Let n,m ∈ R. We show that the family of all operator monotone functions f on an interval (a, b) such that f(t0) = n and f(t0) = m > 0 for t0 ∈ (a, b) 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 a symmetric interval (−a, a). As a consequence, we show that if f is an odd operator monotone function on (−a, a), then f is concave on (−a; 0) and convex on (0; a).

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