Mobius transformations, the Caratheodory metric, and the objects of   complex analysis and potential theory in multiply connected domains
  It is proved that the family of Ahlfors extremal mappings of a multiply connected region in the plane onto the unit disc can be expressed as a rational combination of two fixed Ahlfors mappings in much the same way that the family of Riemann mappings associated to a simply connected region can be expressed in terms of a single such map. The formulas reveal that this family of mappings extends to the double as a real analytic function of both variables. In particular, the infinitesimal Caratheodory metric will be expressed in strikingly simple terms. Similar results are proved for the Green's function, the Poisson kernel, and the Bergman kernel. 
Steven R. Bell