<p>In this paper, we study upper bounds for the 2nd- and 3rd-order Hankel determinants, with a particular focus on estimating the third-order Hankel functional <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathscr {H}_3(1) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">H</mi> <mn>3</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for a newly defined class of bi-univalent functions denoted by <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathcal{O}\mathcal{S}_\Sigma ^*(\lambda , \beta ) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">O</mi> <msubsup> <mi mathvariant="script">S</mi> <mi mathvariant="normal">Σ</mi> <mo>∗</mo> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>λ</mi> <mo>,</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. This class is associated with the function <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\frac{2\sqrt{1 + \zeta }}{1 + e^{-\zeta }} $</EquationSource> <EquationSource Format="MATHML"><math> <mfrac> <mrow> <mn>2</mn> <msqrt> <mrow> <mn>1</mn> <mo>+</mo> <mi>ζ</mi> </mrow> </msqrt> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>ζ</mi> </mrow> </msup> </mrow> </mfrac> </math></EquationSource> </InlineEquation> and is defined in balloon domain. Using analytic techniques and coefficient estimates, we derive meaningful upper bounds for <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathscr {H}_3(1) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">H</mi> <mn>3</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. To obtain these estimates, we also examine upper bounds for the individual Taylor coefficients involved. The geometry of the balloon domain introduces new aspects in the analysis, enriching the structural study of these functions. Several special cases are discussed to illustrate the effectiveness and generality of the results obtained.</p>

Third Hankel determinant for a class of bi-univalent functions defined in balloon-shaped domain

期刊论文
发表日期：

2026年2月2日

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Murat Çağlar,
Mohammad El-Ityan,
Sarah Jalawi Abd,
Adel Salim Tayyah

摘要

In this paper, we study upper bounds for the 2nd- and 3rd-order Hankel determinants, with a particular focus on estimating the third-order Hankel functional $\mathscr {H}_3(1) $ $H_{3} (1)$ for a newly defined class of bi-univalent functions denoted by $\mathcal{O}\mathcal{S}_\Sigma ^*(\lambda , \beta ) $ $O S_{Σ}^{*} (λ, β)$ . This class is associated with the function $\frac{2\sqrt{1 + \zeta }}{1 + e^{-\zeta }} $ $\frac{2 \sqrt{1 + ζ}}{1 + e^{- ζ}}$ and is defined in balloon domain. Using analytic techniques and coefficient estimates, we derive meaningful upper bounds for $\mathscr {H}_3(1) $ $H_{3} (1)$ . To obtain these estimates, we also examine upper bounds for the individual Taylor coefficients involved. The geometry of the balloon domain introduces new aspects in the analysis, enriching the structural study of these functions. Several special cases are discussed to illustrate the effectiveness and generality of the results obtained.