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⩾IZ∗Z⩾I, and let f:[0,∞)→[0,∞)f:[0,∞)→[0,∞) be a concave function. Then, for all symmetric norms‖f(Z∗AZ)‖⩽‖Z∗f(A)Z‖.‖f(Z∗AZ)‖⩽‖Z∗f(A)Z‖.This inequality complements a classical trace inequality of Brown–Kosaki.