Higher order extension of L\"owner's theory: Operator $k$-tone functions

TL;DR

This paper introduces operator $k$-tone functions as a higher order extension of monotone and convex functions, providing their differential properties, characterizations, and integral representations.

Contribution

It presents the concept of operator $k$-tone functions, extending classical theories, with new characterizations, properties, and integral representations for these higher order functions.

Findings

Differential properties of matrix $k$-tone functions established

Characterizations and examples of operator $k$-tone functions provided

Integral representations generalizing known functions derived

Abstract

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.