Matrix Monotone Functions and Convexity

Abstract

Let $(a, b) \subset {\mathbb R}$ be an interval. A function $f: (a, b) \rightarrow {\mathbb R}$ is said to be monotone for $n \times n$ matrices if $f(A) \le f(B)$ whenever $A$ and $B$ are self-adjoint $n \times n$ matrices, $A \le B$ and their eigenvalues are in $(a, b)$. If a function is monotone for every matrix size, then it is called matrix monotone or operator monotone.