Operator monotone functions and Löwner functions of several variables

Jim Agler, John E. McCarthy, Nicholas J. Young

Published 2010-09-20, updated 2013-12-18Version 3

We prove generalizations of L\"owner's results on matrix monotone functions to several variables. We give a characterization of when a function of $d$ variables is locally monotone on $d$-tuples of commuting self-adjoint $n$-by-$n$ matrices. We prove a generalization to several variables of Nevanlinna's theorem describing analytic functions that map the upper half-plane to itself and satisfy a growth condition. We use this to characterize all rational functions of two variables that are operator monotone.