Hermite–Hadamard–Mercer-Type Inequalities for Harmonically Convex Mappings

Xuexiao You,Muhammad Aamir Ali,Hüseyin Budak,Jiraporn Reunsumrit,Thanin Sitthiwirattham

doi:10.3390/math9202556

Abstract

In this paper, we prove Hermite–Hadamard–Mercer inequalities, which is a new version of the Hermite–Hadamard inequalities for harmonically convex functions. We also prove Hermite–Hadamard–Mercer-type inequalities for functions whose first derivatives in absolute value are harmonically convex. Finally, we discuss how special means can be used to address newly discovered inequalities.

Highlights

For some recent studies linked to the Jensen–Mercer inequality, one can consult [6,7]
F ( x ) + f (y) for x, y ∈ [κ1, κ2 ]. Inspired by these ongoing studies, we will establish modified versions of inequalities (11) and (12) for harmonically convex functions because we discovered some flaws in the proof of inequality (11)
We proved some new Hermite–Hadamard–Mercer inequalities for harmonically convex functions and differentiable harmonically convex functions

Summary

Introduction

The well-known Jensen inequality [1] states that if f is a convex function on an interval and contains x1 , x2 , . Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. In the theory of convex functions, the Hermite–Hadamard inequality is very important. The following variant of the Jensen inequality, known as the Jensen–Mercer, was demonstrated by Mercer [4]: Theorem 1. If f is a convex function on [ a, b], the following inequality is true:. Mathematics 2021, 9, 2556 n where ∑ λ j = 1, x j ∈ [ a, b] and λ j ∈ [0, 1]. For some recent studies linked to the Jensen–Mercer inequality, one can consult [6,7]

Harmonic Convexity and Related Inequalities

Main Results

Application to Special Means

Conclusions