Piling Structure of Families of Matrix Monotone Functions and of Matrix Convex Functions

Abstract

Given a nontrivial open interval I on the real line, we consider the set P_n(I) (resp. K_n(I)) of all n-monotone (resp. n-convex) functions defined on I. Then $\begin{array}{lll} \left\{ {P_n \left( I \right)} \right\}_{n = 1}^\infty \left( {resp.\,\left\{ {K_n \left( I \right)} \right\}_{n = 1}^\infty } \right) \end{array}$ is a decreasing sequence of sets “piled” on their intersection, the set of all operator monotone functions P_∞(I) (resp. the set of all operator convex functions K_∞(I)) on I. In this article we give criteria for a function to belong to such a set, and we describe the gap between the sets for n and n + 1. In fact, for every n we provide abundant examples of n-monotone (resp. n-convex) functions in the gap between P_n(I) and P_n+1(I) (resp. K_n(I) and K_n+1(I)). When I is finite we show that this gap contains polynomials of degree 2n–1 and 2n (resp. 2n and 2n +1).