Eigenvalue inequalities for convex and log-convex functions

We give a matrix version of the scalar inequality f(a + b) ? f(a) + f(b) for positive concave functions f on 0, ∞). We show that Choi’s inequality for positive unital maps and operator convex functions remains valid for monotone convex functions at the cost of unitary congruences. Some inequalities for log-convex functions are presented and a new arithmetic-geometric mean inequality for positive matrices is given. We also point out a simple proof of the Bhatia-Kittaneh arithmetic-geometric mean inequality.