On uniqueness of a linear quasiconformal mapping on a starshaped domain. (Russian) Zbl 0694.30017

Let \({\mathcal J}\) be a simply connected domain and its boundary contain more than one point. Let G be a bounded starshaped domain with rectifiable boundary \(\Gamma\). In this paper it is proved that the solution of the equation \[ w_{\bar z}=\mu (z)w_ z+\nu (z)\bar w_{\bar z}, \] homeomorphic mapping from \({\mathcal J}\) to G and satisfying \(w(z_ j)=w_ j(j=0,1)\), is unique. Here \(\mu\) (z) and \(\nu\) (z) are measurable almost everywhere in \({\mathcal J}\) and satisfy the inequality \[ | \mu (z)| +| \nu (z)| \leq k_ 0<1, \] \(z_ 0\) and \(w_ 0\) are certain points on the boundary of \({\mathcal J}\) and \(\Gamma\) respectively, \(z_ 1\) and \(w_ 1\) are given inner points of \({\mathcal J}\) and G respectively.
Reviewer: Xu Zhenyuan


30C62 Quasiconformal mappings in the complex plane