Certain Families of Holomorphic and Sălăgean Type Bi-Univalent Functions Defined by (p,q)-Lucas Polynomials Involving a Modified Sigmoid Activation Function

Ali Mohammed  Ramadhan; and Najah  Ali Jiben Al-Ziadi

doi:10.24297/jam.v21i.9253

Authors

Ali Mohammed Ramadhan Department of Mathematics, College of Education, University of Al-Qadisiyah, Diwaniya-Iraq
and Najah Ali Jiben Al-Ziadi Department of Mathematics, College of Education, University of Al-Qadisiyah, Diwaniya-Iraq

DOI:

https://doi.org/10.24297/jam.v21i.9253

Keywords:

modified sigmoid function, Sălăgean operator, Lucas polynomials, Fekete-Szegӧ inequality, Bi-univalent functions, Holomorphic function

Abstract

The aim of the present paper is to introduce a certain families of holomorphic and Sălăgean type bi-univalent functions by making use (p, q) - Lucas polynomials involving the modified sigmoid activation function Φ(δ)=z/(1+e^-δ) δ>=1 in the open unit disk Λ. For functions belonging to these subclasses, we obtain upper bounds for the second and third coefficients. Also, we debate Fekete-Szegö inequality for these families. Further, we point out several certain special cases for our results.

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References

Ş. Altinkaya, Inclusion properties of Lucas polynomials for bi-univalent functions introduced through the q-analogue of the Noor integral operator, Turkish J. Math., 43(2019), 620-629.

Ş. Altinkaya and S. Yalçin, On the (p,q)-Lucas polynomial coefficient bounds of the bi-univalent functions class σ, Boletin dela Sociedad Matematica Mexicana, (2018), 1-9.

Ş. Altinkaya and S. Yalçin, (p,q)-Lucas polynomials and their applications to bi-univalent functions, Proyecciones, 39(5)(2019), 1093-1105.

N. A. J. Al-Ziadi and A. K. Wanas, Coefficient bounds and Fekete-Szegӧ inequality for a certain families of bi-prestarlike functions defined by (M,N)-Lucas polynomials, Journal of Advances in Mathematics, 20(2021), 121-134.

S. Bulut, Faber polynomial coefficient estimates for a comprehensive subclass of analytic bi-univalent functions, C. R. Acad. Paris. Ser. I., 352(6)(2014), 479-484.

P. L. Duren, Univalent Functions, Vol. 259 of Grundlehren der Mathematischen Wissenschaften, Springer, New York, NY, USA, (1983).

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)(2018). 7-20.

B. A. Frasin, M. K. Aouf, New subclasses of bi-univalent functions. Appl. Math. Lett., 24(2011), 1569-1573.

A. F. Horadam and J. M. Mahon, Pell and Pell-Lucas polynomials, Fibonacci Quart., 23(1985), 7-20.

T. Horzum and E. Gӧkçen Koçer, On some properties of Horadam polynomials, Int. Math. Forum, 4(2009), 1243-1252.

A. Lupas, A guide of Fibonacci and Lucas polynomials, Octogon Math. Mag., 7(1999), 3-12.

G. Ş. Sǎlǎgean, Subclasses of univalent functions, Lecture Notesin Math., Springer, Berlin, 1013(1983), 362-372.

T. G. Shaba and A. K. Wanas, Coefficient bounds for a new family of bi-univalent functions associated with (U,V )-Lucas polynomials, Int. J. Nonlinear Anal. Appl. 13(1) (2022), 615-626.

H. M. Srivastava, S. Gaboury and F. Ghanim, Coefficient estimates for some subclasses of m-fold symmetric bi-univalent functions, Acta Universitatis Apulensis, 41(2015), 153-164.

H. M. Srivastava, A. K. Mishra and P. Gochhayat, Certain subclasses of analytic and bi-univalent functions, Appl. Math. Lett., 23(2010), 1188–1192.

S. R. Swamy, P. K. Mamatha, N. Magesh and J. Yamini, Certain subclasses of bi-univalent functions defined by Sǎlǎgean operator with the (p,q)-Lucas polynomial, Advances in Mathematics: Scientific Journal, 9(8)(2020), 6017-6025.

S. R. Swamy, A. K. Wanas and Y. Sailaja, Some special families of holomorphic and Sălăgean type bi-univalent functions associated with (m,n)-Lucas polynomials, Communications in Mathematics and Applications, 11(4)(2020), 563-574.