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Grunsky inequalities and quasiconformal extension

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Abstract

The Grunsky coefficient inequalities play a crucial role in various problems and are intrinsically connected with the integrable holomorphic quadratic differentials having only zeros of even order. For the functions with quasi-conformal extensions, the Grunsky constant ℵ(f) and the extremal dilatationk(f) are related by ℵ(f)≤k(f). In 1985, Jürgen Moser conjectured that any univalent functionf(z)=z+b 0+b 1 z −1+… on Δ*={|z|>1} can be approximated locally uniformly by functions with ℵ(f)<k(f). In this paper, we prove a theorem confirming Moser’s conjecture, which sheds new light on the features of Grunsky coefficients.

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Author information

Authors and Affiliations

  1. Department of Mathematics, Bar-Ilan University, 52900, Ramat-Gan, Israel

    Samuel Krushkal

  2. Institut für Analysis, Fachbereich Mathematik und Informatik, Martin-Luther Universität Halle-Wittenberg, D-06099, Halle(Saale), Germany

    Reiner Kühnau

Authors
  1. Samuel Krushkal
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  2. Reiner Kühnau
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Corresponding author

Correspondence to Samuel Krushkal.

Additional information

In memory of Jürgen Moser

The research was supported by the RiP program of the Volkswagen-Stiftung in the Mathematisches Forschungsinstitut Oberwolfach.

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Krushkal, S., Kühnau, R. Grunsky inequalities and quasiconformal extension. Isr. J. Math. 152, 49–59 (2006). https://doi.org/10.1007/BF02771975

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  • DOI: https://doi.org/10.1007/BF02771975

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