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<p>In this paper, we investigate several properties of a harmonic mapping <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(f=\mathcal {P}[F]\)</EquationSource> </InlineEquation> on the unit disk with boundary function <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(F\in {L^{p}(\textrm{T})}\)</EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(p\in {[1, \infty ]}\)</EquationSource> </InlineEquation>. The coefficients of <i>f</i> are initially estimated, and these estimates are directly employed in analysis of the Landau theorem. Subsequently, the Schwarz lemma for the harmonic mapping <i>f</i> is established by applying certain properties of Gauss hypergeometric functions. This work generalizes the classical Schwarz lemma to bounded harmonic mappings. As an application, two new versions of the Schwarz-Pick lemmas for the harmonic mapping <i>f</i> are also discussed.</p>

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Estimates on the Schwarz Lemma and Landau Theorem for a Harmonic Mapping with a Given Boundary Function in Lebesgue Space

  • 期刊论文

  • 发表日期：

    2026年1月3日

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  • Qingtian Shi,
  • Xizhen Li,
  • Xiaoli Lian

摘要

In this paper, we investigate several properties of a harmonic mapping \(f=\mathcal {P}[F]\) on the unit disk with boundary function \(F\in {L^{p}(\textrm{T})}\) and \(p\in {[1, \infty ]}\) . The coefficients of f are initially estimated, and these estimates are directly employed in analysis of the Landau theorem. Subsequently, the Schwarz lemma for the harmonic mapping f is established by applying certain properties of Gauss hypergeometric functions. This work generalizes the classical Schwarz lemma to bounded harmonic mappings. As an application, two new versions of the Schwarz-Pick lemmas for the harmonic mapping f are also discussed.