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<p>In this paper, we investigate the properties of hyperbolic metrics on various domains to derive the Schwarz-Pick type inequalities for harmonic quasiconformal mappings (shortly HQC mappings). As applications, we establish versions of the Koebe theorem and hyperbolic distortion results for same class of real and complex harmonic functions defined on the unit disk <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathbb D\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">D</mi> </math></EquationSource> </InlineEquation> and harmonic mappings <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(u: R \rightarrow (-1,1)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>u</mi> <mo>:</mo> <mi>R</mi> <mo stretchy="false">→</mo> <mo stretchy="false">(</mo> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, where <i>R</i> is a Riemann surface with a complete conformal metric of Gaussian curvature bounded below by a negative constant.</p>

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Some Estimates for Hyperbolic Derivative for HQC Mappings and Applications

  • 期刊论文

  • 发表日期：

    2025年11月5日

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  • Miljan Knežević,
  • Miodrag Mateljević,
  • Marek Svetlik

摘要

In this paper, we investigate the properties of hyperbolic metrics on various domains to derive the Schwarz-Pick type inequalities for harmonic quasiconformal mappings (shortly HQC mappings). As applications, we establish versions of the Koebe theorem and hyperbolic distortion results for same class of real and complex harmonic functions defined on the unit disk \(\mathbb D\) D and harmonic mappings \(u: R \rightarrow (-1,1)\) u : R → ( - 1 , 1 ) , where R is a Riemann surface with a complete conformal metric of Gaussian curvature bounded below by a negative constant.