Univalent functions with quasiconformal extensions: Becker’s class and estimates of the third coefficient
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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 $\mathbb {C}$. In particular, we answer a question on estimation of $|a_3|$ raised by Kühnau and Niske [Math. Nachr. 78 (1977), pp. 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), pp. 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 $k<k_*$ 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 $\mathbb {C}$ with $k’<k$.References
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Additional Information
- Pavel Gumenyuk
- Affiliation: Department of Mathematics, Politecnico di Milano, via E. Bonardi 9, 20133 Milan, Italy
- MR Author ID: 706440
- Email: pavel.gumenyuk@polimi.it
- Ikkei Hotta
- Affiliation: Department of Applied Science, Yamaguchi University, 2-16-1 Tokiwadai, Ube 755-8611, Japan
- MR Author ID: 884744
- Email: ihotta@yamaguchi-u.ac.jp
- Received by editor(s): June 17, 2019
- Received by editor(s) in revised form: January 15, 2020
- Published electronically: May 8, 2020
- Additional Notes: The second author was supported by JSPS KAKENHI Grant Number 17K14205.
- Communicated by: Filippo Bracci
- © Copyright 2020 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 148 (2020), 3927-3942
- MSC (2010): Primary 30C62; Secondary 30C35, 30C50, 30C75, 30D05
- DOI: https://doi.org/10.1090/proc/15010
- MathSciNet review: 4127837