Higher order extension of L？wner's theory: Operator $k$-tone functions

The new notion of operator/matrix $k$-tone functions is introduced, which is a higher order extension of operator/matrix monotone and convex functions. Differential properties of matrix $k$-tone functions are shown. Characterizations, properties, and examples of operator $k$-tone functions are presented. In particular, integral representations of operator $k$-tone functions are given, generalizing familiar representations of operator monotone and convex functions.