  Given a function $f: (a,b) \rightarrow \mathbb{R},$ L\"owner's theorem states
$f$ is monotone when extended to self-adjoint matrices via the functional
calculus, if and only if $f$ extends to a self-map of the complex upper half
plane. In recent years, several generalizations of L\"owner's theorem have been
proven in several variables. We use the relaxed Agler, McCarthy and Young
theorem on locally matrix monotone functions in several commuting variables to
generalize results in the noncommutative case. Specifically, we show that a
real free function defined over an operator system must analytically continue
to a noncommutative upper half plane as map into another noncommutative upper
half plane.
