Norm inequalities for positive semidefinite matrices and a question of Bourin

Let A, B, X is an element of M-n(C) such that A and B are positive semidefinite. It is shown that parallel to vertical bar A(t) X B1-t + B-t X* A(1-t) vertical bar parallel to <= parallel to vertical bar AX vertical bar parallel to + parallel to vertical bar XB vertical bar parallel to for t is an element of [0, 1] and for every unitarily invariant norm. This gives an affirmative answer to one of the questions posed by Bourin regarding subadditivity inequalities in the case of the trace norm. New norm inequalities related to Bourin's question are also presented.