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- W2945794927 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 $mathbb C$. In particular, we answer a question on estimation of $|a_3|$ raised by Kuhnau 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 $mathbb C$ with $kleqslant k_*$ 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 $kin(0,1)$, 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 $mathbb C$ with $k'<k$." @default.
- W2945794927 created "2019-05-29" @default.
- W2945794927 creator A5062439594 @default.
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- W2945794927 date "2019-05-21" @default.
- W2945794927 modified "2023-09-23" @default.
- W2945794927 title "Univalent functions with quasiconformal extensions: Becker's class and estimates of the third coefficient" @default.
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- W2945794927 doi "https://doi.org/10.48550/arxiv.1905.08666" @default.
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