Inequalities for the hyperbolic metric and applications to geometric function theory

Abstract

INEQUALITIES FOR THE HYPERBOLIC METRIC AND APPLICATIONS TO GEOMETRIC FUNCTION THEORY David Minda i. Introduction. In [ii] we derived a reflection principle for the hyperbolic metric on Riemann surfaces and presented various applications. This paper is partly an expository version of [ii]. Here, we shall restrict our attention to plane regions and establish the reflection principle only in the special case of reflecting across a straight line or circle. This streamlines the presentation and lets us emphasize the purely aeometric aspects of the reflection principle. The reflection Principle itself is a simnle consequence of a slight modification o6 Ahlfors' Lem~a. The geometric applications of the reflection nrinciDle are clear, esnecially when one does not care about establishing the sharpness of the results. Analytic tools, which can obscure the simple geometric ideas involved, seem to be needed to obtain the sharpness of the results. The reader is referred to [ii] for the analytic details needed to establish the sharpness of various results. Our first appli- cation of the reflection principle is the derivation of various mono- tenicity properties of the hyperbolic metric. One of these, in con- junction with our modified version of Ahlfors' Lemma, leads to a short proof of the sharp form of Landau's Theorem. There is also an inter- pretation of the reflection principle in terms of convexity relative to hyperbolic geometry. This leads to a number of interesting applications for euclidean convex regions. 2. A Modified Version of Ahlfors' Lemma. We reouire a variant of the usual form of Ahlfors' Lemma ([2], [3, p. 13]). Pommerenke ([14],[15]) established a similar result and used it to study normal meromorphic functions. However, he expressed his result in functiontheoretic terms rather than the differential-geometric language which is more suitable for our purposes.