On asymptotic values of meromorphic functions

Abstract

The following theorem ofLindelöf is known: Iff (z) is meromorphic in |z|<1, and iff (z) admits two distinct asymptotic values at some pointP of |z|=1, thenf (z) assumes infinitely often in any neigh borhood ofP all values of the extended complex plane with at most two possible exceptions.

The purpose of this note is to extend Lindelöf's theorem to Riemannian surfaces. Our extension of Lindelöf's theorem includes the fact that the conclusion of Lindelöf's theorem holds, iff (z) is meromorphic in |z|<∞, and iff (z) admits two distinct asymptotic values atz=∞.