A CHARACTERIZATION OF CONVEX FUNCTIONS AND ITS APPLICATION TO OPERATOR MONOTONE FUNCTIONS

We give a characterization of convex functions in terms of difference among values of a function. As an application, we propose an estimation of operator monotone functions: If A > B >= 0 and f is operator monotone on (0, infinity), then f(A)- f(B) >= f(vertical bar vertical bar B vertical bar vertical bar + epsilon)- f(vertical bar vertical bar B k) > 0, where epsilon =vertical bar vertical bar (A-B)(-1) vertical bar vertical bar(-1). Moreover it gives a simple proof to Furuta's theorem: If log A > log B for A, B > 0 and f is operator monotone on (0, infinity), then there exists a beta > 0 such that f(A(alpha)) > f(B-alpha) for all 0 < alpha <= beta.