Higher order extension of Löwner’s theory: Operator 𝑘-tone functions

Abstract

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