Uniqueness of Harmonic Mappings into Strictly Starlike Domains

Let Ω be a bounded simply connected domain containing a point w 0 and whose boundary is locally connected, be the open unit disc, and be an analytic function. It is known that the elliptic differential equation admits a one-to-one solution normalized by f (0) = w 0 , f z (0) > 0, and maps into Ω such that (i) the unrestricted limit exists and belongs to ∂Ω for all but a countable subset E of the unit circle , (ii) f * is a continuous function on and for every e is ∈ E the one-sided limits and 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 and . In this paper it is shown that this solution is unique if Ω is a strictly starlike domain with respect to ω 0 whose boundary is rectifiable.