Hankel determinant for bi-univalent functions with bounded turning associated with tan hyperbolic function
Abstract
In this article, utilizing the concept of subordination, we have developed two new subclasses of
bi-univalent functions that are related to the domain of the hyperbolic tangent function. Our study focuses on
determining the upper bound of the second Hankel determinant for particular new subclasses of bi-univalent
functions in the open unit disk U0.
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References
. R.M. Ali, S.K. Lee, V. Ravichandran, S. Subramaniam, Coefficient estimates for bi-univalent Ma-Minda starlike and
convex functions, Appl. Math. Lett., 25(3), 344–351, (2012).
2. Altınkaya, S. Yal¸cın, Upper bound of second Hankel determinant for bi-Bazilevi˘c functions, Mediterr. J. Math., 13(6),
4081–4090, (2016).
3. M. C¸ a˘glar, E. Deniz, H. M. Srivastava, Second Hankel determinant for certain subclasses of bi-univalent functions,
Turkish J. Math., 41(3), 694–706, (2017).
4. E. Deniz, M. C¸ a˘glar, H. Orhan, Second Hankel determinant for bi-starlike and bi-convex functions of order β, Appl.
Math. Comput., 271, 301–307, (2015).
5. P. L. Duren, Univalent functions, Grundlehren der mathematischen Wissenschaften, 259, Springer, New York, (1983).
6. M. Fekete, G. Szeg¨o, Eine Bemerkung Uber Ungerade Schlichte Funktionen, J. London Math. Soc., 8(2), 85–89, (1933).
7. B. A. Frasin, M. K. Aouf, New subclasses of bi-univalent functions, Appl. Math. Lett., 24(9), 1569–1573, (2011).
8. S. R. Kanas, E. Analouei Audegani, A. Zireh, An unified approach to second Hankel determinant of bi-subordinate
functions, Mediterr. J. Math., 14(6), Paper No. 233, 12 pp, (2017).
9. A. W. Kedzierawski, Some remarks on bi-univalent functions, Ann. Univ. Mariae Curie-Sk lodowska Sect. A, 39, 77–81
(1988).
10. S. S. Kumar, V. Kumar, V. Ravichandran, Estimates for the initial coefficients of bi-univalent functions, Tamsui Oxf.
J. Inf. Math. Sci., 29(4), 487–504, (2013).
11. S. K. Lee, V. Ravichandran, S. Supramaniam, Bounds for the second Hankel determinant of certain univalent functions,
J. Inequal. Appl., 281, 17 pp, (2013).
12. M. Lewin, On a coefficient problem for bi-univalent functions, Proc. Amer. Math. Soc., 18, 63–68, (1967).
13. K. I. Noor, Hankel determinant problem for the class of functions with bounded boundary rotation, Rev. Roumaine
Math. Pures Appl., 28(8), 731–739, (1983).
14. J. W. Noonan, D. K. Thomas, On the second Hankel determinant of areally mean p-valent functions, Trans. Amer.
Math. Soc., 223, 337–346, (1976).
15. H. Orhan, N. Magesh, J. Yamini, Bounds for the second Hankel determinant of certain bi-univalent functions, Turkish
J. Math., 40(3), 679–687, (2016).
P. Sharma, S. Sivasubramanian, N. E. Cho, Initial coefficient bounds for certain new subclasses of bi-bazileviˇc functions
and exponentially bi-convex functions with bounded boundary rotation, Axioms, 13(1), 25, (2024).
17. P. Sharma, S. Sivasubramanian, N. E. Cho, Initial coefficient bounds for certain new subclasses of bi-univalent functions
with bounded boundary rotation, AIMS Math., 8(12), 29535–29554, (2023).
18. H. M. Srivastava, D. Bansal, Coefficient estimates for a subclass of analytic and bi-univalent functions, J. Egyptian
Math. Soc., 23(2), 242–246, (2015).
19. H. M. Srivastava, A. K. Mishra, P. Gochhayat, Certain subclasses of analytic and bi-univalent functions, Appl. Math.
Lett., 23(10), 1188–1192, (2010).
20. Q. H. Xu, Y.-C. Gui, H. M. Srivastava, Coefficient estimates for a certain subclass of analytic and bi-univalent functions,
Appl. Math. Lett., 25(6), 990–994, (2012).
21. Q. H. Xu, H.-G. Xiao, H. M. Srivastava, A certain general subclass of analytic and bi-univalent functions and associated
coefficient estimate problems, Appl. Math. Comput., 218(23), 11461–11465, (2012).
22. P. Zaprawa, On the Fekete-Szeg¨o problem for classes of bi-univalent functions, Bull. Belg. Math. Soc.-Simon Stevin,
21(1), 169–178, (2014).
convex functions, Appl. Math. Lett., 25(3), 344–351, (2012).
2. Altınkaya, S. Yal¸cın, Upper bound of second Hankel determinant for bi-Bazilevi˘c functions, Mediterr. J. Math., 13(6),
4081–4090, (2016).
3. M. C¸ a˘glar, E. Deniz, H. M. Srivastava, Second Hankel determinant for certain subclasses of bi-univalent functions,
Turkish J. Math., 41(3), 694–706, (2017).
4. E. Deniz, M. C¸ a˘glar, H. Orhan, Second Hankel determinant for bi-starlike and bi-convex functions of order β, Appl.
Math. Comput., 271, 301–307, (2015).
5. P. L. Duren, Univalent functions, Grundlehren der mathematischen Wissenschaften, 259, Springer, New York, (1983).
6. M. Fekete, G. Szeg¨o, Eine Bemerkung Uber Ungerade Schlichte Funktionen, J. London Math. Soc., 8(2), 85–89, (1933).
7. B. A. Frasin, M. K. Aouf, New subclasses of bi-univalent functions, Appl. Math. Lett., 24(9), 1569–1573, (2011).
8. S. R. Kanas, E. Analouei Audegani, A. Zireh, An unified approach to second Hankel determinant of bi-subordinate
functions, Mediterr. J. Math., 14(6), Paper No. 233, 12 pp, (2017).
9. A. W. Kedzierawski, Some remarks on bi-univalent functions, Ann. Univ. Mariae Curie-Sk lodowska Sect. A, 39, 77–81
(1988).
10. S. S. Kumar, V. Kumar, V. Ravichandran, Estimates for the initial coefficients of bi-univalent functions, Tamsui Oxf.
J. Inf. Math. Sci., 29(4), 487–504, (2013).
11. S. K. Lee, V. Ravichandran, S. Supramaniam, Bounds for the second Hankel determinant of certain univalent functions,
J. Inequal. Appl., 281, 17 pp, (2013).
12. M. Lewin, On a coefficient problem for bi-univalent functions, Proc. Amer. Math. Soc., 18, 63–68, (1967).
13. K. I. Noor, Hankel determinant problem for the class of functions with bounded boundary rotation, Rev. Roumaine
Math. Pures Appl., 28(8), 731–739, (1983).
14. J. W. Noonan, D. K. Thomas, On the second Hankel determinant of areally mean p-valent functions, Trans. Amer.
Math. Soc., 223, 337–346, (1976).
15. H. Orhan, N. Magesh, J. Yamini, Bounds for the second Hankel determinant of certain bi-univalent functions, Turkish
J. Math., 40(3), 679–687, (2016).
P. Sharma, S. Sivasubramanian, N. E. Cho, Initial coefficient bounds for certain new subclasses of bi-bazileviˇc functions
and exponentially bi-convex functions with bounded boundary rotation, Axioms, 13(1), 25, (2024).
17. P. Sharma, S. Sivasubramanian, N. E. Cho, Initial coefficient bounds for certain new subclasses of bi-univalent functions
with bounded boundary rotation, AIMS Math., 8(12), 29535–29554, (2023).
18. H. M. Srivastava, D. Bansal, Coefficient estimates for a subclass of analytic and bi-univalent functions, J. Egyptian
Math. Soc., 23(2), 242–246, (2015).
19. H. M. Srivastava, A. K. Mishra, P. Gochhayat, Certain subclasses of analytic and bi-univalent functions, Appl. Math.
Lett., 23(10), 1188–1192, (2010).
20. Q. H. Xu, Y.-C. Gui, H. M. Srivastava, Coefficient estimates for a certain subclass of analytic and bi-univalent functions,
Appl. Math. Lett., 25(6), 990–994, (2012).
21. Q. H. Xu, H.-G. Xiao, H. M. Srivastava, A certain general subclass of analytic and bi-univalent functions and associated
coefficient estimate problems, Appl. Math. Comput., 218(23), 11461–11465, (2012).
22. P. Zaprawa, On the Fekete-Szeg¨o problem for classes of bi-univalent functions, Bull. Belg. Math. Soc.-Simon Stevin,
21(1), 169–178, (2014).
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2025-12-05
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