# Inequalities for relative operator entropies and operator means

## TL;DR

This paper derives new bounds for the Tsallis relative operator entropy using Hermite-Hadamard inequality, revealing relations with generalized relative operator entropy and analyzing monotonicity properties of operator means and entropies.

## Contribution

It introduces novel bounds for Tsallis relative operator entropy and explores its relation to generalized entropy, along with monotonicity properties of operator means.

## Key findings

- New bounds for Tsallis relative operator entropy
- Relation established between Tsallis and generalized relative operator entropy
- Monotonicity properties of operator means and entropies analyzed

## Abstract

The main purpose of this article is to study estimates for the Tsallis relative operator entropy, by the use of Hermite-Hadamard inequality. Thus, we obtain alternative bounds for the Tsallis relative operator entropy. In the process to derive these bounds, we established the significant relation between the Tsallis relative operator entropy and the generalized relative operator entropy. In addition, we study the properties on monotonicity for the weight of operator means, and for the parameter of relative operator entropies.