Concavity of certain matrix trace and norm functions

We refine Epstein’s method to prove joint concavity/convexity of matrix trace functions of the extended Lieb type Tr{Φ(Ap)1/2Ψ(Bq)Φ(Ap)1/2}s, where Φ and Ψ are positive linear maps. By the same method combined with majorization technique, similar properties are proved for symmetric (anti-) norm functions of the form ||{Φ(Ap)σΨ(Bq)}s|| involving an operator mean σ. Carlen and Lieb’s variational method is also used to improve the convexity property of norm functions ||Φ(Ap)s||.