  In this paper we establish a multivariable non-commutative generalization of
L\"owner's classical theorem from 1934 characterizing operator monotone
functions as real functions admitting analytic continuation mapping the upper
complex half-plane into itself. The non-commutative several variable theorem
proved here characterizes several variable operator monotone functions, not
assumed to be free analytic or even continuous, as free functions that admit
free analytic continuation mapping the upper operator poly-halfspace into the
upper operator halfspace over an arbitrary Hilbert space. We establish a new
abstract integral formula for them using non-commutative topology, matrix
convexity and LMIs. The formula represents operator monotone and operator
concave free functions as a conditional expectation of a Schur complement of a
linear matrix pencil on a tensor product operator algebra. This formula is new
even in the one variable case. The results can be applied to any of the various
multivariable operator means that has been constructed in the last three
decades or so, including the Karcher mean. Thus we obtain an explicit, closed
formula for these operator means of several positive operators.
