Mathematical Problems in Engineering
Volume 2021 (2021), Article ID 5549258, 8 pages
https://doi.org/10.1155/2021/5549258

Research Article

Hermite–Hadamard-Type Inequalities for -Convex Functions via Katugampola Fractional Integral

Erhan Set and İlker Mumcu

Department of Mathematics, Faculty of Science and Arts, Ordu University, Ordu, Turkey

Correspondence should be addressed to Erhan Set; erhanset@yahoo.com

Received 22 January 2021; Revised 1 March 2021; Accepted 9 April 2021; Published 8 May 2021

Academic Editor: Irfan Kaymaz

Copyright © 2021 Erhan Set and İlker Mumcu. 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.

Abstract.

This article is organized as follows: First, definitions, theorems, and other relevant information required to obtain the main results of the article are presented. Second, a new version of the Hermite–Hadamard inequality is proved for the F-convex function class using a fractional integral operator introduced by Katugampola. Finally, new fractional Hermite–Hadamard-type inequalities are given with the help of F-convexity.

1. Introduction and Preliminaries

First of all, let us recall the concept of convex function that is the fundamental notation of convex analysis.

Definition 1. The function is said to be convex if

for all and . We say that is concave if is convex.
There are many inequalities in the literature for convex functions. But, among these inequalities, perhaps the one which takes the most attention of researchers is the Hermite–Hadamard inequality on which hundreds of studies have been conducted. The classical Hermite–Hadamard integral inequalities are as the follows.

Theorem 1. Assume that is a convex mapping defined on the interval of where The statement

holds and known as Hermite–Hadamard inequality. Both inequalities hold in the reversed direction if is concave.
Convex functions played a significant role in several areas such as engineering, finance, statistics, optimization, and mathematical inequalities. Convex functions have a paramount history and have been an intense study issue for over a century in mathematics. Various generalizations, extensions, and variants of the convex functions have been presented by many researchers. Recently, one of them has been introduced by Samet [1] as follows:
Let be a family of a mappings that satisfy the following axioms:

(A1)If , , then for every , we have

(A2)For every , and , we have

 where is a function that depends on , and it is nondecreasing with respect to the first variable.(A3)For any , , we have

where is a constant that depends only on .

Definition 2. Let , , , be a given function. We say that is a convex function with respect to some (or F-convex function) if

Remark 1. Let , and let , , be an -convex function, that is,

We define the functions by

and by

For

it is explicit that and

that is, is an F-convex function. Particularly, taking , we show that if is a convex function, then is an F-convex function with respect to F defined above.

Remark 2. Let , , be an -convex function, , that is,

We define the functions by

and by

For , it is clear that and

that is, is an -convex function.

Remark 3. Let be a given function which is not identical to 0, where is a interval in such that . Let , , be an h-convex function, that is,

We define the functions by

and by

For , it is clear that and

that is, is an -convex function.
We define the beta function as ([2], p18)

where is the gamma function.
The incomplete Beta function is defined by

Definition 3. Let . The Riemann–Liouville integrals and of order are defined by

respectively, where . Here, .
For , the fractional integral reduces to a classical integral.
Fractional calculus has the great impact in pure and applied sciences. In recent years, fractional analysis has become one of the most frequently used methods to obtain new and different versions of the results available in the literature. In [3], by using Riemann–Liouville fractional integrals, a new version of Hermite–Hadamard’s inequalities was proved by Budak et al. for F-convex function classes as follows.

Theorem 2. Let be an interval and be a differentiable mapping on . If it is on , for some , then we have

and

where .
For results related to F-convex functions, one can see [1, 3–5].
Here, we present some definitions of fractional integrals.

Definition 4 (see [6]). Let be a finite or infinite interval on the half-axis . The Hadamard fractional integrals (left sided and right sided) of order , of a real function are defined by

and

We consider the space , consists of those complex-valued Lebesque measurable functions on for which , with

A new fractional integral operator, which is a generalization of Riemann–Liouville and Hadamard fractional integrals to a single form, is introduced by Katugampola as follows.

Definition 5 (see [7]). Let be a finite interval. Then, the left- and right-side Katugampola fractional integrals of order of are defined:

with and , if the integral exists.
If we take as q = 1 in this definition, the Riemann–Liouville fractional integral operator that is well known in the literature used to describe Riemann–Liouville and Caputo fractional derivatives is obtained [6, 8, 9]. Using L’Hospital rule, when , we have Hadamard fractional integrals of (25) and (26).
For results associated with Katugampola fractional operators, we refer the reader to the some recent papers (see [7, 10, 11]).
Motivated from the studies presenting Hermite–Hadamard-type inequalities obtained with the help of fractional integral operators for the F-convexity class, to obtain more general and new versions of Hermite–Hadamard-type inequalities by using Katugampola fractional integrals is the main purpose of this article.

2. Hermite–Hadamard Inequalities for F-Convexity via Katugampola Fractional Integrals

Now, let us give the Hermite–Hadamard inequality for F-convex functions via Katugampola fractional integrals as follows.

Theorem 3. Let and . Let be a positive function with and . If is F-convex on , for some , then the following inequalities hold:

and

where .

Proof. Let , , . Since is -convex, we can write

For and , we have

Multiplying both sides of the last inequality by , , and using axiom (A3), we have

Integrating the last inequality with respect to over and using axiom (A1), we get

where

This establishes the first inequality. For the proof of the second inequality, since is -convex, we have

By adding these inequalities, we get

Applying axiom (A3) for , we obtain

Integrating over the last inequality and using axiom (A2), we have

and thus, the proof is completed.

Remark 4. In Theorem 3, taking , we obtain inequalities of (23) and (24).

Remark 5. In Theorem 3, taking limit , we get

Corollary 1. If is -convex on [a, b] with , , then we have

where .

Proof. It is known that an -convex is an F-convex. Using (10) with , we have

Using (8), (29), and (42), we can write

So, we get

On the other hand, using (9) with , we have

for . So, from (30) and (45), we obtain

This implies that

The proof is completed.

Remark 6. If we take in Corollary 1, then is convex and we have Theorem 2.1 in [12].

3. Hermite–Hadamard-Type Inequalities for F-Convexity via Katugampola Fractional Integrals

To prove our main results in this section, let us consider the following lemma.

Lemma 1 (see [12]). Let be a differentiable mapping on with . Then, the following equality holds if the fractional integrals exist:

where .

Theorem 4. Let be an interval and be a differentiable mapping on . Suppose that is F-convex on , for some , and the function belongs to . Then, we have the inequality

where and .

Proof. By using F-convexity of , we can write

Using axiom (A3) with , we get

Integrating over and using axiom (A2), we obtain

From Lemma 1, we get

Since is nondecreasing with respect to the first variable, we establish

The proof is completed.

Remark 7. In Theorem 4, taking , we obtain Theorem 2.5 in [3].

Remark 8. In Theorem 4, taking limit , we get

Corollary 2. If is -convex on with , , then we have

where .

Proof. From (10), with , we have

Using (9) with ,

Then, by Theorem 4, we get

So, we get the desired result.

Remark 9. If we take in Corollary 2, then is convex and we have Theorem 2.5 in [12].

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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