Operator log-convex functions and f-divergence functional

We present a characterization of operator log-convex functions by using positive linear mappings. Moreover, we study the non-commutative f-divergence functional of operator log-convex functions. In particular, we prove that f is operator log-convex if and only if the non-commutative f-divergence functional is operator log-convex in its first variable and operator convex in its second variable.