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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 positivelinear maps and $\sigma$ is an operator mean. As applications, we prove jointlyconcavity$/$convexity of matrix trace functions $\Tr\left\{ F(A,B)\right\}$.Moreover, considering positive multi-linear mappings in $F(A,B)$, our study ofthe 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 complementto results of Ando and Lieb concerning the concavity$/$ convexity of mapsinvolving tensor product. In addition, we present Minkowski type operatorinequalities for a unial positive linear map, which is an operator version ofMinkowski type matrix trace inequalities under a more general setting thanCarlen and Lieb, Bekjan, and Ando and Hiai.

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