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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).doi:10.14492/hokmj/1384273390 fatcat:xhn2mlpos5h2jjibvymyidlyby