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
