Hyperbolic Distortion of Conformal Maps at Corners

Abstract

We consider conformal self-maps φ of the unit disk \(\mathbb{D}\) onto simply connected domains. We assume φ is continuous in a neighborhood of a point \(\zeta\in\partial\mathbb{D}\) , with φ(ζ) of modulus one, and that \(\partial\varphi(\mathbb{D})\) has a corner at φ(ζ). We prove that the modulus of the hyperbolic derivative of φ tends to a limit along certain simple curves in the disk that end at ζ non-tangentially. Moreover, we prove that the value of this limit depends only on the geometry of the corner and on the angle of approach to ζ. Our proof is based on a constructive approximation of the domain \(\varphi (\mathbb{D})\) by more special domains.