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<p>In this paper, we establish several sharp versions of Bohr’s inequalities for the class of <i>K</i>-quasiconformal sense-preserving harmonic mappings on the unit disk := {<i>z</i> ∈ ℂ : <i>|z| &lt;</i> 1} by using a sequence <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\({\left\{{\psi }_{n}\left(r\right)\right\}}_{n=0}^{\infty }\)</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mfenced close="}" open="{"> <msub> <mi>ψ</mi> <mi>n</mi> </msub> <mfenced close=")" open="("> <mi>r</mi> </mfenced> </mfenced> <mrow> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mi>∞</mi> </msubsup> </math></EquationSource> </InlineEquation> of nonnegative continuous functions defined on [0<i>,</i> 1) and such that the series <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\({\sum }_{n=0}^{\infty }{\psi }_{n}\left(r\right)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mo>∑</mo> <mrow> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mi>∞</mi> </msubsup> <msub> <mi>ψ</mi> <mi>n</mi> </msub> <mfenced close=")" open="("> <mi>r</mi> </mfenced> </mrow> </math></EquationSource> </InlineEquation> converges locally uniformly in [0<i>,</i> 1)<i>.</i> As an application, we deduce several well-known results, as well as numerous improved and refined Bohr’s inequalities for harmonic mappings in the unit disk 𝔻<i>.</i></p>

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Bohr’s Inequalities Associated with the Set of All Sequences of Nonnegative Continuous Functions

  • 期刊论文

  • 发表日期：

    2025年12月16日

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  • Raju Biswas,
  • Rajib Mandal

摘要

In this paper, we establish several sharp versions of Bohr’s inequalities for the class of K-quasiconformal sense-preserving harmonic mappings on the unit disk := {z ∈ ℂ : |z| < 1} by using a sequence \({\left\{{\psi }_{n}\left(r\right)\right\}}_{n=0}^{\infty }\) ψ n r n = 0 ∞ of nonnegative continuous functions defined on [0, 1) and such that the series \({\sum }_{n=0}^{\infty }{\psi }_{n}\left(r\right)\) ∑ n = 0 ∞ ψ n r converges locally uniformly in [0, 1). As an application, we deduce several well-known results, as well as numerous improved and refined Bohr’s inequalities for harmonic mappings in the unit disk 𝔻.