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.