Functions of bounded characteristic in n-connected domains

Abstract

In this paper we define a metric d on the Nevanlinna class N(G) of an n-connected domain G. We show that ( N, d) is a complete translation invariant metric space in which multiplication by a constant is not continuous. We also show that N(G) is disconnected. Our main theorem characterizes the component of the origin of N(G), extending earlier work of J. Shapiro and A. Shields and of J. Roberts for the unit disk. We use the factorization theorem of R. Coifman and G. Weiss to show that the component of the origin of N(G) is equal to the set of all functions f in N(G) for which the positive part of the singular measure of the least harmonic majorant of log ¦ f¦ is continuous.