Rotfel'd type inequalities for norms

In this note, we consider some norm inequalities related to the Rotfel'd Trace Inequality Tr f(vertical bar A + B vertical bar) <= Tr f(vertical bar A vertical bar) + f(vertical bar B vertical bar) for concave functions f : [0, infinity) -> [0, infinity) and arbitrary n-by-n matrices. For instance we show that for a large class of non-negative concave functions f (t) and for all symmetric norms we have parallel to f(vertical bar A + B vertical bar)parallel to <= root 2 parallel to f(vertical bar A vertical bar) + f(vertical bar B vertical bar)parallel to and we conjecture that this holds for all non-negative concave functions. (C) 2010 Elsevier Inc. All rights reserved.