Note on L\"owner's theorem on matrix monotone functions in several   commuting variables of Agler, McCarthy and Young
  In this brief note, we show that the hypotheses of L\"owner's theorem on matrix monotonicity in several commuting variables as proved by Agler, McCarthy and Young can be significantly relaxed. Specifically, we extend their theorem from continuously differentiable locally matrix monotone functions to arbitrary locally matrix monotone functions using mollification techniques. 
J. E. Pascoe
math.FA math.CV