Fourth Hankel and Toeplitz determinant estimates for certain analytic functions associated with Four Leaf function
DOI:
https://doi.org/10.17398/Keywords:
Hankel determinants, Starlike functions, Coefficient inequalities, Four Leaf domain, Toeplitz determinantsAbstract
The objective of this paper is to establish initial coefficient inequalities, Upper bounds to the Hankel and Toeplitz determinants for certain normalized univalent functions defined on the open unit disk D in the complex plane related to the analytic function ϕ4L (z) = 1 + 5z/6 + z5/6 that maps the open unit disk in the complex plane onto the interior of four leaf shaped domain in the right half of the complex plane.
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M.F. Ali, D.K. Thomas, A. Vasudevarao, Toeplitz determinants whose elements are the coefficients of analytic and Univalent functions, Bull. Aust. Math. Soc. 97 (2) (2018), 253 – 264. doi:10.1017/S0004972717001174
M. Arif, L. Rani, M. Raza, P. Zaprawa, Fourth Hankel Determinant for the family of functions with bounded turning, Bull. Korean Math. Soc. 55 (6) (2018), 1703 – 1711.
M. Arif, M. Raza, H. Tang, S. Hussain, H. Khan, Hankel determinant of order three for familiar subsets of analytic functions related with sine function, Open Math. 17 (2019) 1615 – 1630.
K.O. Babalola, On H3 (1) Hankel determinant for some classes of univalent functions, Inequality Theory and Applications 6 (2010), 1 – 7.
S. Gandhi, Radius estimates for three leaf function and convex combination of starlike functions, in “Mathematical Analysis I: Approximation Theory”, Springer Proceedings in Mathematics and Statistics, Vol. 306, Springer Singapore, Springer, 2020, 173 – 184. doi.org/10.1007/978-981-15-1153-0 15
K. Ganesh, R. Bharavi Sharma, K. Saroja, K. Yakaiah, Fourth Hankel and Toeplitz determinants for a subclass of analytic functions subordinate to 1 + sin z, Math. Appl. (Warsaw) 51 (2) (2023), 321 – 338.
A.W. Goodman, “Univalent functions, Vol. I, Vol. II”, Mariner Publishing Co., Inc., Tampa, FL, 1983.
S. Gunasekar, B. Sudharsanan, M. Ibrahim, T. Bulboacǎ, Subclasses of analytic functions subordinated to the four-leaf function, Axioms 13 (3), p. 155, 2024.
W.K. Hayman, On the second Hankel determinant of mean univalent functions, Proc. London Math. Soc. (3) 18 (1968), 77 – 94.
G. Kaur, G. Singh, Fourth Hankel determinant for a new subclass of Bounded turning functions, Jnanabha 52 (1) (2022), 229 – 233.
M.G. Khan, B. Ahmad, J. Sokól, Z. Muhammad, W.K. Mashwani, R. Chinram, P. Petchkaew, Coefficient problems in a class of functions with bounded turning associated with Sinefunction, Eur. J. Appl. Math. 14 (1) (2021), 53 – 64.
M.G. Khan, N.E. Cho, T.G. Shaba, B. Ahmad. W.K. Mashwani, Coefficient functionals for a class of bounded turning functions related to modified sigmoid function, AIMS Math. 7 (2) (2022), 3133 – 3149.
B. Kowalczyk, A. Lecko, The sharp bound of the third Hankel determinant for functions of bounded turning, Bol. Soc. Mat. Mex. (3) 27 (2021), Paper No. 69, 13 pp. doi.org/10.1007/s40590-021-00383-7
B. Kowalczyk, A. Lecko, Y.J. Sim, The sharp bound for the Hankel determinant of the third kind for convex functions, Bull. Aust. Math. Soc. 97 (2018), 435 – 445. doi.org/10.1017/S0004972717001125
B. Kowalczyk, A. Lecko, D.K. Thomas, The sharp bound of the third Hankel determinant for starlike functions, Forum Math. 34 (5) (2022), 1249 – 1254. doi.org/10.1515/forum-2021-0308
R.J. Libera, E.J. Ziotkiewicz, Coefficient bounds for the inverse of a function with derivatives in P, Proc. Amer. Math. Soc. 87 (2) (1983), 251 – 257.
W. Ma, Generalized Zalcman conjecture for starlike and typically real functions, J. Math. Anal. Appl. 234 (1) (1999), 328 – 339. doi.org/10.1006/jmaa.1999.6378
W.C. Ma, D. Minda, A unified treatment of some special classes of univalent functions, in “Proceedings of the Conference on Complex Analysis (Tianjin, 1992), International Press, Cambridge, MA, 1994, 157 – 169.
S. Mandal, P.P. Roy, M.B. Ahmad, Hankel and Toeplitz determinants of logarithmic coefficients of inverse functions for certain classes of univalent functions, Iran. J. Sci. 49 (2025), 243 – 252.
J.W. Noonan, D.K. Thomas, On the second Hankel determinant of areally mean p-valent functions, Tran. Amer. Math. Soc. 223 (1976), 337 – 346.
K.I. Noor, On the Hankel determinants of close-to-convex univalent functions, Internat. J. Math. Math. Sci. 3 (3) (1980), 477 – 481.
Ch. Pommerenke, On the coefficients and Hankel determinants of univalent functions, J. London Math. Soc. 41 (1966), 111 – 122.
Ch. Pommerenke, On the Hankel determinants of univalent functions, Mathematika 14 (1967), 108 – 112.
Ch. Pommerenke, “Univalent functions”, Studia Math./Math. Lehrbücher, Band XXV, Vandenhoeck & Ruprecht, Göttingen, 1975, 376 pp.
V. Radhika, S. Sivasubramanian, G. Murugusundaramoorthy, J.M. Jahangiri, Toeplitz matrices whose elements are the coefficients of functions with bounded boundary rotation, J. Complex Anal. 2016, Art. ID 4960704, 4 pp.
V. Ravichandran, S. Verma, Bound for the fifth coefficient of certain starlike functions, C. R. Math. Acad. Sci. Paris 353 (2015), 505 – 510.
T.G. Shaba, S. Araci, B.O. Adebesin, Fekete-Szegö problem and second Hankel determinant for a subclass of bi-univalent functions sssociated with four leaf domain, Asia Pac. J. Math. 10 (2023), Paper No. 21, 16 pp. apjm.apacific.org/PDFs/10-21.pdf
K. Sharma, N.K. Jain, V. Ravichandran, Starlike functions associated with a cardioid, Afr. Mat. 27 (2016), 923 – 939.
H.M. Srivastava, Q.Z. Ahmad, N. Khan, B. Khan, Hankel and Toeplitz determinants for a subclass of q-starlike functions associated with a general conic domain, Mathematics 7 (2) (2019), paper no. 181, 15 pp. www.mdpi.com/2227-7390/7/2/181
H.M. Srivastava, G. Kaur, G. Singh, Estimates of the fourth Hankel determinant for a class of analytic functions with bounded turnings involving cardioid domains, J. Nonlinear Convex Anal. 22 (3) (2021), 511 – 526.
H.M. Srivastava, N. Khan, M.A. Bah, A. Alahmade, Ferdous M.O. Tawfiq, Z. Syed, Fourth order Hankel determinants for certain subclasses of modified sigmoid-activated analytic functions involving the trigonometric sine function, J. Inequal. Appl. 2024, Paper No. 84, 18 pp. doi.org/10.1186/s13660-024-03150-0
P. Sunthrayuth, Y. Jawarneh, M. Naeem, N. Iqbal, J. Kafle, Some Sharp on coefficient estimate problems for four-leaf-type bounded turning functions, J. Funct. Spaces 2022, Art. ID 8356125, 10 pp. doi.org/10.1155/2022/8356125
D.K. Thomas, S.A. Halim, Toeplitz matrices whose elements are the coefficients of Starlike and Close-to-convex functions, Bull. Malays. Math. Sci. Soc. 40 (4) (2017), 1781 – 1790.
S.P. Vijayalakshmi, T.V. Sudharsan, T. Bulboaca, Symmetric toeplitz determinants for classes defined by post quantum operators subordinated to the limaçon function, Stud. Univ. Babes-Bolyai Math. 69 (2) (2024), 299 – 316.
K. Yakaiah, R. Bharavi Sharma, Fourth Hankel and Toeplitz Determinants for a class of Analytic Univalent Functions subordinated to cos z, Stochastic Modeling and Applications, Vol. 26, No. 3 pp. 619 – 623. (January– June, Special Issue 2022 Part - 9).
Zhang, Hai-Yan, and Huo Tang, Fourth Toeplitz determinants for starlike functions defined by using the sine function, J. Funct. Spaces 2021, Art. ID 4103772, 7 pp. doi.org/10.1155/2021/4103772
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