New Fractional Hermite–Hadamard–Mercer Inequalities for Harmonically Convex Function

Saad Ihsan Butt,Saba Yousaf,Hamid Reza Moradi,Khuram Ali Khan,Atifa Asghar,Huo Tang

doi:10.1155/2021/5868326

Abstract

In 2003, Mercer presented an interesting variation of Jensen’s inequality called Jensen–Mercer inequality for convex function. In the present paper, by employing harmonically convex function, we introduce analogous versions of Hermite–Hadamard inequalities of the Jensen–Mercer type via fractional integrals. As a result, we introduce several related fractional inequalities connected with the right and left differences of obtained new inequalities for differentiable harmonically convex mappings. As an application viewpoint, new estimates regarding hypergeometric functions and special means of real numbers are exemplified to determine the pertinence and validity of the suggested scheme. Our results presented here provide extensions of others given in the literature. The results proved in this paper may stimulate further research in this fascinating area.

Highlights

The definition of convexity has been improved, generalized, and expanded in several directions in recent years
Using Lemma 13, we present the following fractional integral inequality for jφ′jq ∈ HKðIÞ as follows
We present the (H-H-M) inequalities involving (RL) fractional integrals for the class of harmonic convex function and established some integral inequalities connected with the right and left sides of fractional (H-H-M) type inequalities for differentiable mappings whose derivatives in absolute value are harmonically convex

Summary

Introduction

The definition of convexity has been improved, generalized, and expanded in several directions in recent years. If jφ′jq ∈ HKð1⁄2θ, ΘÞ for some fixed q > 1 and φ′ ∈ L1⁄2θ, Θ along with assumption A1 and 0 < α ≤ 1, the following inequality for fractional integrals holds: Iφðħ ; α, x, yÞ ≤. If jφ′jq ∈ HKð1⁄2θ, ΘÞ for some fixed q > 1 and φ′ ∈ L1⁄2θ, Θ along with assumption A1 and 0 < α ≤ 1, following inequality for fractional integrals holds:. Lemma 13 and Lemma 9, using the Hölder inequality and jφ′jq ∈ HK ð1⁄2θ, ΘÞ, we find y−x 2xy y−x 2xy ð1ð1

A2ζq hφ

Application to Special Means

Conclusion