Rotfel’d type inequalities for norms

In this note, we consider some norm inequalities related to the Rotfel’d Trace InequalityTrf(|A+B|)⩽Trf(|A|)+f(|B|)for concave functions f:[0,∞)→[0,∞)f:[0,∞)→[0,∞) and arbitrary n-by-n   matrices. For instance we show that for a large class of non-negative concave functions f(t)f(t) and for all symmetric norms we have‖f(|A+B|)‖⩽2‖f(|A|)+f(|B|)‖and we conjecture that this holds for all non-negative concave functions.