New sharp inequalities for operator means

TL;DR

This paper introduces new sharp inequalities for operator means, extending existing inequalities and bounds for related operator functions, with implications for operator entropy and monotone functions.

Contribution

It presents novel sharp multiplicative reverses of operator mean inequalities and extends the Pólya-Szegö inequality to all operator means.

Findings

Extended operator Pólya-Szegö inequality to arbitrary means

Derived new bounds for Tsallis relative operator entropy

Provided sharp reverses for operator mean inequalities

Abstract

New sharp multiplicative reverses of the operator means inequalities are presented, with a simple discussion of squaring an operator inequality. As a direct consequence, we extend the operator P\'olya-Szeg\"o inequality to arbitrary operator means. Furthermore, we obtain some new lower and upper bounds for the Tsallis relative operator entropy, operator monotone functions and positive linear maps.