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Quasiconformal extensions, Loewner chains, and the \(\lambda \)-Lemma

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Abstract

Becker (J Reine Angew Math 255:23–43, 1972) discovered a sufficient condition for quasiconformal extendibility of Loewner chains. Many known conditions for quasiconformal extendibility of holomorphic functions in the unit disk can be deduced from his result. We give a new proof of (a generalization of) Becker’s result based on Slodkowski’s Extended \(\lambda \)-Lemma. Moreover, we characterize all quasiconformal extensions produced by Becker’s (classical) construction and use that to obtain examples in which Becker’s extension is extremal (i.e. optimal in the sense of maximal dilatation) or, on the contrary, fails to be extremal.

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Authors and Affiliations

  1. Institutt for matematikk og fysikk, Universitetet i Stavanger, 4036, Stavanger, Norway

    Pavel Gumenyuk

  2. Department of Mathematics and Statistics, University of Helsinki, P.O. Box 68 (Gustaf Hällströmin katu 2b), 00014, Helsinki, Finland

    István Prause

Authors
  1. Pavel Gumenyuk
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  2. István Prause
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Correspondence to István Prause.

Additional information

In memory of Alexander Vasil’ev.

Pavel Gumenyuk is partially supported by Ministerio de Economía y Competitividad (Spain) Project MTM2015-63699-P. István Prause is supported by Academy of Finland Grants 1266182 and 1303765.

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Gumenyuk, P., Prause, I. Quasiconformal extensions, Loewner chains, and the \(\lambda \)-Lemma. Anal.Math.Phys. 8, 621–635 (2018). https://doi.org/10.1007/s13324-018-0247-3

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  • DOI: https://doi.org/10.1007/s13324-018-0247-3

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