Factorizations of analytic self-maps of the upper half-plane
  We extend a factorization due to Krein to arbitrary analytic functions from the upper half-plane to itself. The factorization represents every such function as a product of fractional linear factors times a function which, generally, has fewer zeros and singularities than the original one. The result is used to construct functions with given zeros and poles on the real line. 
Hari Bercovici and Dan Timotin
math.CV