A concavity inequality for symmetric norms

We review some recent convexity results for Hermitian matrices and we add a new one to the list: Let A be semidefinite positive, let Z be expansive, Z*Z I, and let f:[0,∞)→[0,∞) be a concave function. Then, for all symmetric norms f(Z*AZ) Z*f(A)Z .This inequality complements a classical trace inequality of Brown–Kosaki.