Mathematische Zeitschrift

, Volume 285, Issue 3–4, pp 1063–1089 | Cite as

Chordal Loewner chains with quasiconformal extensions

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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.

Keywords

Univalent function Quasiconformal extension Loewner chain Chordal Loewner equation Evolution family Loewner range 

Mathematics Subject Classification

Primary 30C62 Secondary 30C35 30D05 

Notes

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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© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Institutt for matematikk og naturvitenskapUniversitetet i StavangerStavangerNorway
  2. 2.Department of Applied ScienceYamaguchi UniversityUbeJapan

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