the inequality become aswhere .

Proof. When , the above inequality is trivially true, i.e., reduce to official definition of interval-valued function. Now, we suppose that inequality is true for ; then, considerThus, the result is proved by using mathematical induction.

4. Conclusions

We introduced the notion of h-Godunova–Levin interval-valued functions and established Hermite–Hadamard and Jensen-type inequalities for the introduced class of functions. Our results are extension of many existing results. It is interesting for the researchers to establish fractional version of established inequalities for the h-Godunova–Levin interval-valued functions.

Data Availability

All data needed for this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors have equally contributed to this paper.

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Copyright © 2022 Xiaoju Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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