Uniqueness of Harmonic Mappings into Strictly Starlike Domains

Abstract

Let Ω be a bounded simply connected domain containing a point w ₀ and whose boundary is locally connected, $\mathbb{D} = \left\{ {z:\left| z \right| < 1} \right\}$ be the open unit disc, and $\omega :\mathbb{D} \to \mathbb{D}$ be an analytic function. It is known that the elliptic differential equation ${{\overline {f_{\bar z} } } \mathord{\left/ {\vphantom {{\overline {f_{\bar z} } } {f_z = \omega }}} \right. \kern-\nulldelimiterspace} {f_z = \omega }}$ admits a one-to-one solution normalized by f(0) = w ₀, f _z(0) > 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 e ^is ∈ 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 e ^is ∈ 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.