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Univalent functions with quasiconformal extensions: Becker's class and estimates of the third coefficient.
- Source :
-
Proceedings of the American Mathematical Society . Sep2020, Vol. 148 Issue 9, p3927-3942. 16p. - Publication Year :
- 2020
-
Abstract
- We investigate univalent functions ƒ(z) = z+a2z2 + a3z3 + . . . in the unit disk \mathbb{D} extendible to k-q.c.(=quasiconformal) automorphisms of C. In particular, we answer a question on estimation of |a3|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* ∈ (0,1] such that every univalent function ƒ in D having a k-q.c. extension to 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 ∈ (0,1), to each finite Blaschke product there corresponds a univalent function ƒ in D that admits a Becker k-q.c. extension but no k'-q.c. extensions to C with k' < k. [ABSTRACT FROM AUTHOR]
- Subjects :
- *UNIVALENT functions
*ESTIMATES
Subjects
Details
- Language :
- English
- ISSN :
- 00029939
- Volume :
- 148
- Issue :
- 9
- Database :
- Academic Search Index
- Journal :
- Proceedings of the American Mathematical Society
- Publication Type :
- Academic Journal
- Accession number :
- 144503625
- Full Text :
- https://doi.org/10.1090/proc/15010