A new proof of the Ahlfors five islands theorem

Abstract. We deduce the Ahlfors five islands theorem from a corresponding result of Nevanlinna concerning perfectly branched values, a rescaling lemma for non-normal families and ' an existence theorem for quasiconformal mappings. We also give a proof of Nevanlinna's result based on the rescaling lemma and a version of Schwarz's lemma. 1 Results of Ahlfors, Nevanlinna and Zalcman Let D j, j = 1,..., 5, be simply-connected domains on the Riemann sphere with piecewise analytic boundary and pairwise disjoint closures. Let D C C be a domain and denote by 3rA(D) = fa (D, {Dj}~=I) the family of all meromorphic functions f: D ~ C with the property that no subdomain of D is mapped conformally onto one of the domains Dj by f. (If there is such a subdomain, then it is called a simple island over D s.) The Ahlfors five islands theorem can now be stated as follows. Theorem A.1.-~A (D, {Ds}~=I) is normal. Closely related is the following result. 5 Theorem A.2. 3VA (C, {Dj}S=I) contains only the constant functions. These results come from the Ahlfors theory of covering surfaces; see [5]