Some operator convex functions of several variables

TL;DR

This paper investigates operator convexity and concavity of multivariable functions using perspectives of operator mappings, with applications to the convexity properties of specific power functions' Fréchet differentials.

Contribution

It introduces methods to establish operator convexity and concavity of multivariable functions via perspectives of operator mappings, extending understanding of these properties.

Findings

Proves operator concavity and convexity for certain functions of two or three variables.

Establishes convexity/concavity of Fréchet differentials for specific power functions.

Provides new tools for analyzing operator functions in multivariable settings.

Abstract

We obtain operator concavity (convexity) of some functions of two or three variables by using perspectives of regular operator mappings of one or several variables. As an application, we obtain, for $ 0<p < 1,$ concavity, respectively convexity, of the Frech\'{e}t differential mapping associated with the functions $t \to t^{1+p} $ and $t \to t^{1-p}.$