Global Mapping Properties of Some Functions of Class S

Abstract

The Lemma of Schwarz is one of the most surprising results in complex analysis in the sense that some very weak conditions on an analytic function in the unit disk |z| < 1 imply a very strict behavior of that function in the respective disk. What about the behavior of the function outside the unit disk? This is the question we deal with in this paper. The theory we presented in some previous publications was about univalent functions, not necessarily in the unit disk, but in the most general setting, namely in the fundamental domains of arbitrary analytic functions. Naturally, connections can be expected between the two fields of complex analysis. The purpose of this paper is to explore these connections and take advantage of the well established theory of univalent functions in order to advance the theory of fundamental domains.