Ahlfors Five Island Theorem

Let  be a transcendental meromorphic function, and let , , ...,  be five simply connected domains in  with disjoint closures (Ahlfors 1932). Then there exists  and, for any , a simply connected domain such that  is a conformal mapping of  onto . If  has only finitely many poles, then "five" may be replaced by "three" (Ahlfors 1933).

REFERENCES:

Ahlfors, L. "Sur les fonctions inverses des fonctions méromorphes." Comptes Rendus Acad. Sci. Paris 194, 1145-1147, 1932. Reprinted in Lars Valerian Ahlfors: Collected Papers Volume 1, 1929-1955 (Ed. R. M. Shortt). Boston, MA: Birkhäuser, 149-151, 1982.

Ahlfors, L. "Über die Kreise die von einer Riemannschen Fläche schlicht überdeckt werden." Comm. Math. Helv. 5, 28-38, 1933. Reprinted in Lars Valerian Ahlfors: Collected Papers Volume 1, 1929-1955 (Ed. R. M. Shortt). Boston, MA: Birkhäuser, 163-173, 1982.

Bergweiler, W. "Iteration of Meromorphic Functions." Bull. Amer. Math. Soc. (N. S.) 29, 151-188, 1993.

Hayman, W. K. Meromorphic Functions. Oxford, England: Oxford University Press, 1964.

Nevanlinna, R. Analytic Functions. New York: Springer-Verlag, 1970.