In this paper, the class of starlike functions of complex order γ (γ ∈ ℂ − {0}) is extended from the case on unit disk <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10473_2025_417_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="159" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb{U}=\{z\in \mathbb{C}:|z|<1\}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi mathvariant="double-struck">U</mi> </mrow> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>z</mi> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mo>:</mo> <mrow> <mo stretchy="false">∣</mo> </mrow> <mi>z</mi> <mrow> <mo stretchy="false">∣</mo> </mrow> <mo><</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> </math></EquationSource> </InlineEquation> to the case on the unit ball B in a complex Banach space or the unit polydisk <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10473_2025_417_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb{U}^n$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msup> <mrow> <mi mathvariant="double-struck">U</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation> in ℂn. Let g be a convex function in <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10473_2025_417_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb{U}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi mathvariant="double-struck">U</mi> </mrow> </math></EquationSource> </InlineEquation>. We mainly establish the sharp bounds of all terms of homogeneous polynomial expansions for a subclass of g-parametric starlike mappings of complex order γ on B (resp. <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10473_2025_417_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb{U}^n$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msup> <mrow> <mi mathvariant="double-struck">U</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation>) when the mappings f are k-fold symmetric, k ∈ ℕ. Our results partly solve the Bieberbach conjecture in several complex variables and generalize some prior works.

Estimates of all terms of homogeneous polynomial expansions for the subclasses of g-parametric starlike mappings of complex order in several complex variables

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发表日期：

2025年4月8日

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Liangpeng Xiong,
Qingchao Wang,
Xiaoying Sima

摘要

In this paper, the class of starlike functions of complex order γ (γ ∈ ℂ − {0}) is extended from the case on unit disk $\mathbb{U}=\{z\in \mathbb{C}:|z|<1\}$ $U = {z \in C : ∣ z ∣ < 1}$ to the case on the unit ball B in a complex Banach space or the unit polydisk $\mathbb{U}^n$ $U^{n}$ in ℂⁿ. Let g be a convex function in $\mathbb{U}$ $U$ . We mainly establish the sharp bounds of all terms of homogeneous polynomial expansions for a subclass of g-parametric starlike mappings of complex order γ on B (resp. $\mathbb{U}^n$ $U^{n}$ ) when the mappings f are k-fold symmetric, k ∈ ℕ. Our results partly solve the Bieberbach conjecture in several complex variables and generalize some prior works.