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<p>In this paper, we study the concepts of normal functions and <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="605_2025_2093_Article_IEq1.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="TEX">\(\varphi \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>φ</mi> </math></EquationSource> </InlineEquation>-normal functions in the framework of planar harmonic mappings. We establish the harmonic mapping counterpart of the well-known Zalcman-Pang lemma and as a consequence, we prove that a harmonic mapping whose spherical derivative is bounded away from zero is normal. Furthermore, we introduce the concept of the extended spherical derivative for harmonic mappings and establish several sufficient conditions for a harmonic mapping to be <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="605_2025_2093_Article_IEq1.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="TEX">\(\varphi \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>φ</mi> </math></EquationSource> </InlineEquation>-normal. The results in this paper improve and generalize some recent results obtained by Deng et al. (Monatsh Math 193 (2020), 605-621) and Bohra et al. Monatsh Math. 206, 797–808 (2025).</p>

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Results on normal harmonic and \(\varphi \)-normal harmonic mappings

  • 期刊论文

  • 发表日期：

    2025年6月21日

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  • Nikhil Bharti,
  • Nguyen Van Thin

摘要

In this paper, we study the concepts of normal functions and \(\varphi \) φ -normal functions in the framework of planar harmonic mappings. We establish the harmonic mapping counterpart of the well-known Zalcman-Pang lemma and as a consequence, we prove that a harmonic mapping whose spherical derivative is bounded away from zero is normal. Furthermore, we introduce the concept of the extended spherical derivative for harmonic mappings and establish several sufficient conditions for a harmonic mapping to be \(\varphi \) φ -normal. The results in this paper improve and generalize some recent results obtained by Deng et al. (Monatsh Math 193 (2020), 605-621) and Bohra et al. Monatsh Math. 206, 797–808 (2025).