Uniqueness of Harmonic Mappings into Strictly Starlike Domains

Abstract

Let Ω be a bounded simply connected domain containing a point w0 and whose boundary is locally connected, \(\mathbb{D} = \left\{ {z:\left| z \right| 0, and maps \(\mathbb{D}\) into Ω such that (i) the unrestricted limit \(f^* \left( {e^{it} } \right) = \lim _{z \to e^{it} } f(z)\) exists and belongs to ∂Ω for all but a countable subset E of the unit circle \(\mathbb{T} = \partial \mathbb{D}\), (ii) f* is a continuous function on \(\mathbb{T}\backslash E\) and for every eis ∈ E the one-sided limits \(\lim _{t \to s^ + } f^* \left( {e^{it} } \right)\) and \(\lim _{t \to s^ - } f^* \left( {e^{it} } \right)\) exist, belong to ∂Ω, and are distinct, and (iii) the cluster set of f at eis ∈ E is the straight line segment joining the one-sided limits \(\lim _{t \to s^ + } f^* \left( {e^{it} } \right)\) and \(\lim _{t \to s^ - } f^* \left( {e^{it} } \right)\). In this paper it is shown that this solution is unique if Ω is a strictly starlike domain with respect to ω0 whose boundary is rectifiable.