Modulus of convexity for operator convex functions

Abstract

Given an operator convex function $f(x)$, we obtain an operator-valued lower bound for $cf(x) + (1-c)f(y) - f(cx + (1-c)y)$, $c \in [0,1]$. The lower bound is expressed in terms of the matrix Bregman divergence. A similar inequality is shown to be false for functions that are convex but not operator convex.Comment: 5 pages, change of title. The new version shows that the main result of the original paper cannot be extended to convex functions that are not operator convex