Convex inequalities and their applications to relative operator entropies

TL;DR

This paper introduces new inequalities involving log-convex and geometrically convex functions, leading to refined versions of Young's and Jensen's inequalities, and explores their applications to operator inequalities and relative operator entropy.

Contribution

It presents novel inequalities for convex functions and extends Jensen's inequality to operators, with applications to relative operator entropy.

Findings

Refined Young's and Jensen's inequalities derived.

Operator Jensen's inequality developed for conditioned functions.

Applications to inequalities involving relative operator entropy.

Abstract

A considerable amount of literature in the theory of inequality is devoted to the study of Jensen's and Young's inequality. This article presents a number of new inequalities involving the log-convex functions and the geometrically convex functions. As their consequences, we derive the refinements for Young's inequality and Jensen's inequality. In addition, the operator Jensen's type inequality is also developed for conditioned two functions. Utilizing these new inequalities, we investigate the operator inequalities related to the relative operator entropy.