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

*is 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.


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

Definition 1.
e function Z: [u, v]⊆R ⟶ R is said to be convex if for all x, y ∈ [u, v] and s ∈ [0, 1]. We say that Z is concave if (− Z) is convex. ere 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. e classical Hermite-Hadamard integral inequalities are as the follows. Theorem 1. Assume that Z: I ⊆ R ⟶ R is a convex mapping defined on the interval I of R where u < v. e statement holds and known as Hermite-Hadamard inequality. Both inequalities hold in the reversed direction if f 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 F be a family of a mappings F: R × R × R ×[0, 1] ⟶ R that satisfy the following axioms: (A2) For every τ ∈ L 1 (0, 1), ψ ∈ L ∞ (0, 1) and where T F,ψ : R × R × R ⟶ R is a function that depends on (F, ψ), and it is nondecreasing with respect to the first variable.
where L ψ ∈ R is a constant that depends only on ψ.
We say that f is a convex function with respect to some F ∈ F (or F-convex function) if Remark 1. Let ε ≥ 0, and let f: We define the functions F: and For it is explicit that F ∈ F and that is, f is an F-convex function. Particularly, taking ε � 0, we show that if f is a convex function, then f is an F-convex function with respect to F defined above.
, that is, We define the functions F: and For L ψ � 0, it is clear that F ∈ F and that is, f is an F-convex function.
We define the functions F: and that is, f is an F-convex function.
We define the beta function as ( [2], p18) where Γ is the gamma function. e incomplete Beta function is defined by Mathematical Problems in Engineering For μ � 1, 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 I⊆R be an interval and f: I
and For results related to F-convex functions, one can see [1,[3][4][5].
Here, we present some definitions of fractional integrals.
Definition 4 (see [6]). Let (a, b)(0 ≤ a < b < ∞) be a finite or infinite interval on the half-axis R + . e Hadamard fractional integrals (left sided and right sided) of order α ∈ C, Re(α) > 0 of a real function f ∈ L(a, b) are defined by and We consider the space X 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.
en, the left-and right-side Katugampola fractional integrals of order (α > 0) of f ∈ X p c (a, b) are defined: with a < x < b and ρ > 0, 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 ρ ⟶ 0 + , 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.

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.  F-convex on [a, b], for some F ∈ F, then the following inequalities hold: and (32) Multiplying both sides of the last inequality by αρt αρ− 1 , α > 0, and using axiom (A3), we have Integrating the last inequality with respect to t over [0, 1] and using axiom (A1), we get is establishes the first inequality. For the proof of the second inequality, since f is F-convex, we have By adding these inequalities, we get Applying axiom (A3) for ψ(t) � αρt αρ− 1 , we obtain Integrating over [0, 1] the last inequality and using axiom (A2), we have and thus, the proof is completed.

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.
Proof. By using F-convexity of |f ′ |, we can write Using axiom (A3) with ψ(t) � |(1 − t ρ ) α − t ρα |, we get Integrating over [0, 1] and using axiom (A2), we obtain From Lemma 1, we get Since T F,ψ is nondecreasing with respect to the first variable, we establish e proof is completed.
Mathematical Problems in Engineering