# Convexity of parameter extensions of some relative operator entropies with a perspective approach

## TL;DR

This paper introduces two parameterized extensions of relative operator entropies and demonstrates their joint convexity or concavity using a perspective approach, expanding the theoretical framework of operator entropy measures.

## Contribution

It proposes new parametric forms of relative operator entropies and establishes their convexity or concavity properties through a novel perspective method.

## Key findings

- Defined two new parameterized operator entropies
- Proved joint convexity or concavity under specific conditions
- Extended the theoretical understanding of operator entropy functions

## Abstract

In this paper, we introduce two notions of a relative operator $(\alpha, \beta)$-entropy and a Tsallis relative operator $(\alpha, \beta)$-entropy as two parameter extensions of the relative operator entropy and the Tsallis relative operator entropy. We apply a perspective approach to prove the joint convexity or concavity of these new notions, under certain conditions concerning $\alpha$ and $\beta$. Indeed, we give the parametric extensions, but in such a manner that they remain jointly convex or jointly concave.