# Peierls-Bogolyubov's inequality for deformed exponentials

## TL;DR

This paper explores convexity and concavity of trace functions involving deformed logarithms and exponentials, deriving new inequalities that generalize Peierls-Bogolyubov's inequality and improve bounds for Tsallis relative entropy.

## Contribution

It introduces new trace inequalities for deformed exponentials, extending Peierls-Bogolyubov's inequality and enhancing bounds for Tsallis relative entropy.

## Key findings

- Derived new trace inequalities for deformed exponentials.
- Generalized Peierls-Bogolyubov's inequality.
- Improved lower bounds for Tsallis relative entropy.

## Abstract

We study convexity or concavity of certain trace functions for the deformed logarithmic and exponential functions, and obtain in this way new trace inequalities for deformed exponentials that may be considered as generalizations of Peierls-Bogolyubov's inequality. We use these results to improve previously known lower bounds for the Tsallis relative entropy.