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

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.

What is novel here is that we convincingly demonstrate how our techniques can be used to give simple proofs for the old and new theorems for the functions that are relevant to quantum statistics