On the Teichmüller-Wittich-Belinskij theorem (Q1099286)

This paper is a survey on the history of the Teichmüller-Wittich- Belinskij theorem (Theorem A) and its impact on value-distribution theory. As a first application the author presents his method of producing entire functions with preasigned asymptotic values on a given family of curves consistent with Ahlfors' theorem. By quasiconformal (qc) modifications a qc solution g is constructed and, using Theorem A, conditions for \(f=g\circ \phi\) to solve the problem are given, where \(\phi^{-1}\) is a qc homeomorphism \({\mathbb{C}}\to {\mathbb{C}}\) with the same complex dilatation as g a.e. The result is compared with Hayman's solution. Further following topics are treated: the connection of Theorem A with the inverse problem of Nevanlinna's theory and author's full solution by means of qc modification and of a generalization of Theorem A obtained in a joint work with A. Weitsman; another ``inverse'' application where using the qc modification the author determines the nature of a function F of finite order with \(\sum \delta (a,F)=2\); results on the behaviour of functions in angles, e.g. that obtained by A. E. Eremenko's on Valiron deficiencies.