We refine Epstein's method to prove joint concavity/convexity of matrix trace functions of the extended Lieb type $Tr{ \Phi(A^p)^{ 1/2}\Psi(B^q)\Phi(A^p)^{ 1/2}}^s$, where $\Phi$ and $\Psi$ are positive linear maps. By the same method combined with majorization technique, similar properties are proved for symmetric (anti-) norm functions of the form $||{ \Phi(A^p)\sigma\Psi(B^q)}^s||$ involving an operator mean $\sigma$. Carlen and Lieb's variational method is also used to improve the convexity property of norm functions $||\Phi(A^p)^s||$.
Categories: math.FA math-ph math.MP math.OA