Quasiconformal reflections and mirrors (Q2782549)

The paper under review deals with quasiconformal reflections across general subsets \(E\) of the extended complex plane \(\widehat C\). A quasiconformal reflection across \(E\) is an orientation reversing quasiconformal self-map of \(\widehat C\) which the set \(E\) fixed pointwise. \textit{R. Kühnau} [Complex Analysis, Artic. dedicated to Albert Pfluger, 139-156 (1998; Zbl 0659.30016)] posed the question for which sets \(E\) a quasiconformal reflection exists. The main result of the paper is that a quasiconformal reflection exists if \(E\) is a subset of a quasicircle \(L\) and it holds \(Q_L\leq Q^4_E\), where \(Q_L\) and \(Q_E\) are the quasiconformal dilatation of \(L\) and \(E\). For the main result see also: \textit{S. L. Krushkal} [Sib. Mat. Zh. 40, No. 4, 880-892 (1999; Zbl 0933.30014)].NEWLINENEWLINEFor the entire collection see [Zbl 0980.00031].