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abstract: 'We investigate univalent functions $f(z)=z+a_2z^2+a_3z^3+\ldots$ in the unit disk ${\mathbb{D}}$ extendible to $k$-q.c.(=quasiconformal) automorphisms of $\C$. In particular, we answer a question on estimation of $|a_3|$ raised by Kühnau and Niske \[Math. Nachr. [**78**]{} (1977) 185–192\]. This is one of the results we obtain studying univalent functions that admit q.c.-extensions via a construction, based on Loewner’s parametric representation method, due to Becker \[J. Reine Angew. Math. [**255**]{} (1972) 23–43\]. Another problem we consider is to find the maximal $k_*\in(0,1]$ such that every univalent function $f$ in ${\mathbb{D}}$ having a $k$-q.c. extension to $\C$ with $k\le k_*$ admits also a Becker q.c.-extension, possibly with a larger upper bound for the dilatation. We prove that $k_*>1/6$. Moreover, we show that in some cases, Becker’s extension turns out to be the optimal one. Namely, given any $k\in(0,1)$, to each finite Blaschke product there corresponds a univalent function $f$ in ${\mathbb{D}}$ that admits a Becker $k$-q.c. extension but no $k''$-q.c. extensions to $\C$ with $k''<k$.'
address:
- 'Institutt for matematikk og fysikk, Universitetet i Stavanger, 4036 Stavanger, '
- 'Department of Applied Science, Yamaguchi University, 2-16-1 Tokiwadai, Ube 755-8611, '
author:
- Pavel Gumenyuk
- 'Ikkei Hotta$^\dag$'
title: 'Univalent functions with quasiconformal 