<p>Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathcal {K}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">K</mi> </math></EquationSource> </InlineEquation> be the familiar class of normalized convex univalent functions in the unit disk. Let <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$f(z)=z+\sum \limits _{m=2}^\infty a_mz^m \in \mathcal {K}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>z</mi> <mo>+</mo> <munderover> <mo movablelimits="false">∑</mo> <mrow> <mi>m</mi> <mo>=</mo> <mn>2</mn> </mrow> <mi>∞</mi> </munderover> <msub> <mi>a</mi> <mi>m</mi> </msub> <msup> <mi>z</mi> <mi>m</mi> </msup> <mo>∈</mo> <mi mathvariant="script">K</mi> </mrow> </math></EquationSource> </InlineEquation>. Kowalczyk, Lecko and Sim proved the following sharp estimate: <Equation ID="Equ16"> <EquationSource Format="TEX">$\begin{aligned} |H_{3,1}(f)|\le \frac{4}{135}, \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <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>135</mn> </mfrac> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq3"> <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 <Equation ID="Equ17"> <EquationSource Format="TEX">$\begin{aligned} H_{3,1}(f)= \begin{vmatrix} a_1&a_2&a_3 \\ a_2&a_3&a_4 \\ a_3&a_4&a_5 \end{vmatrix}. \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <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> <mo>=</mo> <mfenced close="|" open="|"> <mrow> <mtable> <mtr> <mtd> <msub> <mi>a</mi> <mn>1</mn> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mn>2</mn> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mn>3</mn> </msub> </mtd> </mtr> <mtr> <mtd> <mrow> <mrow /> <msub> <mi>a</mi> <mn>2</mn> </msub> </mrow> </mtd> <mtd> <msub> <mi>a</mi> <mn>3</mn> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mn>4</mn> </msub> </mtd> </mtr> <mtr> <mtd> <mrow> <mrow /> <msub> <mi>a</mi> <mn>3</mn> </msub> </mrow> </mtd> <mtd> <msub> <mi>a</mi> <mn>4</mn> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mn>5</mn> </msub> </mtd> </mtr> </mtable> </mrow> </mfenced> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>In this paper, we generalize the above result to a subclass of quasi-convex mappings defined on the unit ball in a complex Banach space.</p>

The Sharp Bound of the Third Hankel Determinant for Quasi-Convex Mappings in Complex Banach Spaces

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2025年12月22日

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Qinghua Xu,
Renata Długosz,
Piotr Liczberski

摘要

Let $\mathcal {K}$ $K$ be the familiar class of normalized convex univalent functions in the unit disk. Let $f(z)=z+\sum \limits _{m=2}^\infty a_mz^m \in \mathcal {K}$ $f (z) = z + \sum_{m = 2}^{\infty} a_{m} z^{m} \in K$ . Kowalczyk, Lecko and Sim proved the following sharp estimate: $\begin{aligned} |H_{3,1}(f)|\le \frac{4}{135}, \end{aligned}$ $\begin{matrix} | H_{3, 1} (f) | \leq \frac{4}{135}, \end{matrix}$ where $H_{3,1}(f)$ $H_{3, 1} (f)$ is the third Hankel determinant $\begin{aligned} H_{3,1}(f)= \begin{vmatrix} a_1&a_2&a_3 \\ a_2&a_3&a_4 \\ a_3&a_4&a_5 \end{vmatrix}. \end{aligned}$ $\begin{matrix} H_{3, 1} (f) = |\begin{matrix} a_{1} & a_{2} & a_{3} \\ a_{2} & a_{3} & a_{4} \\ a_{3} & a_{4} & a_{5} \end{matrix}| . \end{matrix}$ In this paper, we generalize the above result to a subclass of quasi-convex mappings defined on the unit ball in a complex Banach space.