Around Operator Monotone Functions

We show that the symmetrized product AB + BA of two positive operators A and B is positive if and only if f(A + B) = f(A)+ f(B) for all non-negative operator monotone functions f on [0, infinity) and deduce an operator inequality. We also give a necessary and sufficient condition for that the composition f circle g of an operator convex function f on [0, infinity) and a non-negative operator monotone function g on an interval (a, b) is operator monotone and present some applications.