Operator log-convex functions and operator means

Tsuyoshi Ando, Fumio Hiai

Published 2009-11-27, updated 2014-12-20Version 5

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.