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.