International Journal of Nonlinear Analysis and Applications

Coefficient bounds for a new family of bi-univalent functions associated with $(U,V)$-Lucas polynomials

Document Type : Research Paper

Authors

1 Department of Mathematics, University of Ilorin, P. M. B. 1515, Ilorin, Nigeria

2 University of Al-Qadisiyah, College of science, Department of Mathematics, Al Diwaniyah, Al-Qadisiyah, Iraq

Abstract

The aim of this paper is to use (U,V)-Lucas polynomials to introduce and study a new family of holomorphic and bi-univalent functions defined in the open unit disk which involve q-derivative operator. We investigate upper bounds for the Taylor-Maclaurin coefficients |d2| and |d3| and Fekete- Szego ̈ problem for functions belongs to this new family. Some interesting consequences of the results established here are indicated.

Keywords

[1] A. Akgul, (P, Q)-Lucas polynomial coefficient inequalities of the bi-univalent function class, Turk. J. Math. 43 (2019) 2170—2176.
[2] A. Akgul and F.M. Sakar, certain subclass of bi-univalent analytic functions introduced by means of the q-analogue of Noor integral operator and Horadam polynomials, Turk. J. Math. 43 (2019) 2275-–2286.
[3] H. Aldweby and M. Darus, A subclass of harmonic univalent functions associated with q-analogue of DziokSrivastava operator, ISRN Math. Anal. 2013 (2013), Article ID 382312, 6 pages.
[4] H. Aldweby and M. Darus, Coefficient estimates for initial taylor-maclaurin coefficients for a subclass of analytic and bi-univalent functions associated with q-derivative operator, Recent Trends Pure Appl. Math. 2017 (2017).
[5] R.M. Ali, S.K. Lee, V. Ravichandran and S. Supramanian, Coefficient estimates for bi-univalent Ma-Minda starlike and convex functions, Appl. Math. Lett. 25(3) (2012) 344—351.
[6] S. Altinkaya and S. Yalcin, On the (p, q)-Lucas polynomial coefficient bounds of the bi-univalent function class, Bol. Soc. Mat. Mex. 25 (2019) 567—575.
[7] S. Altinkaya and S. Yalcin, The (p, q)-Chebyshev polynomial bounds of a general bi-univalent function class, Bol. Soc. Mat. Mex. 26 (2019) 341—348.
[8] S. Altinkaya and S. Yal¸cin, Some application of the (p, q)-Lucas polynomials to the bi-univalent function class Σ,Math. Sci. Appl. E-Notes 8(1) (2020) 134—141.
[9] A. Aral, V. Gupta and R.P. Agarwal, Application of q-Calculus in Operator Theory, Springer, New York, USA, 2013.
[10] K.O. Babalola, On λ-pseudo-starlike functions, J. Class. Anal. 3(2) (2013) 137–147.
[11] S. Bulut, Certain subclasses of analytic and bi-univalent functions involving the q-derivative operator, Communications Faculty of Sciences University of Series A1: Math. Stat. 66(1) (2017) 108–114.
[12] M. Caglar, E. Deniz and H.M. Srivastava, Second Hankel determinant for certain subclasses of bi-univalent functions, Turk. J. Math. 41 (2017) 694–706.
[13] M. Caglar, H. Orhan and N. Yagmur, Coefficient bounds for new subclasses of bi-univalent functions, Filomat 27(7) (2013) 1165—1171.
[14] P.L. Duren, Univalent Functions, Grundlehren der Mathematischen Wissenschaften, Springer, New York, 1983.
[15] P. Filipponi and A.F. Horadam, Derivative sequences of Fibonacci and Lucas polynomials, Applications of Fibonacci Numbers, Springer, 1991, pp. 99-–108.
[16] P. Filipponi and A.F. Horadam, Second derivative sequences of Fibonacci and Lucas polynomials. Fibonacci Q. 31(3) (1993) 194-–204.
[17] P. Goswami, B.S. Alkahtani and T. Bulboaca, Estimate for initial Maclaurin coefficients of certain subclasses of bi-univalent functions, (2015) arXiv:1503.04644v1 [math.CV]
[18] H. O Guney, G. Murugusundaramoorthy and K. Vijaya, Coefficient bounds for subclasses of biunivalent functions associated with the chebyshev polynomials, J. Complex Anal. (2017) Article ID: 4150210, 7 pages.
[19] F.H. Jackson, On q-functions and a certain difference operator, Trans. Roy. Soc. Edinburgh 46 (1908) 253–281.
[20] F.H. Jackson, On q-definite integrals, Quart. J. Pure Appl. Math. 41 (1910) 193–203.
[21] G.Y. Lee and M. Asci, Some properties of the (p, q)-Fibonacci and (p, q)-Lucas polynomials, J. Appl. Math. (2012) Article ID: 264842, 1–18.
[22] M. Lewin, On a coefficient problem for bi-univalent functions, Proc. Am. Math. Soc. 18(1) (1967) 63-–68.
[23] A. Lupas, A guide of Fibonacci and Lucas polynomials, Octagon Math. Mag. 7 (1999) 2–12.
[24] G. Murugusundaramoorthy and S. Yal¸cin, On λ-Pseudo bi-starlike functions related (p, q)-Lucas polynomial, Lib. Math. (N.S.) 39(2) (2019) 79–88.
[25] G. Murugusundaramoorthy, K. Vijaya and H. O. Guney,  On λ-pseudo bi-starlike functions with respect to symmetric points associated to shell-like curves, Kragujevac J. Math. 45 (1) (2021) 103–114.
[26] F. Muge Sakar and M. O. Guney, Coefficient estimates for certain subclasses of m-mold symmetric bi-univalent functions defined by the q-derivative operator, Konuralp J. Math. 6 (2) (2018) 279–285.
[27] H. Orhan and H. Arikan, Lucas polynomial coefficients inequalities of bi-univalent functions defined by the combination of both operators of Al-Oboudi and Ruscheweyh, Afr. Mat. 32 (2021) 589–598.
[28] A. Ozkoc and A. Porsuk, A note for the (  p, q)-Fibonacci and Lucas quaternion polynomials, Konuralp J. Math. 5(2) (2017) 36–46.
[29] A.B. Patil and T.G. Shaba, On sharp Chebyshev polynomial bounds for general subclassof bi-univalent functions, Appl. Sci. 23 (2021) 109–117.
[30] T.M. Seoudy and M.K. Aouf, Convolution properties for C certain classes of analytic functions defined by qderivative operator, Abstr. Appl. Anal. 2014 (2014), Article ID 846719, 7 pages.
[31] T.M. Seoudy and M.K. Aouf, Coefficient estimates of new classes of q-starlike and q-convex functions of complex order, J. Math. Inequal. 10 (1) (2016) 135–145.
[32] T. G. Shaba, Subclass of bi-univalent functions satisfying subordinate conditions defined by Frasin differential operator, Turkish J. Inequal. 4 (2) (2020) 50–58.
[33] T.G. Shaba, On some new subclass of bi-univalent functions associated with Opoola differential operator, Open J. Math. Anal. 4(2) (2020) 74—79.
[34] T.G. Shaba, Certain new subclasses of analytic and bi-univalent functions using Salagean operator, Asia Pac. J. Math. 7(29) (2020) 1—11.
[35] T.G. Shaba, A.A. Ibrahim and A.A. Jimoh, On a new subclass of bi-pseudo-starlike functions defined by frasin differential operator, Adv. Math. Sci. J. 9(7) (2020) 4829-–4841.
[36] H.M. Srivastava, S. Altınkaya and S. Yal¸cın, Certain subclasses of bi-univalent functions associated with the Horadam polynomials. Iran. J. Sci. Technol. Trans. Sci. 43 (2019) 1873–1879.
[37] H. M. Srivastava, S. Bulut, M. Caglar and N. Yagmur, Coefficient estimates for a general subclass of analytic and bi-univalent functions, Filomat 27 (2013) 831—842.
[38] H.M. Srivastava and S.S. Eker, Some applications of a subordination theorem for a class of analytic functions, Appl. Math. Lett. 21 (2008) 394—399.
[39] H.M. Srivastava, S.S. Eker and R.M. Ali, Coefficient bounds for a certain class of analytic and bi-univalent functions, Filomat 29(8) (2015) 1839—1845.
[40] H.M. Srivastava, S.S. Eker and S.G. Hamidi and J.M. Jahangiri, Faber polynomial coefficient estimates for biunivalent functions defined by the Tremblay fractional derivative operator, Bull. Iran. Math. Soc. 44 (2018) 149-–157.
[41] 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.
[42] H. M. Srivastava, S. Hussain, A. Raziq and M. Raza, The Fekete-Szego functional for a subclass of analytic functions associated with quasi-subordination, Carpathian J. Math. 34 (2018) 103—113.
[43] H.M. Srivastava, S. Khan, Q.Z. Ahmad and N. Khan and S. Hussain, The Faber polynomial expansion method and its application to the general coefficient problem for some subclasses of bi-univalent functions associated with a certain q-integral operator, Stud. Univ. Babes¸-Bolyai Math. 63 (2018) 419-–436.
[44] H.M. Srivastava, N. Magesh and J. Yamini, Initial coefficient estimates for bi-λ-convex and bi-µ-starlike functions connected with arithmetic and geometric means, Electron. J. Math. Anal. Appl. 2(2) (2014) 152-–162.
[45] H.M. Srivastava, A.K. Mishra and M.K. Das, The Fekete-Szego problem for a subclass of close-to-convex functions,Complex Variables Theory Appl. 44 (2001) 145—163.
[46] H.M. Srivastava, A.K. Mishra and P. Gochhayat, Certain subclasses of analytic and bi-univalent functions, Appl. Math. Lett. 23(10) (2010) 1188—1192.
[47] 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, Math. 8 (2020) Article ID 172, 1—12.
[48] H.M. Srivastava, F.M. Sakar and H.O. G¨uney, Some general coefficient estimates for a new class of analytic and bi-univalent functions defined by a linear combination, Filomat 34 (2018) 1313–1322.
[49] 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 (2019) 493–503.
[50] H. Tang, H.M. Srivastava, S. Sivasubramanian and P. Gurusamy, The Fekete-Szego functional problems for some subclasses of m-fold symmetric bi-univalent functions, J. Math. Inequal. 10(4) (2016) 1063-–1092.
[51] P. Vellucci and A.M. Bersani, The class of Lucas-Lehmer polynomials, Rend. Mat. Appl. 37 (2016) 43–62.
[52] A.K. Wanas, Application of (M, N)-Lucas polynomials for holomorphic and bi-univalent functions, Filomat 39(10) (2020) 3361–3368.
[53] T. Wang and W. Zhang, Some identities involving Fibonacci, Lucas polynomials and their applications, Bull. Math. Soc. Sci. Math. Roum. 55(1) (2012) 95–103.
[54] S. Yal¸cin, K. Muthunagai and G. Saravanan, A subclass with bi-univalent involving The (p, q)-Lucas polynomials and its coefficient bounds, Bol. Soc. Mat. Mex. 26 (2020) 1015–1022.
International Journal of Nonlinear Analysis and Applications
Volume 13, Issue 1
March 2022
Pages 615-626
  • Receive Date: 10 June 2021
  • Revise Date: 23 September 2021
  • Accept Date: 24 September 2021