A matrix subadditivity inequality for f(A + B) and f(A) + f(B)

In 1999 Ando and Zhan proved a subadditivity inequality for operator concave functions. We extend it to all concave functions: Given positive semidefinite matrices A, B and a non-negative concave function f on [0,∞),‖f(A+B)‖⩽‖f(A)+f(B)‖‖f(A+B)‖⩽‖f(A)+f(B)‖for all symmetric norms (in particular for all Schatten p  -norms). The case f(t)=t is connected to some block-matrix inequalities, for instance the operator norm inequalityAX∗XB∞⩽maxX|‖∞;‖|B|+|X∗|‖∞}for any partitioned Hermitian matrix.