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

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.