Abstract

We aim at introducing two new families of holomorphic and bi-univalent functions in the open unit disc D by making use of Horadam polynomials, which are known to generalize some potentially useful polynomials such as the Lucas polynomials, the Pell polynomials and the Chebyshev polynomials of the second kind. For functions belonging to the defined families, the coefficient inequalities and the Fekete-Szeg¨o problem are discussed. Some interesting consequences of the result found here are also presented.

Keywords

References

[1] A. G. Alamoush, Certain subclasses of bi-univalent functions involving the Poisson distribution associated with Haradam polynomials, Malaya Journal of Matematik, 7 (4) (2019), 618-624. doi.org/10.26637/MJM0704/0003.
[2] A. G. Alamoush, Coefficient estimates for certain subclass of bi-functions associated the Horadam polynomials, arXiv: 1812 .10589v1 [math.CV] 22 Dec 2018, 7 pages.
[3] A. Alsoboh and M. Darus, On Fekete-Szeg¨o problem associated with q-derivative operator, IOP conf. Series: Journal of Physics: Conf. Series 1212 (2019) 012003. doi: 10.1088/1742-6596/1212/1/012003.
[4] S¸¸ Altınkaya and S. Yalc¸ın, On the Chebyshev polynomial coefficient problem of some subclasses of bi-univalent functions, Gulf J. Math., 5(3) (2017), 34-40.
[5] D. A. Brannan and T. S. Taha, On some classes of bi-univalent functions, In: S. M. Mazhar, A. Hamoui, N.S. Faour (eds) Mathematical analysis and its applications. Kuwait, pp 53-60, KFAS Proceedings Series, Vol. 3 (1985), Pergamon Press (Elsevier Science Limited), Oxford, 1988; see also Studia Univ. Babes¸-Bolyai Math., 31(2) (1986), 70-77.
[6] S. Bulut, Faber polynomial coefficient estimates for a comprehensive subclass of analytic bi-univalent functions, C R Acad. Sci. Paris S´er I(352) (2014), 479-484.
[7] M. C¸ a˘glar, E. Deniz and H. M. Srivastava, S˙econd Hankel determinant for certain subclasses of bi-univalent functions, Turk. J. Math., 41 (2017), 694-706.
[8] P. L. Duren, Univalent Functions, Grundlehren der Mathematischen Wissenschaften, Band 259. Springer-Verlag, New York, (1983).
[9] M. Fekete and G. Szego¨, Eine Bemerkung U¨ ber Ungerade Schlichte Funktionen, J. Lond. Math. Soc., 89 (1933), 85-89.
[10] P. Filipponi and A.F. Horadam, Derivative sequences of Fibonacci and Lucas polynomials. In: G. E. Bergum, A. N. Philippou, A. F. Horadam (eds) Applications of Fibonacci Numbers, vol 4 (1990), pp 99-108, Proceedings of the fourth international conference on Fibonacci numbers and their applications,Wake Forest University,Winston- Salem, North Carolina; Springer (Kluwer Academic Publishers), Dordrecht, Boston and London, 4 (1991), 99- 108.
[11] P. Filipponi and A.F. Horadam, Second derivative sequence of Fibonacci and Lucas polynomials, Fibonacci Quart., 31 (1993), 194-204.
[12] A. F. Horadam and J. M. Mahon, Pell and Pell - Lucas polynomials, Fibonacci Quart., 23 (1985), 7-20.
[13] T. H¨orc¸um and E. G¨okc¸en Koc¸er, On some properties of Horadam polynomials, Int. Math. Forum., 4 (2009), 1243-1252.
[14] M. Lewin, On a coefficient problem for bi-univalent functions, Proc. Amer. Math. Soc., 18 (1967), 63-68.
[15] A. I. Lupas,¸ A guide of Fibonacci and Lucas polynomials, Octagon Math. Mag., 7 (1999), 2-12.
[16] M. H. Mohd and M. Darus, Fekete-Szeg¨o problems for Quasi-Subordination classes, Abst. Appl. Anal.,Article ID 192956, 14 pages (2012), doi: 10.1155/2012/192956.
[17] S. R. Swamy, Ruscheweyh derivative and a new generalized multiplier differential operator, Annal. Pure. Appl. Math.,10(2) (2015), 229-238.
[18] H. M. Srivastava, A. K. Mishra and P. Gochhayat, Certain subclasses of analytic and bi-univalent functions, Appl. Math. Lett., 23 (2010), 1188-1192.
[19] H. M. Srivastava, S¸¸ Altınkaya and S. Yalc¸ın, Certain Subclasses of bi-Univalent functions associated with the Horadam polynomials, Iran J. Sci. Technol. Trans. Sci., (2018). doi.org/10.1007/s 40995-018-0647-0.
[20] A. K. Wanas and A. A. Lupas, Applications of Horadam polynomials on Bazilevic bi- univalent function satisfying subordinate conditions, IOP Conf. Series: Journal of Physics: Conf. Series 1294 (2019) 032003, doi:10.1088/1742-6596/1294/3/032003.
[21] T-T. Wang and W-P. Zhang, Some identities involving Fibonacci, Lucas polynomials and their applications. Bull. Math. Soc. Sci. Math. Roumanie (New Ser.), 55 (103) (2012), 95-103.