On the Operator Jensen-Mercer Inequality

TL;DR

This paper introduces new operator inequalities related to Mercer and Jensen-Mercer inequalities, including a log-convex version, refining existing bounds and extending the scope beyond convexity assumptions.

Contribution

It presents a Mercer-type inequality for operators without convexity assumptions and introduces a log-convex operator inequality, expanding the theoretical framework.

Findings

Refined Mercer-type inequalities for operators

Established a log-convex operator inequality

Improved bounds for quasi-arithmetic means of operators

Abstract

Mercer inequality for convex functions is a variant of Jensen's inequality, with an operator version that is still valid without operator convexity. This paper is two folded. First, we present a Mercer-type inequality for operators without assuming convexity nor operator convexity. Yet, this form refines the known inequalities in the literature. Second, we present a log-convex version for operators. We then use these results to refine some inequalities related to quasi-arithmetic means of Mercer's type for operators.