A note on Ahlfors’ theory of covering surfaces

Abstract

In his results on the theory of covering surfaces, Ahlfors [1] obtained theorems which are analogues of Nevanlinna's first and second fundamental theorems in the theory of meromorphic functions. For a given function f meromorphic in the plane, Ahlfors' theory unfortunately gives relatively little information about the behavior of the restriction of f to I zj < r for certain exceptional values of r. While observing that his second fundamental theorem did imply Picard's theorem, Ahlfors remarked that the existence of these exceptional r-values seemed to make it impossible to deduce Nevanlinna's second fundamental theorem from his by integration. Nevanlinna [3] in his treatise on meromorphic functions also stated that the exceptional r-values of Ahlfors' theory prevented one from obtaining the integrated form of the second fundamental theorem from the unintegrated form. Two attempts to derive the classical result from Ahlfors' second fundamental theorem have met with partial success. Both however lead to versions of the classical theorem for which there are exceptional r-values even for functions of finite order. It is the purpose of this note to show that Ahlfors' theory implies a form of the classical second fundamental theorem having no exceptional r-values for functions of finite order. The proof is in fact extremely elementary, yet seems to have been overlooked. We remind the reader of one form of Ahlfors' second fundamental theorem.