The sharp bounds of the third-order Hankel determinant of inverse functions for certain univalent functions

Dong Guo; Xin Wang; Xi Luo; Dong Guo; Xin Wang; Xi Luo

doi:10.3934/math.20261235

AIMS Mathematics

2026, Volume 11, Issue 9: 31267-31293. doi: 10.3934/math.20261235

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Research article

The sharp bounds of the third-order Hankel determinant of inverse functions for certain univalent functions

Dong Guo ^{1
,
,},
Xin Wang ²,
Xi Luo ³

1.
School of Mathematical Sciences, Yangzhou Polytechnic University, Yangzhou 225009, Jiangsu, China
2.
Department of Mathematics, Shenzhen Polytechnic University, Shenzhen 518055, Guangdong, China
3.
School of Mathematics, Jiaying University, Jiaying 514015, Guangdong, China

Received: 16 July 2026 Revised: 06 September 2026 Accepted: 16 September 2026 Published: 23 September 2026
MSC : 30C45, 30C80

In this paper, we obtain the sharp bounds of third Hankel determinants $ |\mathcal H_{3, 1}(\kappa)| $ of inverse functions for $ \kappa\in \mathcal S^{*}_{e} $, $ \kappa\in \mathcal C_{e} $, $ \kappa\in\mathcal S^{*}_\mathcal{L} $, and $ \kappa\in\mathcal C_\mathcal{L} $, which elevates the function categories of the existing results to the inverse function category. In addition, we extend the second-order Hankel determinant of the logarithmic coefficient of the function class $ \kappa\in \mathcal S^{*}_{e} $ to the class $ \mathcal{S}_{\mathcal{L}}^{*} $ and obtain precise results. Most notably, we improve the third-order Hankel determinant of the function class $ \mathcal C_{car} $ from $ \frac{319}{4374} $ to $ \frac{1}{81} $, which resolves the conjecture posed in ^[10].
- Hankel determinants,
- starlike functions,
- convex functions,
- inverse functions,
- cardioid domain,
- exponential function
Citation: Dong Guo, Xin Wang, Xi Luo. The sharp bounds of the third-order Hankel determinant of inverse functions for certain univalent functions[J]. AIMS Mathematics, 2026, 11(9): 31267-31293. doi: 10.3934/math.20261235

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Abstract

In this paper, we obtain the sharp bounds of third Hankel determinants $ |\mathcal H_{3, 1}(\kappa)| $ of inverse functions for $ \kappa\in \mathcal S^{*}_{e} $, $ \kappa\in \mathcal C_{e} $, $ \kappa\in\mathcal S^{*}_\mathcal{L} $, and $ \kappa\in\mathcal C_\mathcal{L} $, which elevates the function categories of the existing results to the inverse function category. In addition, we extend the second-order Hankel determinant of the logarithmic coefficient of the function class $ \kappa\in \mathcal S^{*}_{e} $ to the class $ \mathcal{S}_{\mathcal{L}}^{*} $ and obtain precise results. Most notably, we improve the third-order Hankel determinant of the function class $ \mathcal C_{car} $ from $ \frac{319}{4374} $ to $ \frac{1}{81} $, which resolves the conjecture posed in ^[10].

References

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