<p>Kowalczyk et al.(Forum Math. 34(5): 1249C1254, 2022) and Rath et al.(Complex Anal.Oper.Theory. 16(5), Paper No. 65, 2022) established the sharp inequalities <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1711_Article_IEq1.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="94" /> </InlineMediaObject> <EquationSource Format="TEX">$|H_{3,1}(f)|\le \frac{4}{9}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">|</mo> </mrow> <msub> <mi>H</mi> <mrow> <mn>3</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> <mrow> <mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">|</mo> <mo>≤</mo> </mrow> <mfrac> <mn>4</mn> <mn>9</mn> </mfrac> </mrow> </math></EquationSource> </InlineEquation> for starlike functions, and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1711_Article_IEq2.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="94" /> </InlineMediaObject> <EquationSource Format="TEX">$|H_{3,1}(f)|\le \frac{1}{9}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">|</mo> </mrow> <msub> <mi>H</mi> <mrow> <mn>3</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> <mrow> <mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">|</mo> <mo>≤</mo> </mrow> <mfrac> <mn>1</mn> <mn>9</mn> </mfrac> </mrow> </math></EquationSource> </InlineEquation> for starlike functions of order <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1711_Article_IEq3.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="8" /> </InlineMediaObject> <EquationSource Format="TEX">$\frac{1}{2}$</EquationSource> <EquationSource Format="MATHML"><math> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </math></EquationSource> </InlineEquation>, respectively, where <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1711_Article_IEq4.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="53" /> </InlineMediaObject> <EquationSource Format="TEX">$H_{3,1}(f)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>H</mi> <mrow> <mn>3</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is the third Hankel determinant of <i>f</i>. In this paper, we generalize the works of Kowalczyk et al. and Rath et al. to the unit ball in a complex Banach space.</p>

The Sharp Bounds of the Third Hankel Determinant for Starlike Mappings and Starlike Mappings of Order 1/2 in Complex Banach Spaces

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2025年5月22日

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Qinghua Xu,
Peng He,
Zhenyu Xu

摘要

Kowalczyk et al.(Forum Math. 34(5): 1249C1254, 2022) and Rath et al.(Complex Anal.Oper.Theory. 16(5), Paper No. 65, 2022) established the sharp inequalities $|H_{3,1}(f)|\le \frac{4}{9}$ $| H_{3, 1} (f) | \leq \frac{4}{9}$ for starlike functions, and $|H_{3,1}(f)|\le \frac{1}{9}$ $| H_{3, 1} (f) | \leq \frac{1}{9}$ for starlike functions of order $\frac{1}{2}$ $\frac{1}{2}$ , respectively, where $H_{3,1}(f)$ $H_{3, 1} (f)$ is the third Hankel determinant of f. In this paper, we generalize the works of Kowalczyk et al. and Rath et al. to the unit ball in a complex Banach space.