Operator log-convex functions and operator means

TL;DR

This paper characterizes operator log-convex functions on positive reals, showing they are precisely the operator monotone decreasing functions, and explores their relation to operator means and log-concavity.

Contribution

It provides a complete characterization of operator log-convex functions and links them to operator monotone decreasing functions and operator means.

Findings

Operator log-convex functions are exactly the operator monotone decreasing functions.

Several equivalent conditions involving operator means are established.

Operator log-concave functions are also analyzed.

Abstract

We study operator log-convex functions on $(0,\infty)$, and prove that a continuous nonnegative function on $(0,\infty)$ is operator log-convex if and only if it is operator monotone decreasing. Several equivalent conditions related to operator means are given for such functions. Operator log-concave functions are also discussed.