The Sharp Form of the Principle of Hyperbolic Measure

Abstract

Schwarz's Lemma and its generalizations, the Lindel6f Principle and the Principle of Hyperbolic Measure, have provided an important tool for the study of the theory of functions. From a knowledge of the range of values of an analytic function F(z) in a given domain one is able, with their aid, to obtain estimates on the range of values which the function and its derivative can assume in any subdomain of the given domain. However, these estimates are not in general sharp when the domain of definition of the analytic function F(z) is multiply-connected. Thus considerable interest has been aroused in the problem of finding sharp estimates corresponding to the estimates of these fundamental principles. The work of Teichmiiller [13], Grunsky [6] and Ahlfors [1] has led to a method which yields the precise estimates and the associated extremal functions in the case where bounds on the modulus of the function F(z) are given. This method consists in applying Green's formula to the harmonic function log I F(z) 1. However, if the values of F(z) are assumed to lie in a multiply-connected domain, rather than in a circle corresponding to an upper bound on i F(z) 1, the above method for obtaining sharp estimates breaks down. The difficulty lies in the fact that the Green's function of a multiply-connected domain has critical points. In this paper we attack the general problem of obtaining estimates on F(z) when its domain of definition and range of values are both known multiplyconnected domains. We utilize the technique of Teichmfiller, Grunsky and Ahlfors, while at the same time we are able to avoid the difficulties inherent in the general problem by using the variational method developed by Schiffer [11, 12]. Our results will underline the fact that Schiffer's method is by no means restricted to the study of schlicht or p-valued functions. Our approach will also show that many extremal problems in the conformal mapping of multiply-connected domains can be treated in almost the same manner as one treats similar problems for simply-connected domains.