A Complementary Inequality to the Information Monotonicity for Tsallis Relative Operator Entropy

TL;DR

This paper introduces a reverse inequality for Tsallis relative operator entropy involving positive linear maps, extends and reverses known inequalities like Ando's and L"owner-Heinz, and explores related consequences in inner product spaces.

Contribution

It provides new reverse inequalities and extensions for Tsallis entropy and classical operator inequalities, enhancing understanding of their relationships and applications.

Findings

Established a reverse inequality for Tsallis relative operator entropy.

Presented a converse of Ando's inequality for various parameters.

Extended and reversed the L"owner-Heinz inequality under specific conditions.

Abstract

We establish a reverse inequality for Tsallis relative operator entropy involving a positive linear map. In addition, we present converse of Ando's inequality, for each parameter. We give examples to compare our results with the known results by Furuta and Seo. In particular, we establish an extension and a reverse of the L\"owner-Heinz inequality under certain condition. Some interesting consequences of inner product spaces and norm inequalities are also presented.