Hermite–Hadamard-Type Inequalities for -Convex Functions via Katugampola Fractional Integral
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 iffor 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 statementholds 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  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 havewhere 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 byand byForit is explicit that andthat 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 byand byFor , it is clear that andthat 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 byand byFor , it is clear that andthat is, is an -convex function.
We define the beta function as (, 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 byrespectively, 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 , 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 haveandwhere .
For results related to F-convex functions, one can see [1, 3–5].
Here, we present some definitions of fractional integrals.
Definition 4 (see ). 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 byandWe consider the space , consists of those complex-valued Lebesque measurable functions on for which , withA 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 ). 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:andwhere .
Proof. Let , , . Since is -convex, we can writeFor and , we haveMultiplying both sides of the last inequality by , , and using axiom (A3), we haveIntegrating the last inequality with respect to over and using axiom (A1), we getwhereThis establishes the first inequality. For the proof of the second inequality, since is -convex, we haveBy adding these inequalities, we getApplying axiom (A3) for , we obtainIntegrating over the last inequality and using axiom (A2), we haveand thus, the proof is completed.
Remark 5. In Theorem 3, taking limit , we get
Corollary 1. If is -convex on [a, b] with , , then we havewhere .
Proof. It is known that an -convex is an F-convex. Using (10) with , we haveUsing (8), (29), and (42), we can writeSo, we getOn the other hand, using (9) with , we havefor . So, from (30) and (45), we obtainThis implies thatThe proof is completed.
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 ). 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 inequalitywhere and .
Proof. By using F-convexity of , we can writeUsing axiom (A3) with , we getIntegrating over and using axiom (A2), we obtainFrom Lemma 1, we getSince is nondecreasing with respect to the first variable, we establishThe proof is completed.
Remark 8. In Theorem 4, taking limit , we get
Corollary 2. If is -convex on with , , then we havewhere .
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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