Schwarz’s lemma and the Szegö kernel function

This paper is concerned with extremal problems in the family of bounded analytic functions in a multiply-connected domain D, and it is concerned with extremal problems for the mean modulus ^c of meromorphic functions f in D taken over the boundary C of D. These two types of problems are shown to be closely related, and solutions are obtained simultaneously for both types by a method of contour integration. An altogether analogous method was exploited by Grunsky in his thesis Thus it is interesting to remark that the fundamental distortion theorems of schlicht conformai mapping theory have been developed by Grunsky by a method which we are able to apply here to obtain the fundamental distortion theorems for bounded functions. It will appear, then, that the generalization of Schwarz's lemma to multiply-connected domains and the generalization of the Koebe distortion theorem can be carried out by a unified technique(3).