Nevanlinna Theory in an Annulus

Abstract

A concrete presentation of Nevanlinna theory in a domain z: ¦z¦≥R has been offered by Bieberbach. He applied Green’s formula to prove the first main theorem and the lemma of the logarithmic derivative for meromorphic functions outside a disc of radius R. Apart from this work, Nevanlinna theory outside a disc has been considered in the form of brief remarks only in various articles. The purpose of this paper is to collect these comments into a coherent presentation, and to generalize these results for functions meromorphic in an open annulus. We define annulus versions of the Nevanlinna functions allowing accumulation of poles also to the inner boundary, and prove analogues of Nevanlinna’s main theorems including the lemma of the logarithmic derivative. Instead of using Green’s formula, we base our reasoning on a theorem due to Valiron.