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<p>In this paper, we study the uniqueness of the generalized Riemann mapping theorem for harmonic mappings of the unit disk onto polygons with <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(n+2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>+</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> vertices with Blaschke dilatations <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(B_n(z)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>B</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> of order <i>n</i>, where <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(n=1,2,\ldots \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> </mrow> </math></EquationSource> </InlineEquation>. Our main results show the uniqueness of the mapping for the special case where <i>n</i> is an odd number and for the cases <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(n=3,4,5\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>=</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>5</mn> </mrow> </math></EquationSource> </InlineEquation>. In addition, we provide numerical experiments to support the uniqueness of the mapping for the general case of harmonic mappings onto polygons.</p>

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On Uniqueness of Harmonic Mappings with Blaschke Dilatations

  • 期刊论文

  • 发表日期：

    2026年2月12日

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  • Daoud Bshouty,
  • Jie Huang,
  • Antti Rasila,
  • Terry Wallace

摘要

In this paper, we study the uniqueness of the generalized Riemann mapping theorem for harmonic mappings of the unit disk onto polygons with \(n+2\) n + 2 vertices with Blaschke dilatations \(B_n(z)\) B n ( z ) of order n, where \(n=1,2,\ldots \) n = 1 , 2 , … . Our main results show the uniqueness of the mapping for the special case where n is an odd number and for the cases \(n=3,4,5\) n = 3 , 4 , 5 . In addition, we provide numerical experiments to support the uniqueness of the mapping for the general case of harmonic mappings onto polygons.