  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.
