Coefficient estimates and Fekete-Szegö functional for subclasses of bi-univalent functions with respect to symmetric points associated
with Gegenbauer polynomials
Received November 25, 2024. Published online February 25, 2025.
Abstract: In the present article, the authors introduce two new subclasses of holomorphic and bi-univalent functions with respect to the symmetric points defined in the domain of open unit disk $\Delta:=\{z \in\mathbb{C} |z|<1\}$ by making use of subordination between two analytic functions and also using the Gegenbauer polynomials. We investigate bounds of some of the initial Taylor-Maclaurin coefficients belonging to this newly constructed holomorphic and bi-univalent function class. Moreover, we derive the well-known Fekete-Szegö functional for the above said classes. Some of the corollaries of the main results are pointed out.
References: [1] W. G. Atshan, I. A. R. Rahman, A. A. Lupaş: Some results of new subclasses for bi-univalent functions using quasi-subordination. Symmetry 13 (2021), Article ID 1653, 12 pages. DOI 10.3390/sym13091653
[2] A. A. Attiya, A. M. Albalahi, T. S. Hassan: Coefficient estimates for certain families of analytic functions associated with Faber polynomial. J. Funct. Spaces 2023 (2023), Article ID 4741056, 6 pages. DOI 10.1155/2023/4741056 | MR 4546480 | Zbl 1516.30013
[3] D. A. Brannan, J. G. Clunie (Eds.): Aspects of Contemporary Complex Analysis: Proceedings of the NATO Advanced Study Institute held at the University of Durham, Durham, July 1-20, 1979. Academic Press, London (1980). MR 0623462 | Zbl 0483.00007
[4] M. Çağlar, H. Orhan, H. M. Srivastava: Coefficient bounds for $q$-starlike functions and associated with $q$-Bernoulli numbers. J. Appl. Anal. Comput. 13 (2023), 2354-2364. DOI 10.11948/20220566 | MR 4618403 | Zbl 07920427
[5] P. L. Duren: Univalent Functions. Grundlehren der Mathematischen Wissenschaften 259. Springer, New York (1983). MR 0708494 | Zbl 0514.30001
[6] S. M. El-Deeb, S. Bulut: Faber polynomial coefficient estimates of bi-univalent functions connected with the $q$-convolution. Math. Bohem. 148 (2023), 49-64. DOI 10.21136/MB.2022.0173-20 | MR 4536309 | Zbl 1538.30038
[7] M. Fekete, G. Szegö: Eine Bemerkung über ungerade schlichte Funktionen. J. Lond. Math. Soc. 8 (1933), 85-89. (In German.) DOI 10.1112/jlms/s1-8.2.85 | MR 1574865 | JFM 59.0347.04
[8] R. M. Goel, B. S. Mehrok: A subclass of starlike functions with respect to symmetric points. Tamkang J. Math. 13 (1982), 11-24. MR 0678080 | Zbl 0498.30013
[9] A. Janteng, S. A. Halim: A subclass of convex functions with respect to symmetric points. Proceeding of the 16th National Symposium on Science Mathematical (2008).
[10] M. Lewin: On a coefficient problem for bi-univalent functions. Proc. Am. Math. Soc. 18 (1967), 63-68. DOI 10.1090/S0002-9939-1967-0206255-1 | MR 0206255 | Zbl 0158.07802
[11] S. S. Miller, P. T. Mocanu: Differential Subordination: Theory and Applications. Pure and Applied Mathematics, Marcel Dekker 225. Marcel Dekker, New York (2000). DOI 10.1201/9781482289817 | MR 1760285 | Zbl 0954.34003
[12] S. K. Mohapatra, T. Panigrahi: Coefficient estimates for bi-univalent functions defined by $(p,q)$ analogue of the Salagean differential operator related to the Chebyshev polynomial. J. Math. Fund. Sci. 53 (2021), 49-66. DOI 10.5614/j.math.fund.sci.2021.53.1.4
[13] E. Netanyahu: The minimial 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. DOI 10.1007/BF00247676 | MR 0235110 | Zbl 0186.39703
[14] T. Panigrahi, G. Murugusundaramoorthy: Coefficient bounds for bi-univalent analytic functions associated with Hohlov operator. Proc. Jangjeon Math. Soc. 16 (2013), 91-100. MR 3059287
[15] T. Panigrahi, J. Sokół: Generalized Laguerre polynomial bounds for subclass of bi-univalent functions. Jordan J. Math. Stat. 14 (2021), 127-140. MR 4245841 | Zbl 1474.30105
[16] K. Sakaguchi: On a certain univalent mapping. J. Math. Soc. Japan 11 (1959), 72-75. DOI 10.2969/jmsj/01110072 | MR 0107005 | Zbl 0085.29602
[17] H. M. Srivastava: Operators of basic (or $q$-) calculus and fractional $q$-calculus and their applications in geometric function theory of complex analysis. Iran. J. Sci. Technol. Trans. A, Sci. 44 (2020), 327-344. DOI 10.1007/s40995-019-00815-0 | MR 4064730
[18] H. M. Srivastava, A. K. Mishra, P. Gochhayat: Certain subclasses of analytic and bi-univalent functions. Appl. Math. Lett. 23 (2010), 1188-1192. DOI 10.1016/j.aml.2010.05.009 | MR 2665593 | Zbl 1201.30020
[19] H. M. Srivastava, A. K. Wanas: Initial Maclaurin coefficient bounds for new subclasses of analytic and $m$-fold symmetric bi-univalent function defined by a linear combination. Kyungpook Math. J. 59 (2019), 493-503. DOI 10.5666/KMJ.2019.59.3.493 | MR 4020441 | Zbl 1435.30064
[20] H. M. Srivastava, A. K. Wanas, R. Srivastava: Applications of the $q$-Srivastava-Attiya operator invovling a certain family of bi-univalent functions associated with the Horadam polynomial. Symmetry 13 (2021), Article ID 1230, 14 pages. DOI 10.3390/sym13071230
[21] Z.-G. Wang, C.-Y. Gao, S.-M. Yuan: On certain subclasses of close-to-convex and quasi-convex functions with respect to $k$-symmetric points. J. Math. Anal. Appl. 322 (2006), 97-106. DOI 10.1016/j.jmaa.2005.08.060 | MR 2238151 | Zbl 1102.30015
Affiliations: Trailokya Panigrahi, Eureka Pattnayak, Institute of Mathematics and Applications, Andharua, Bhubaneswar-751029, Odisha, India, e-mail: trailokyap6@gmail.com, pattnayakeureka99@gmail.com; Rabha Mohamed El-Ashwah (corresponding author), Department of Mathematics, Faculty of Science, Damietta University, New Damietta 34517, Egypt, e-mail: r_elashwah@yahoo.com
