Third Hankel determinant for a subclass of bi-univalent functions related to balloon-shaped domain

Mohammad El-Ityan; Tariq Al-Hawary; Basem Aref Frasin; Ibtisam Aldawish; Mohammad El-Ityan; Tariq Al-Hawary; Basem Aref Frasin; Ibtisam Aldawish

doi:10.3934/math.20251127

AIMS Mathematics

2025, Volume 10, Issue 11: 25452-25468. doi: 10.3934/math.20251127

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Research article

Third Hankel determinant for a subclass of bi-univalent functions related to balloon-shaped domain

1.
Department of Mathematics, Faculty of Science, Al-Balqa Applied University, Salt 19117, Jordan
2.
Department of Applied Science, Ajloun College, Al-Balqa Applied University, Ajloun 26816, Jordan
3.
Department of Mathematics, Faculty of Science, Al Al-Bayt University, Mafraq 25113, Jordan
4.
Mathematics and Statistics Department, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh, Saudi Arabia

Received: 26 July 2025 Revised: 24 October 2025 Accepted: 27 October 2025 Published: 05 November 2025
MSC : 30C45

In this paper, we consider a subclass of bi-univalent functions, denoted by $ \mathbb{CS}_{\beth}^{\ast}(\gamma) $, which is defined via a balloon-shaped domain associated with the function $ \frac{2\sqrt{1+\varsigma}} {1+e^{-\varsigma}} $. We prove that this class is non-empty using illustrative mappings. We investigate upper bounds for the second- and third-order Hankel determinants, focusing on the functional $ H_{3}(1) $. In addition, we estimate the Taylor coefficients up to order $ l = 5 $, which are essential in obtaining bounds for the determinants. The balloon shape introduces new analytic features that enrich the behavior of the functions in this class. Several examples and special cases are offered to illustrate the flexibility of the outcomes.
- Hankel determinant,
- unit disk $ \mathcal{U} $,
- analytic,
- bi-univalent functions,
- Fekete-Szegö inequality
Citation: Mohammad El-Ityan, Tariq Al-Hawary, Basem Aref Frasin, Ibtisam Aldawish. Third Hankel determinant for a subclass of bi-univalent functions related to balloon-shaped domain[J]. AIMS Mathematics, 2025, 10(11): 25452-25468. doi: 10.3934/math.20251127

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Abstract

In this paper, we consider a subclass of bi-univalent functions, denoted by $ \mathbb{CS}_{\beth}^{\ast}(\gamma) $, which is defined via a balloon-shaped domain associated with the function $ \frac{2\sqrt{1+\varsigma}} {1+e^{-\varsigma}} $. We prove that this class is non-empty using illustrative mappings. We investigate upper bounds for the second- and third-order Hankel determinants, focusing on the functional $ H_{3}(1) $. In addition, we estimate the Taylor coefficients up to order $ l = 5 $, which are essential in obtaining bounds for the determinants. The balloon shape introduces new analytic features that enrich the behavior of the functions in this class. Several examples and special cases are offered to illustrate the flexibility of the outcomes.

References

[1]	A. S. Tayyah, W. G. Atshan, A class of bi-Bazilevič and bi-pseudo-starlike functions involving Tremblay fractional derivative operator, Probl. Anal. Issues Anal., 14 (2025), 145–161. http://dx.doi.org/10.15393/j3.art.2025.16750 doi: 10.15393/j3.art.2025.16750
[2]	M. El-Ityan, Q. A. Shakir, T. Al-Hawary, R. Buti, D. Breaz, L. I. Cotîrlă, On the third Hankel determinant of a certain subclass of bi-univalent functions defined by $(p, q)$-derivative operator, Mathematics, 13 (2025), 1269. https://doi.org/10.3390/math13081269 doi: 10.3390/math13081269
[3]	M. Al-Ityan, M. A. Sabri, S. Hammad, B. Frasin, T. Al-Hawary, F. Yousef, Third-order Hankel determinant for a class of bi-univalent functions associated with sine function, Mathematics, 13 (2025), 2887. https://doi.org/10.3390/math13172887 doi: 10.3390/math13172887
[4]	S. H. Hadi, Y. H. Saleem, A. A. Lupaş, K. M. K. Alshammari, A. Alatawi, Toeplitz determinants for inverse of analytic functions, Mathematics, 13 (2025), 676. https://doi.org/10.3390/math13040676 doi: 10.3390/math13040676
[5]	S. S. Alhily, Some results on Koebe one-quarter theorem and Koebe distortion theorem, J. Iraqi Al-Khwarizmi Soc., 2 (2018), 29–38.
[6]	H. M. Srivastava, A. K. Mishra, P. Gochhayat, Certain subclasses of analytic and bi-univalent functions, Appl. Math. Lett., 23 (2010), 1188–1192. http://dx.doi.org/10.1016/j.aml.2010.05.009 doi: 10.1016/j.aml.2010.05.009
[7]	E. Deniz, Certain subclasses of bi-univalent functions satisfying subordinate conditions, J. Class. Anal., 2 (2013), 49–60. http://dx.doi.org/10.7153/jca-02-05 doi: 10.7153/jca-02-05
[8]	M. Lewin, On a coefficient problem for bi-univalent functions, Proc. Amer. Math. Soc., 18 (1967), 63–68. http://dx.doi.org/10.1090/S0002-9939-1967-0206255-1 doi: 10.1090/S0002-9939-1967-0206255-1
[9]	D. Brannan, J. Clunie, Aspects of contemporary complex analysis, London: Academic Press, 1981.
[10]	E. Natanyahu, The minimal distance of the image boundary from the origin and the second coefficient of a univalent function in $\|z\| < 1$, Arch. Ration. Mech. Anal., 32 (1969), 100–112. https://doi.org/10.1007/BF00247676 doi: 10.1007/BF00247676
[11]	D. A. Brannan, T. S. Taha, On some classes of bi-univalent functions, In: Mathematical analysis and its applications, Kuwait, 1985, 53–60. https://doi.org/10.1016/B978-0-08-031636-9.50012-7
[12]	A. S. Tayyah, W. G. Atshan, Starlikeness and bi-starlikeness associated with a new Carathéodory function, J. Math. Sci., 290 (2025), 232–256. https://doi.org/10.1007/s10958-025-07604-8 doi: 10.1007/s10958-025-07604-8
[13]	M. Fekete, G. Szegö, Eine bemerkung über ungerade schlichte funktionen, J. Lond. Math. Soc., s1-8 (1933), 85–89. https://doi.org/10.1112/jlms/s1-8.2.85 doi: 10.1112/jlms/s1-8.2.85
[14]	S. Singh, R. Singh, Subordination by univalent functions, Proc. Amer. Math. Soc., 82 (1981), 39–47. https://doi.org/10.2307/2044313 doi: 10.2307/2044313
[15]	J. Soköł, J. Stankiewicz, Radius of convexity of some subclasses of strongly starlike functions, Zesz. Nauk. Politech. Rzesz. Mat., 19 (1996), 101–105.
[16]	L. A. Wani, A. Swaminathan, Starlike and convex functions associated with a nephroid domain, Bull. Malays. Math. Sci. Soc., 44 (2021), 79–104. http://dx.doi.org/10.1007/s40840-020-00935-6 doi: 10.1007/s40840-020-00935-6
[17]	J. W. Noonan, D. K. Thomas, On the second Hankel determinant of a really mean $p$-valent functions, Trans. Amer. Math. Soc., 223 (1976), 337–346. http://dx.doi.org/10.2307/1997533 doi: 10.2307/1997533
[18]	A. M. Darweesh, W. G. Atshan, A. H. Battor, M. S. Mahdi, On the third Hankel determinant of certain subclass of bi-univalent functions, Math. Model. Eng. Probl., 10 (2023), 1087–1095. https://doi.org/10.18280/mmep.100345 doi: 10.18280/mmep.100345
[19]	M. Illafe, M. H. Mohd, F. Yousef, S. Supramaniam, Bounds for the second Hankel determinant of a general subclass of bi-univalent functions, Int. J. Math. Eng. Manag. Sci., 9 (2024), 1226–1239. http://dx.doi.org/10.33889/IJMEMS.2024.9.5.065 doi: 10.33889/IJMEMS.2024.9.5.065
[20]	Q. A. Shakir, A. S. Tayyah, D. Breaz, L. I. Cotîrlă, E. Rapeanu, F. M. Sakar, Upper bounds of the third Hankel determinant for bi-univalent functions in crescent-shaped domains, Symmetry, 16 (2024), 1281. https://doi.org/10.3390/sym16101281 doi: 10.3390/sym16101281
[21]	H. M. Srivastava, Ş. Altınkaya, S. Yalçın, Hankel determinant for a subclass of bi-univalent functions defined by using a symmetric $q$-derivative operator, Filomat, 32 (2018), 503–516. https://doi.org/10.2298/FIL1802503S doi: 10.2298/FIL1802503S
[22]	I. Al-Shbeil, J. Gong, S. Khan, N. Khan, A. Khan, M. F. Khan, et al., Hankel and symmetric Toeplitz determinants for a new subclass of $q$-starlike functions, Fractal Fract., 6 (2022), 658. https://doi.org/10.3390/fractalfract6110658 doi: 10.3390/fractalfract6110658
[23]	I. Al-Shbeil, T. G. Shaba, A. Cătaş, Second Hankel determinant for the subclass of bi-univalent functions using $q$-Chebyshev polynomial and Hohlov operator, Fractal Fract., 6 (2022), 186. https://doi.org/10.3390/fractalfract6040186 doi: 10.3390/fractalfract6040186
[24]	A. Ahmad, J. Gong, I. Al-Shbeil, A. Rasheed, A. Ali, S. Hussain, Analytic functions related to a balloon-shaped domain, Fractal Fract., 7 (2023), 865. https://doi.org/10.3390/fractalfract7120865 doi: 10.3390/fractalfract7120865
[25]	B. Khan, J. Gong, M. G. Khan, F. Tchier, Sharp coefficient bounds for a class of symmetric starlike functions involving the balloon shape domain, Heliyon, 10 (2024), e38838. http://dx.doi.org/10.1016/j.heliyon.2024.e38838 doi: 10.1016/j.heliyon.2024.e38838
[26]	W. Ma, D. Minda, A unified treatment of some special cases of univalent functions, In: Proceeding of conference on complex analytic, New York: International Press, 1994,157–169.
[27]	R. K. Raina, J. Soköł, On coefficient estimates for a certain class of starlike functions, Hacet. J. Math. Stat., 44 (2015), 1427–1433. http://dx.doi.org/10.15672/HJMS.2015449676 doi: 10.15672/HJMS.2015449676
[28]	K. Sharma, N. K. Jain, V. Ravichandran, Starlike functions associated with a cardioid, Afr. Math., 27 (2016), 923–939. http://dx.doi.org/10.1007/s13370-015-0387-7 doi: 10.1007/s13370-015-0387-7
[29]	N. E. Cho, V. Kumar, S. Kumar, V. Ravichandran, Radius problems for starlike functions associated with the sine function, Bull. Iran. Math. Soc., 45 (2019), 213–232. http://dx.doi.org/10.1007/s41980-018-0127-5 doi: 10.1007/s41980-018-0127-5
[30]	S. S. Kumar, K. Arora, Starlike function associated with a petal shaped domain, arXiv preprint, 2020, arXiv: 2010.10072. http://dx.doi.org/10.48550/arXiv.2010.10072
[31]	P. L. Duren, Univalent functions, New York: Springer, 1983.
[32]	U. Grenander, G. Szegö, Toeplitz forms and their applications, Berkeley: University of California Press, 1958.

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