Mond and Pe$\check{c}$ari$\acute{c}$ inequality for $h$-convex functions with applications

TL;DR

This paper develops new operator inequalities for h-convex functions, including Jensen's and Hermite-Hadamard's inequalities, with refinements and applications to various classes of convex functions.

Contribution

It introduces operator versions and refinements of Jensen's and Hermite-Hadamard inequalities for h-convex functions, expanding their applicability.

Findings

Operator Jensen's inequality and its converse established.

Refined Jensen-type inequality for h-convex functions provided.

Hermite-Hadamard type inequality for h-convex functions proved.

Abstract

In this paper, we prove an operator version of the Jensen's inequality and its converse for $h$-convex functions. We provide a refinement of the Jensen type inequality for $h$-convex functions. Moreover, we prove the Hermite-Hadamard's type inequality and a multiple operator version of the Jensen's inequality for $h$-convex functions. In particular, a result for convex, $P$-class, $s$-convex, Godunova-Levin, and $s$-Godunova-Levin functions can be deduced.