Abstract
In 1972, Becker (J Reine Angew Math 255:23–43, 1972), discovered a construction of quasiconformal extensions making use of the classical radial Loewner chains. In this paper we develop a chordal analogue of Becker’s construction. As an application, we establish new sufficient conditions for quasiconformal extendibility of holomorphic functions and give a simplified proof of one well-known result by Becker and Pommerenke (J Reine Angew Math 354:74–94, 1984) for functions in the half-plane.
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Notes
A one-parameter semigroup \((\phi _t)\) is called trivial if \(\phi _t=\mathsf{id}_\mathbb {D}\) for all \(t\ge 0\).
These definitions and basic facts mentioned below can be found in [47, Chapter 6].
That is a ball w.r.t. the hyperbolic distance in D.
The same can be done in the context of Theorem 5.6. However, it would not lead to any improvement.
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Acknowledgments
The authors are grateful to Professors Dmitri Prokhorov and Toshiyuki Sugawa for interesting discussions concerning Corollary 5.17. We would also like to thank the anonymous referee for thorough reading of the manuscript and helpful comments.
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P. Gumenyuk: Partially supported by the FIRB grant Futuro in Ricerca “Geometria Differenziale Complessa e Dinamica Olomorfa” n. RBFR08B2HY.
I. Hotta: Supported by JSPS KAKENHI Grant Numbers 13J02250, 26800053.
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Gumenyuk, P., Hotta, I. Chordal Loewner chains with quasiconformal extensions. Math. Z. 285, 1063–1089 (2017). https://doi.org/10.1007/s00209-016-1738-2
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DOI: https://doi.org/10.1007/s00209-016-1738-2
Keywords
- Univalent function
- Quasiconformal extension
- Loewner chain
- Chordal Loewner equation
- Evolution family
- Loewner range