# Coefficient Bounds and Fekete-Szeg¨o inequality for a Certain Families of Bi-Prestarlike Functions Defined by (M,N)-Lucas Polynomials

## DOI:

https://doi.org/10.24297/jam.v20i.8989## Keywords:

Bi-Univalent function, Bi-Prestarlike function, Lucas Polynomials## Abstract

In the current work, we use the (M,N)-Lucas Polynomials to introduce a new families of holomorphic and bi-Prestarlike functions defined in the unit disk O and establish upper bounds for the second and third coefficients of the Taylor-Maclaurin series expansions of functions belonging to these families. Also, we debate Fekete-Szeg¨o problem for these

families. Further, we point out several certain special cases for our results.

### Downloads

## References

C. Abirami, N. Magesh and J. Yamini, Initial bounds for certain classes of bi-univalent functions defined by Horadam polynomials, Abstr. Appl. Anal., 2020, Art. ID 7391058, (2020), 1–8.

[2] A. Akg¨ul, (P,Q)-Lucas polynomial coefficient inequalities of the bi-univalent function class, Turkish Journal of Mathematics, 43(2019), 2170–2176.

A. G. Al-Amoush, Certain subclasses of bi-univalent functions involving the Poisson distribution associated with Horadam polynomials, Malaya J. Mat., 7(2019), 618–624.

S. 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.

S. Altinkaya and S. Yal¸cin, Coefficient estimates for two new subclasses of bi-univalent functions with respect to symmetric points, J. Funct. Spaces, 2015, Art. ID 145242, (2015), 1–5.

S. Altinkaya and S. Yal¸cin, On the (p,q)-Lucas polynomial coefficient bounds of the biunivalent function class σ, Bolet´inde la Sociedad Matem´atica Mexicana, 25(2019), 567–575.

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

A. Amourah, Initial bounds for analytic and bi-univalent functions by means of (p,q)-Chebyshev polynomials defined by differential operator, General Letters in Mathematics, 7 (2019), 45–51.

A. Amourah, B. A. Frasin, G. Murugusundaramoorthy and T. Al-Hawary, Bi-Bazileviˇc functions of order ϑ + iδ associated with (p,q)-Lucas polynomials, AIMS Mathematics, 6(5)(2021), 4296–4305.

S. Bulut, Faber polynomial coefficient estimates for a subclass of analytic bi-univalent functions, Filomat, 30(2016), 1567–1575.

M. Caglar, E. Deniz and H. M. Srivastava, Second Hankel determinant for certain subclasses of bi-univalent functions, Turkish J. Math., 41(2017), 694–706.

P. L. Duren, Univalent Functions, Grundlehren der Mathematischen Wissenschaften, Band 259, Springer Verlag, New York, Berlin, Heidelberg and Tokyo, 1983.

M. Fekete and G. Szeg¨o, Eine bemerkung uber ungerade schlichte funktionen, J. London Math. Soc., 2(1933), 85–89.

P. Filipponi and AF. Horadam, Derivative sequences of Fibonacci and Lucas polynomials, Applications of Fibonacci Numbers, 4(1991), 99–108.

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

H. O. G¨uney, G. Murugusundaramoorthy and J. Sok´o l, Subclasses of bi-univalent functions related to shell-like curves connected with Fibonacci numbers, Acta Univ. Sapient. Math., 10(2018), 70–84.

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

S. Joshi, S. Joshi and H. Pawar, On some subclasses of bi-univalent functions associated with pseudo-starlike functions, J. Egyptian Math. Soc., 24(2016), 522–525.

GY. Lee and M. A¸sci, Some properties of the (p,q)-Fibonacci and (p,q)-Lucas polynomials, J. Appl. Math., 2012(2012), 1–18.

A. Lupas, A guide of Fibonacci and Lucas polynomials, Octagon Math. Mag., 7(1999), 2–12.

N. Magesh and S. Bulut, Chebyshev polynomial coefficient estimates for a class of analytic bi-univalent functions related to pseudo-starlike functions, Afr. Mat., 29(2018), 203–209.

E. P. Mazi and T. O. Opoola, On some subclasses of bi-univalent functions associating pseudo-starlike functions with Sakaguchi type functions, General Mathematics, 25(2017), 85–95.

G. Murugusundaramoorthy and S. Yal¸cin, On the λ-Psedo-bi-starlike functions related to (p,q)-Lucas polynomial, Libertas Mathematica (newseries), 39(2019), 79–88.

H. Orhan and H. Arikan, (P,Q)-Lucas polynomial coefficient inequalities of bi-univalent functions defined by the combination of both operators of Al-Aboudi and Ruscheweyh, Afr. Mat. (2020). https://doi.org/10.1007/s13370-020-00847-5

S. Ruscheweyh, Linear operators between classes of prestarlike functions, Comment. Math. Helv., 52(4), (1977), 497-509.

H. M. Srivastava and D. Bansal, Coefficient estimates for a subclass of analytic and biunivalent functions, J. Egyptian Math. Soc., 23(2015), 242–246.

H. M. Srivastava, S. Bulut, M. Caglar and N. Yagmur, Coefficient estimates for a general subclass of analytic and bi-univalent functions, Filomat, 27(5)(2013), 831–842.

H. M. Srivastava, S. S. Eker and R. M. Ali, Coefficient bounds for a certain class of analytic and bi-univalent functions, Filomat, 29(2015), 1839–1845.

H. M. Srivastava, S. S. Eker, S. G. Hamidi and J. M. Jahangiri, Faber polynomial coefficient estimates for bi-univalent functions defined by the Tremblay fractional derivative operator, Bull. Iranian Math. Soc., 44(1)(2018), 149–157.

H. M. Srivastava, S. Gaboury and F. Ghanim, Coefficient estimates for some general subclasses of analytic and bi-univalent functions, Afr. Mat., 28(2017), 693–706.

H. M. Srivastava, S. Gaboury and F. Ghanim, Coefficient estimates for a general subclass of analytic and bi-univalent functions of the Ma-Minda type, Rev. Real Acad. Cienc. Exactas F´ıs. Natur. Ser. A Mat. (RACSAM), 112(2018), 1157–1168.

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

H. M. Srivastava, A. Motamednezhad and E. A. Adegani, Faber polynomial coefficient estimates for bi-univalent functions defined by using differential subordination and a certain fractional derivative operator, Mathematics, 8(2020), Art. ID 172, 1–12.

H. M. Srivastava, N. Raza, E. S. A. AbuJarad, G. Srivastava and M. H. AbuJarad, FeketeSzeg¨o inequality for classes of (p, q)-starlike and (p, q)-convex functions, Rev. Real Acad. Cienc. Exactas F´ıs. Natur. Ser. A Mat. (RACSAM), 113(2019), 3563–3584.

[35] H. M. Srivastava and A. K. Wanas, Initial Maclaurin coefficient bounds for new subclasses of analytic and m-fold symmetric bi-univalent functions defined by a linear combination, Kyungpook Math. J., 59(3)(2019), 493–503.

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

P. Vellucci and AM. Bersani, The class of Lucas-Lehmer polynomials, Rend Math. Appl., 37(2016), 43–62.

F. Yousef, T. Al-Hawary and G. Murugusundaramoorthy, Fekete-Szeg¨o functional problems for some subclasses of bi-univalent functions defined by Frasin differential operator, Afr. Mat., 30(2019), 495–503.

F. Yousef, B. Frasin and T. Al-Hawary, Fekete-Szeg¨o inequality for analytic and bi-univalent functions subordinate to Chebyshev polynomials, Filomat, 32(2018), 3229–3236.

T. Wang and W. Zhang, Some identities involving Fibonacci, Lucas polynomials and their applications, Bull Math. Soc. Sci. Math. Roum., 55(2012), 95–103.

A. Zireh, E. Analouei Adegani and S. Bulut, Faber polynomial coefficient estimates for a comprehensive subclass of analytic bi-univalent functions defined by subordination, Bull.

## Downloads

## Published

## How to Cite

*JOURNAL OF ADVANCES IN MATHEMATICS*,

*20*, 121–134. https://doi.org/10.24297/jam.v20i.8989

## Issue

## Section

## License

Copyright (c) 2021 Najah Ali Jiben Al-Ziadi, Abbas Kareem Wanas

This work is licensed under a Creative Commons Attribution 4.0 International License.

All articles published in *Journal of Advances in Linguistics* are licensed under a Creative Commons Attribution 4.0 International License.