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).
Ahlfors Five Island Theorem
See also
Meromorphic Function, Transcendental FunctionExplore with Wolfram|Alpha
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.Referenced on Wolfram|Alpha
Ahlfors Five Island TheoremCite this as:
Weisstein, Eric W. "Ahlfors Five Island Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/AhlforsFiveIslandTheorem.html