<p>This paper, motivated by the previous one [<CitationRef CitationID="CR12">12</CitationRef>], presents some new achievements in estimating Hankel determinants for the class <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathcal {S}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">S</mi> </math></EquationSource> </InlineEquation> of univalent functions. With the help of the Grunsky inequalities, we improve earlier results for the bound of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$H_3(1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>H</mi> <mn>3</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> in <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathcal {S}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">S</mi> </math></EquationSource> </InlineEquation>. It is shown that this bound is less than 1. Moreover, we obtain the bounds of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$H_3(1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>H</mi> <mn>3</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for univalent functions with the second or the third coefficient vanishing. In particular, the estimate of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$H_3(1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>H</mi> <mn>3</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for odd univalent functions is derived.</p>

An improved estimate of the third Hankel determinant for univalent functions

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发表日期：

2026年3月2日

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Milutin Obradović,
Nikola Tuneski,
Paweł Zaprawa

摘要

This paper, motivated by the previous one [12], presents some new achievements in estimating Hankel determinants for the class $\mathcal {S}$ $S$ of univalent functions. With the help of the Grunsky inequalities, we improve earlier results for the bound of $H_3(1)$ $H_{3} (1)$ in $\mathcal {S}$ $S$ . It is shown that this bound is less than 1. Moreover, we obtain the bounds of $H_3(1)$ $H_{3} (1)$ for univalent functions with the second or the third coefficient vanishing. In particular, the estimate of $H_3(1)$ $H_{3} (1)$ for odd univalent functions is derived.