Some special families of holomorphic and Sălăgean type bi-univalent functions associated with Horadam polynomials involving a modified sigmoid activation function
The aim of this paper is to introduce some special families of holomorphic and S\u{a}l\u{a}gean type bi-univalent functions by making use of Horadam polynomials involving the modified sigmoid activation function $\phi(s)=\frac{2}{1+e^{-s} },\,s\geq0$ in the open unit disc $\mathfrak{D}$. We investigate the upper bounds on initial coefficients for functions of the form $g_{\phi}(z)=z+\sum\limits_{j=2}^{\infty}\phi(s)d_jz^j$, in these newly introduced special families and also discuss the Fekete-Szegö problem. Some interesting consequences of the results established here are also indicated.
___
- [1] A.G. Alamoush, Coefficient estimates for certain subclass of bi-univalent functions
associated the Horadam polynomials, arXiv: 1812 .10589v1 [math.CV].
- [2] A.G. Alamoush, Certain subclasses of bi-univalent functions involving the Poisson
distribution associated with Haradam polynomials, Malaya J. Mat. 7 (4), 618–624,
2019.
- [3] Ş. Altınkaya and S. Yalçın, On the Chebyshev polynomial coefficient problem of some
subclasses of bi-univalent functions, Gulf J. Math. 5 (3), 34–40, 2017.
- [4] 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, KFAS Proceedings Series 3, 53–60, 1985, Pergamon Press (Elsevier Science
Limited), Oxford, 1988, see also Studia Univ. Babeş-Bolyai Math. 31 (2), 70–77,
1986.
- [5] S. Bulut, Faber polynomial coefficient estimates for a comprehensive subclass of analytic
bi-univalent functions, C R Acad. Sci. Paris Sér. I (352), 479–484, 2014.
- [6] P.L. Duren, Univalent Functions, Grundlehren der Mathematischen Wissenschaften,
Band 259. Springer-Verlag, New York, 1983.
- [7] O.A. Fadipe-Joseph, B.B. Kadir, S.E. Akinwumi and E.O. Adeniran, Polynomial
bounds for a class of univalent function involving sigmoid function, Khayyam J. Math.
4 (1), 88–101, 2018.
- [8] M. Fekete and G. Szegö, Eine Bemerkung Über Ungerade Schlichte Funktionen, J.
Lond. Math. Soc. 89, 85–89, 1933.
- [9] 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 4, 99-108, Springer, Dordrecht, 1991.
- [10] P. Filipponi and A.F. Horadam, Second derivative sequence of Fibonacci and Lucas
polynomials, Fibonacci Quart. 31, 194–204, 1993.
- [11] A.F. Horadam and J.M. Mahon, Pell and Pell - Lucas polynomials, Fibonacci Quart.
23, 7–20, 1985.
- [12] T. Hörzum and E. Gökçen Koçer, On some properties of Horadam polynomials, Int.
Math. Forum. 4, 1243–1252, 2009.
- [13] M. Lewin, On a coefficient problem for bi-univalent functions, Proc. Amer. Math.
Soc. 18, 63–68, 1967.
- [14] G.S. Sălăgean, Subclasses of Univalent Functions, Lecture notes in Mathematics
1013, 362–372, Springer, Berlin, 1983.
- [15] H.M. Srivastava, Ş. Altınkaya and S. Yalçın, Certain Subclasses of bi-univalent functions
associated with the Horadam polynomials, Iran J. Sci. Technol. Trans. Sci. 43,
1873–1879, 2019.
- [16] H.M. Srivastava, A.K. Mishra and P. Gochhayat, Certain subclasses of analytic and
bi-univalent functions, Appl. Math. Lett. 23, 1188–1192, 2010.
- [17] S.R. Swamy and Y. Sailaja, Horadam polynomial coefficient estimates for two families
of holomorphic and bi-univalent functions, International Journal of Mathematics
Trends and Technology 66 (8), 131–138, 2020.
- [18] 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, 032003, 2019. doi:10.1088/1742-6596/1294/3/032003.
- [19] 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),
95–103, 2012.