Uniform estimates for the hyperbolic metric and Euclidean distance to the boundary.

Abstract

Throughout this article,D is a proper subdomain of the complex plane C possessing at least two finite boundary points, usually termed a hyperbolic domain. Each suchD carries constant negative curvature metrics, and we let λD denote the scale factor or density for the maximal constant curvature −1 metric. We call λD the Poincare hyperbolic metric for D; it can be defined by λD(z) = λB(ζ)/|p′(ζ)| = 2/(1− |ζ|2)|p′(ζ)|, where z = p(ζ) and p : B→ D is any holomorphic covering projection from the unit disk B = {|ζ| < 1} onto D. See [BP; HM; M1; M2] and their references for basic properties of the Poincare metric. An elementary exercise using Schwarz’s lemma shows thatλD satisfies a domain monotonicity property, from which we easily conclude that