Jointly convex mappings related to the Lieb's functional and Minkowski type operator inequalities

TL;DR

This paper investigates joint convexity and concavity of multivariable operator functions using operator log-convexity, leading to new matrix trace inequalities and generalizations of classical results like Lieb's and Minkowski inequalities.

Contribution

It introduces new joint convexity/concavity results for multivariable operator functions and extends Minkowski type inequalities in the context of positive linear maps.

Findings

Proves joint convexity/concavity of matrix trace functions involving operator means.

Generalizes and complements results of Ando and Lieb on tensor product maps.

Establishes Minkowski type operator inequalities under broader conditions.

Abstract

Employing the notion of operator log-convexity, we study joint concavity$/$ convexity of multivariable operator functions: $(A,B)\mapsto F(A,B)=h\left[ \Phi(f(A))\ \sigma\ \Psi(g(B))\right]$, where $\Phi$ and $\Psi$ are positive linear maps and $\sigma$ is an operator mean. As applications, we prove jointly concavity$/$convexity of matrix trace functions $\Tr\left\{ F(A,B)\right\}$. Moreover, considering positive multi-linear mappings in $F(A,B)$, our study of the joint concavity$/$ convexity of $(A_1,\cdots,A_k)\mapsto h\left[ \Phi(f(A_1),\cdots,f(A_k))\right]$ provides some generalizations and complement to results of Ando and Lieb concerning the concavity$/$ convexity of maps involving tensor product. In addition, we present Minkowski type operator inequalities for a unial positive linear map, which is an operator version of Minkowski type matrix trace inequalities under a more general setting than Carlen and Lieb, Bekjan, and Ando and Hiai.