On Some Trace Inequalities

In this paper we consider some generalizations of Ando's inequality

|||f(A) − f(B)||| ≤ |||f(|A − B|)|||

with the "weight" (A − B)^p. More precisely, for p ≥ 1 such that (−1)^p = −1 and non-negative functions function f on [0, ∞) such that f (0) = 0, we study the following inequality:

Tr ((A − B)^p (f(A) − f(B))) ≥ Tr(|A − B|^pf (|A − B|)),

whenever A and B are positive semidefinite matrices.