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Abstract <p>In this paper, Loewner chain theory and criteria of univalency and quasiconformal extensibility for analytic functions on the unit disk are investigated. By replacing the pre-Schwarzian derivative with the Schwarzian derivative, we obtain results analogous to those of Hotta [Publ. Math. Debrecen. 82 (2013), pp. 473-483]. Furthermore, by constructing various Loewner chains, we generalize and provide uniform proofs of several known criteria of univalency and quasiconformal extensibility. As an application of Loewner chains, we make a modification of the Ahlfors’s criterion involving a complex constant <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11957_2025_7239_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="11" /> </InlineMediaObject> <EquationSource Format="TEX">\(c\)</EquationSource> <!--ContMath2460199Guo-m1--> </InlineEquation>.</p>

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Applications of Loewner Chains in Univalent Functions and Quasiconformal Extensions

  • 期刊论文

  • 发表日期：

    2025年10月10日

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  • S. L. Guo,
  • J. H. Fan

摘要

Abstract

In this paper, Loewner chain theory and criteria of univalency and quasiconformal extensibility for analytic functions on the unit disk are investigated. By replacing the pre-Schwarzian derivative with the Schwarzian derivative, we obtain results analogous to those of Hotta [Publ. Math. Debrecen. 82 (2013), pp. 473-483]. Furthermore, by constructing various Loewner chains, we generalize and provide uniform proofs of several known criteria of univalency and quasiconformal extensibility. As an application of Loewner chains, we make a modification of the Ahlfors’s criterion involving a complex constant \(c\) .