A normal family of operator monotone functions

Abstract

We show that the family of all operator monotone functions f on (-1,1) such that f(0) = 0 and f′(0) = 1 is a normal family and investigate some properties of odd operator monotone functions. We also characterize the odd operator monotone functions and even operator convex functions on (-1,1). As a consequence, we show that if f is an odd operator monotone function on (-1,1), then f is concave on (-1,0) and convex on (0,1).