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Proyecciones (Antofagasta)

versión impresa ISSN 0716-0917

Proyecciones (Antofagasta) vol.38 no.5 Antofagasta dic. 2019 


(p, q)-Lucas polynomials and their applications to bi-univalent functions

1Bursa Uludağ University, Dept. of Mathematics, Bursa, Turkey. E-mail:

2Bursa Uludağ University, Dept. of Mathematics, Bursa, Turkey.


In the present paper, by using the L p,q,n (x) functions, our methodology intertwine to yield the Theory of Geometric Functions and that of Special Functions, which are usually considered as very different fields. Thus, we aim at introducing a new class of bi-univalent functions defined through the (p, q)-Lucas polynomials. Furthermore, we derive coefficient inequalities and obtain Fekete-Szegö problem for this new function class.

Keywords: (p, q)-Lucas polynomials; Coefficient bounds; Bi-univalent functions

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[1] Ş. Altınkaya and S. Yalçın, “Bornes des coefficients des développements en polynômes de Faber d'une sous-classe de fonctions bi-univalentes”. Comptes rendus mathematique, vol. 353, no. 12, pp. 1075-1080, Dec. 2015, doi: 10.1016/j.crma.2015.09.003 [ Links ]

[2] D. Brannan and J. Clunie, Eds., Aspects of contemporary complex analysis. London: Academic Press, 1980. [ Links ]

[3] D. Brannan and T. Taha, “On some classes of bi-univalent functions”, Studia universitatis babes-bolyai. Mathematica, vol. 31, pp. 70-77, 1986. [ Links ]

[4] P. Duren, Univalent functions. New York, NY: Springer, 1983. [ Links ]

[5] M. Fekete and G. Szegö, “Eine bemerkung über ungerade schlichte funktionen”, Journal of the london mathematical society, vol. s1-8, no. 2, pp. 85-89, Apr. 1933, doi: 10.1112/jlms/s1-8.2.85. [ Links ]

[6] M. Lewin, “On a coefficient problem for bi-univalent functions”, Proceedings of the American mathematical society, vol. 18, no. 1, pp. 63-68, Feb. 1967, doi: 10.1090/S0002-9939-1967-0206255-1. [ Links ]

[7] G. Lee and M. Asci, “Some properties of the (p, q)-Fibonacci and (p, q)- Lucas polynomials”, Journal of applied mathematics, vol. 2012, Art. ID 264842, 2012, doi: 10.1155/2012/264842. [ Links ]

[8] A. Lupas, “A guide of Fibonacci and Lucas polynomials”. Octogon mathematical magazine, vol. 7, no. 1, pp. 2-12, 1999. [ Links ]

[9] E. Netanyahu, “The minimal distance of the image boundary from the origin and the second coefficient of a univalent function in |z| < 1”, Archive for rational mechanics and analysis, vol. 32, no. 2, pp. 100-112, Jan. 1969, doi: 10.1007/BF00247676. [ Links ]

[10] A. Özkoç and A. Porsuk, “A note for the (p, q)-Fibonacci and Lucas quarternion polynomials”, Konuralp journal of mathematics, vol. 5, no. 2, pp. 36-46, 2017. [On line]. Available: ]

[11] P. Filipponi and A. Horadam, “Derivative sequences of Fibonacci and Lucas polynomials”, in Applications of Fibonacci numbers, G. Bergum, A. Philippou, andA. Horadam , Eds. Dordrecht: Springer, 1991, pp. 99-108, doi: 10.1007/978-94-011-3586-3_12. [ Links ]

[12] P. Filipponi andA. Horadam , “Second derivative sequences of Fibonacci and Lucas polynomials”, Fibonacci quartery, vol. 31, no. 3, pp. 194-204, 1993. [On line]. Available: ]

[13] H. Srivastava, A. Mishra and P. Gochhayat, “Certain subclasses of analytic and bi-univalent functions”, Applied mathematics letters, vol. 23, no. 10, pp. 1188-1192, Oct. 2010, doi: 10.1016/j.aml.2010.05.009. [ Links ]

[14] P. Vellucci and A. Bersani, “The class of Lucas-Lehmer polynomials”, Rendiconti di matematica e delle sue applicazioni, vol. 37, no. 1-2, pp. 43-62, 2016. [On line]. Available: ]

[15] P. Vellucci and A. Bersani, “Orthogonal polynomials and Riesz bases applied to the solution of Love’s equation”. Mathematics and mechanics of complex systems, vol. 4, no. 1, pp. 55-66, 2016, doi: 10.2140/memocs.2016.4.55. [ Links ]

[16] P. Vellucci andA. Bersani , “Ordering of nested square roots of 2 according to the Gray code”. The ramanujan journal, vol. 45, no. 1, pp. 197-210, Jan. 2018, doi: 10.1007/s11139-016-9862-5. [ Links ]

[17] T. Wang and W. Zhang, “Some identities involving Fibonacci, Lucas polynomials and their applications”, Bulletin mathématique de la société des sciences mathématiques de Roumanie, vol. 55, no. 1, pp. 95-103, 2012. [On line]. Available: Links ]

Received: November 30, 2018; Accepted: October 30, 2019

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