# Inequalities related to some types of entropies and divergences

## TL;DR

This paper explores mathematical properties and bounds of various extended entropies and divergences, including Tsallis, biparametrical, and quantum entropies, using inequalities like Hermite-Hadamard.

## Contribution

It introduces new bounds and inequalities for extended entropies and divergences, including biparametrical and quantum types, expanding theoretical understanding.

## Key findings

- New bounds for Tsallis quasilinear entropy and divergence.
- Bounds for biparametrical extended entropies and divergences.
- Inequalities for extended Lin's divergence and characterizations of quantum entropies.

## Abstract

The aim of this paper is to discuss new results concerning some kinds of parametric extended entropies and divergences. As a result of our studies for mathematical properties on entropy and divergence, we give new bounds for the Tsallis quasilinear entropy and divergence by applying the Hermite-Hadamard inequality. We also give bounds for biparametrical extended entropies and divergences which have been given in \cite{7}. In addition, we study $(r,q)$-quasilinear entropies and divergences as alternative biparametrical extended entropy and divergence, and then we give bounds for them. Finally we obtain inequalities for an extended Lin's divergence and some characterizations of Fermi-Dirac entropy and Bose-Einstein entropy.