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

Chahn Yong Jung,Muhammad Yussouf,Shin Min Kang,Ghulam Farid,Yu-Ming Chu,Sei Ichiro Ueki

doi:10.1155/2020/8245324

Abstract

In this paper, we define a new function, namely, harmonically α , h − m -convex function, which unifies various kinds of harmonically convex functions. Generalized versions of the Hadamard and the Fejér–Hadamard fractional integral inequalities for harmonically α , h − m -convex functions via generalized fractional integral operators are proved. From presented results, a series of fractional integral inequalities can be obtained for harmonically convex, harmonically h − m -convex, harmonically α , m -convex, and related functions and for already known fractional integral operators.

Highlights

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, Gruss, Ostrowski–Gruss, 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 Fejer–Hadamard inequalities for fractional integral operators for harmonically (α, h − m)-convex functions
We begin from the fractional integral operators defined by Andricet al. in [2] containing an extended generalized Mittag-Leffler function in their kernels
A function f: J⊆R+ ⟶ R is said to be harmonically (α, h − m)-convex if f􏼠 mxy 􏼡 ≤ h tα􏼁f(x) + mh 1 − tα􏼁f(y), mty +(1 − t)x holds for all x, y ∈ J, t, α ∈ [0, 1], and m ∈

Summary

Introduction

Introduction and Preliminary ResultsFractional 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, Gruss, Ostrowski–Gruss, 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 Fejer–Hadamard inequalities for fractional integral operators for harmonically (α, h − m)-convex functions. E aim of this paper is to present the Hadamard and the Fejer–Hadamard inequalities for fractional integral operators for harmonically (α, h − m)-convex functions. By using this new definition, we will prove Hadamard and Fejer–Hadamard inequalities for generalized fractional integral operators involving an extended generalized Mittag-Leffler function.

Results

Conclusion