Conditional $h$-convexity with applications

TL;DR

This paper introduces conditional $h$-convex functions, establishing an operator Jensen inequality and refining several classical inequalities using this new concept.

Contribution

The paper presents the novel concept of conditional $h$-convexity and extends classical inequalities through this framework.

Findings

Established an operator Jensen inequality for conditional $h$-convex functions.

Provided refinements for Ky-Fan's, arithmetic-geometric mean, Chrystal, and H"older-McCarthy inequalities.

Showed that many other inequalities can be refined using the new notion.

Abstract

In this paper, we introduce the notion of conditional $h$-convex functions and we prove an operator version of the Jensen inequality for conditional $h$-convex functions. Using this type of functions, we give some refinements for Ky-Fan's inequality, arithmetic-geometric mean inequality, Chrystal inequality, and H$\ddot{o}$lder-McCarthy inequality. Many of the other inequalities can be refined by applying this new notion.