On the extremality for Teichmüller mappings
A quasiconformal mapping \(f\) of the unit disk \(E\) is called a Teichmüller map if the complex dilatation of \(f\) coincides (almost everywhere) on \(E\) with a function of the form \(k \overline \varphi/ |\varphi |\), where \(0 < k < 1\), and \(\varphi\) is a holomorphic function on \(E\). There have been many investigations about the question of whether a Teichmüller mapping is extremal for its boundary values, and, if so, whether it is the unique extremal in that class. Under certain technical growth assumptions on \(\varphi\), the author shows that the corresponding Teichmüller mapping is extremal. Earlier results of this type are due to Reich and Strebel, and others (see the references). The author applies his result to show that there exist functions \(\varphi\) that are not members of the Bers-Nehari Banach space, but for which the Teichmüller mapping is extremal.
scientific article; zbMATH DE number 784739