Computational Methods and Function Theory 8 (2008), No. 2, 433--446
Copyright Heldermann Verlag 2008
Daoud Bshouty
daoud@tx.technion.ac.il , Technion, Department of Mathematics, Haifa 32000, Israel.
Abdallah Lyzzaik
lyzzaik@aub.edu.lb , American University of Beirut, Department of Mathematics, Beirut, Lebanon.
Let $\Omega$ be a bounded simply connected domain containing a point $w_0$ and whose boundary is locally connected, $\mathbb{D}=\{z\colon |z|<1\}$ be the open unit disc, and $\omega: \mathbb{D}\to \mathbb{D}$ be an analytic function. It is known that the elliptic differential equation $\overline{f_{\overline{z}}}/f_z=\omega$ admits a one-to-one solution normalized by $f(0)=w_0,$ $f_z(0)>0,$ and maps $\mathbb{D}$ into $\Omega$ such that (i) the unrestricted limit $f^*(e^{it})=\lim_{z\to e^{it}}f(z)$ exists and belongs to $\partial \Omega$ for all but a countable subset $E$ of the unit circle $\mathbb{T}=\partial \mathbb{D},$ (ii) $f^*$ is a continuous function on $\mathbb{T}\setminus E$ and for every $e^{is}\in E$ the one-sided limits $\lim_{t\to s^+}f^*(e^{it})$ and $\lim_{t\to s^-}f^*(e^{it})$ exist, belong to $\partial \Omega,$ and are distinct, and (iii) the cluster set of $f$ at $e^{is}\in E$ is the straight line segment joining the one-sided limits $\lim_{t\to s^+}f^*(e^{it})$ and $\lim_{t\to s^-}f^*(e^{it})$. In this paper it is shown that this solution is unique if $\Omega$ is a strictly starlike domain with respect to $w_0$ whose boundary is rectifiable.
Keywords: Harmonic mapping, analytic dilatation, strictly starlike domains, elliptic differential equation.
MSC 2000: Primary 30C62; Secondary 30G12, 30G20.
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