Two trace inequalities for operator functions

In this paper we show that for a non-negative operator monotone function $f$ on $[0, \infty)$ such that $f(0)= 0$ and for any positive semidefinite matrices $A$ and $B$, $$Tr((A-B)(f(A)-f(B))) \le Tr(|A-B|f(|A-B|)).$$ When the function $f$ is operator convex on $[0, \infty)$, the inequality is reversed.