Received April 20, 2025. Published online April 14, 2026.
Abstract: Using the $q$-derivative operator, we define a new subclass of bi-univalent functions associated with Horadam polynomials. Furthermore, we obtain estimates for the initial Taylor-Maclaurin coefficients of functions in this class and an estimate for the Fekete-Szegö problem for the same class.
References: [1] C. Abirami, N. Magesh, J. Yamini: Initial bounds for certain classes of bi-univalent functions defined by Horadam polynomials. Abstr. Appl. Anal. 2020 (2020), Article ID 7391058, 8 pages. DOI 10.1155/2020/7391058 | MR 4062163 | Zbl 1436.30010
[2] T. Al-Hawary, A. Amourah, B. A. Frasin: Fekete-Szegö inequality for bi-univalent functions by means of Horadam polynomials. Bol. Soc. Mat. Mex., III. Ser. 27 (2021), Article ID 79, 12 pages. DOI 10.1007/s40590-021-00385-5 | MR 4322463 | Zbl 1482.30026
[3] T. Al-Hawary, B. Frasin, D. Breaz, L.-I. Cotîrlă: Inclusive subclasses of bi-univalent functions defined by error functions subordinate to Horadam polynomials. Symmetry 17 (2025), Article ID 211, 15 pages. DOI 10.3390/sym17020211
[4] S. Altınkaya, S. Yalcin: On the Chebyshev polynomial coefficient problem of some subclasses of bi-univalent functions. Gulf J. Math. 5 (2017), 34-40. DOI 10.56947/gjom.v5i3.105 | MR 3718695 | Zbl 1386.30009
[5] P. L. Duren: Univalent Functions. Grundlehren der Mathematischen Wissenschaften 259. Springer, New York (1983). MR 0708494 | Zbl 0514.30001
[6] B. A. Frasin, M. K. Aouf: New subclasses of bi-univalent functions. Appl. Math. Lett. 24 (2011), 1569-1573. DOI 10.1016/j.aml.2011.03.048 | MR 2803711 | Zbl 1218.30024
[7] A. F. Horadam, J. M. Mahon: Pell and Pell-Lucas polynomials. Fibonacci Q. 23 (1985), 7-20. DOI 10.1080/00150517.1985.12429849 | MR 0786356 | Zbl 0557.10011
[8] T. Horzum, E. G. Koçer: On some properties of Horadam polynomials. Int. Math. Forum 4 (2009), 1243-1252. MR 2545119 | Zbl 1246.11032
[9] F. H. Jackson: On $q$-functions and a certain difference operator. Earth Environ. Sci. Trans. R. Soc. Edinb. 46 (1909), 253-281. DOI 10.1017/S0080456800002751
[10] F. H. Jackson: On $q$-definite integrals. Quart. J. Pure Appl. Math. 41 (1910), 193-203. JFM 41.0317.04
[11] H. M. Srivastava, Ş. Altınkaya, S. Yalçın: Certain subclasses of bi-univalent functions associated with the Horadam polynomials. Iran. J. Sci. Technol., Trans. A, Sci. 43 (2019), 1873-1879. DOI 10.1007/s40995-018-0647-0 | MR 3978601
[12] 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
[13] H. M. Srivastava, A. K. Wanas, G. Murugusundaramoorthy: A certain family of bi-univalent functions associated with the Pascal distribution series based upon the Horadam polynomials. Surv. Math. Appl. 16 (2021), 193-205. MR 4224411 | Zbl 1470.30017
[14] S. R. Swamy, D. Breaz, L.-I. Cotîrlă, K. Venugopal: Bi-univalent function subclasses with $(p,q)$-derivative operator linked to Horadam polynomials. Kragujevac J. Math. 50 (2026), 1279-1296. DOI 10.46793/KgJMat2608.1279S
[15] A. K. Wanas, S. Yalçın: Horadam polynomials and their applications to new family of bi-univalent functions with respect to symmetric conjugate points. Proyecciones 40 (2021), 107-116. DOI 10.22199/issn.0717-6279-2021-01-0007 | MR 4260846 | Zbl 1478.30007
Affiliations: Basem Aref Frasin (corresponding author), Department of Mathematics, Faculty of Science, Al al-Bayt University, Mafraq 25113, Jordan, e-mail: bafrasin@yahoo.com; Nandini Parashivamurthy, Latha Satyanarayana, Department of Mathematics, JSS Academy of Technical Education, Dr.Vishnuvardhan Road, Uttarahalli - Kengeri Main Road, Srinivaspura, Bengaluru 560 060, India, e-mail: pnandinimaths@gmail.com, drlatha@gmail.com
