Generalized Fractional Hadamard and Fejér–Hadamard Inequalities for Generalized Harmonically Convex Functions

Department of Business Administration, Gyeongsang National University, Jinju 52828, Republic of Korea Department of Mathematics, University of Sargodha, Sargodha, Pakistan Department of Mathematics, Huzhou University, Huzhou 313000, China Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering, Changsha University of Science & Technology, Changsha 410114, China Department of Mathematics, COMSATS University Islamabad, Attock, Pakistan Center for General Education, China Medical University, Taichung 40402, Taiwan


Introduction and Preliminary Results
Fractional integral inequalities are the generalizations and extensions of classical integral inequalities using fractional integral/derivative operators. ere are many well-known inequalities which have been extended for fractional calculus operators: for example, Hadamard, Minkowski, Ostrowski, Grüss, Ostrowski-Grüss, and Chebyshev inequalities have been extensively studied in recent decades, see [1][2][3][4][5][6][7][8]. e aim of this paper is to present the Hadamard and the Fejér-Hadamard inequalities for fractional integral operators for harmonically (α, h − m)-convex functions. We begin from the fractional integral operators defined by Andrić et al. in [2] containing an extended generalized Mittag-Leffler function in their kernels.
e Hadamard inequality gives the bounds of integral mean of a convex function on a closed interval. It is equivalently defined by the convex function. Convex functions have a significant role in the field of mathematical inequalities.
e Hadamard inequality is stated in the following theorem.
Theorem 1. Let f: [a, b] ⟶ R be a convex function. en, the following inequality holds: e Fejér-Hadamard inequality proved by Fejér in [10] generalizes the Hadamard inequality, and it is given as follows: e definition of the convex function motivates the researchers to define new notions and their consequences. A notion of the harmonically convex function is defined in [11].
Definition 5. Let I be an interval such that I⊆R + ; a function f: I ⟶ R is said to be harmonically convex if holds for all a, b ∈ I and t ∈ [0, 1]. is notation is further extended to the harmonically (α, m)-convex function in [12] and harmonically h-convex function in [13].
In the upcoming section, we define a new generalized notion, namely, harmonically (α, h − m)-convex function, which unifies all known definitions of harmonically convex functions. By using this new definition, we will prove Hadamard and Fejér-Hadamard inequalities for generalized fractional integral operators involving an extended generalized Mittag-Leffler function. Also, results for harmonically (h − m)-convex functions and harmonically (α, m)-convex functions are deduced.

Main Results
First, we define the harmonically (α, h − m)-convex function as follows: holds for all x, y ∈ J, t, α ∈ [0, 1], and m ∈ (0, 1]. It unifies the definitions of harmonically (α, m)-convexity and harmonically h-convexity of functions. For different specific choices of α, h, and m, almost all kinds of well-known harmonically convex functions can be obtained, see the remark as follows.
Now, we apply Definition 8 and operators (4) and (5) to find the Hadamard and Fejér-Hadamard inequalities.
In the whole paper, we have used the following notations frequently: Theorem 3. Let f, g:  (4) and (5), we have where Proof. Since f is harmonically (α, h − m)-convex, for all x, y ∈ [a, b], the following inequality holds: ).