Kwong matrices and operator monotone functions on $(0,1)$

Abstract

In this paper we study positive operator monotone functions on $(0, 1)$ which have some differences from those on $(0, \infty):$ we show that for a concave operator monotone function $f$ on $(0, 1),$ the Kwong matrices $K_f(s_1, \dots, s_n)$ are positive semidefinite for all $n$ and $s_i \in (0, 1),$ and $f(s^p)^{1/p}$ for $p \in (0,1]$ and $s/f(s)$ are operator monotone. We also give a sufficient condition for the Kwong matrices to be positive semidefinite.