Abstract
Let Ω be a hyperbolic domain in the complex plane C, let ρΩ be the density of the Poincaré metric in Ω, and let βΩ=1/ρΩ. For g analytic in Ω we set ||g||Ω=sup βΩ(w)|g(w)|, w∈Ω. Let S(Ω) be the family of functions f analytic and univalent in Ω. Criteria in terms of the partial derivatives of βΩ for Ω to satisfy sup ||f''/f'||Ω<+∞, where f ranges over S(Ω), are given. For example, sup βΩ(w)|(βΩ)ww(w)|<+∞, w∈Ω. If f∈S(Ω) is isolated in the sense that there is an ε>0 such that 0<||f''/f'−g''/g'||Ω<ε for no g∈S(Ω), then C{\backslash}f(Ω) is of zero area. The domain Ω is simply connected if sup βΩ(w)|(βΩ)ww(w)|{≤}1, w∈Ω, and Ω is convex (hence simply connected) if and only if sup |(βΩ)w(w)|=1, w∈Ω.
