Concavity of certain matrix trace and norm functions. II

We refine Epstein’s method to prove joint concavity/convexity of matrix trace functions of Lieb type $\mathrm{T}\mathrm{r}f(\mathrm{\Phi }(A^p{)}^{1\mathrm{/}2}\mathrm{\Psi }(B^q)\mathrm{\Phi }(A^p{)}^{1\mathrm{/}2})$ and symmetric (anti-) norm functions of the form $\mathrm{\parallel }f(\mathrm{\Phi }(A^p)\sigma \mathrm{\Psi }(B^q))\mathrm{\parallel }$, where $\mathrm{\Phi }$ and $\mathrm{\Psi }$ are positive linear maps, $\sigma$ is an operator mean, and $f(x^{\gamma })$ with a certain power $\gamma$ is an operator monotone function on $(0,\mathrm{\infty })$. Moreover, the variational method of Carlen, Frank and Lieb is extended to general non-decreasing convex/concave functions on $(0,\mathrm{\infty })$ so that we prove joint concavity/convexity of more trace functions of Lieb type.