On $(P,Q)-$Lucas Polynomial Coefficients for a New Class of Bi-Univalent Functions Associated with q-Analogue of Ruscheweyh Operator

Yıl 2019, Cilt: 2 Sayı: 1, 13 - 17, 30.10.2019

Arzu Akgül

Öz

Recently, Fibonacci polynomials, Chebyshev polynomials, Lucas polynomials, Pell polynomials, Lucas–Lehmer polynomials, orthogonal polynomials and other special polynomials became more and more important in the field of Geometric Function Theory. The Theory of Geometric Functions and that of Special Functions are usually considered as very different fields. In this study, by using Lucas polynomials of the second kind, subordination and Ruschewey differential operator,these different fields were connected and a new class of bi-univalent functions was introduced. Also coefficient estimates were obtained for this new class.

Anahtar Kelimeler

(P;Q)-Lucas Polynomial, Coefficient bounds, Bi- Univalent functions

Kaynakça

• [1] A. Akgül, Identification of initial Taylor-Maclaurin coefficients for generalized subclasses of bi-univalent functions, Sahand Communications in Mathematical Analysis (SCMA), University of Maragheh, 11(1)(2018), 133-143.
• [2] A. Akgül, The Fekete-Szego coefficient inequality for a new class of m-fold symmetri c bi -uni valent functi ons satisfying subordination condition, Honam Math. J., Korea Science, 70(4)(2018), 733-748.
• [3] A. Akgül, Coefficient estimates for certain subclass of bi-univalent functions obtained with polylogarithms, Mathemati cal Sciences and Applicati ons E-Notes, An International Electronic Journal, 6(1)(2018), 70-76.
• [4] A. Akgül, Certain inequalities for a general class of analytic and bi-univalent functions, Sahand Communications in Mathematical Analysis (SCMA)14(1) (2019), 1-13.
• [5] A. Akgül, Second-order differential subordinations on a class of analytic functions defined by the Rafid-Operator, Ukrainian Math. J., 70(5), October, 2018 (Ukrainian Original Vol. 70, No. 5, May, 2018),673-686.
• [6] A. Akgül, (P, Q)-Lucas polynomial coefficient inequalities of the bi-univalent function class, Turkish J. Math., (in press).
• [7] A. Özkoç, A. Porsuk, A note for the (p, q)-Fibonacci and Lucas quarternion polynomials Konuralp J. Math. 5 (2017), 36-46 .
• [8] S. Altinkaya, S. Yalçi n, On the (p, q)-Lucas polynomial coefficient bounds of the bi-univalent function class, Boletín de la Sociedad Matemática Mexicana, (2018), 1-9.
• [9] D. A. Brannan, T. S. Taha, On some classes of bi-univalent functions, Studia Univ. On some classes of bi-univalent functions, Studia Univ. Babe¸s-Bolyai Math., 31 (1986), 70-77.
• [10] H. Aldweby, M. Darus, Some Subordination Results on q-Analogue of Ruscheweyh Differential Operator, Abstr. Appl. Anal.,2014(2014), 6 pages.
• [11] P. L. Duren, Univalent Functions, Grundlehren der Mathematischen Wissenschaften Springer, New York, USA 259 (1983).
• [12] P. Filipponi, A.F Horadam, Derivative sequences of Fibonacci and Lucas polynomials In: Bergum, G.E., Philippou, A.N., Horadam, A.F. (eds.) Applications of Fibonacci Numbers, 4 pp. 99–108. Kluwer Academic Publishers, Dordrecht (1991)
• [13] P. Filipponi, A.F. Horadam, Second derivative sequences of Fibonacci and Lucas polynomials Fibonacci Q. 31 (1993), 194-204.
• [14] F. H. Jackson, On q-functions and a certain difference operator, Trans. Roy. Soc. Edinburgh, 46 (1908), 253-281.
• [15] D. O. Jackson, T. Fukuda, O. Dunn, E. Majors, On q-definite integrals Quart. J. Pure Appl. Math., 41 (1910), 193-203.
• [16] G.Y.Lee, M. A¸scı, Some properties of the (p,q)-Fibonacci and (p, q)-Lucas polynomials J. Appl. Math. 2012 (2012), 1-18 .
• [17] M. Lewin, On a coefficient problem for bi-univalent functions Proc. Am.Math. Soc. 18 (1967), 63-68.
• [18] A. Lupas, A guide of Fibonacci and Lucas polynomials Octag. Math. Mag. 7(1)(1999), 2-12.
• [19] W. C. Ma, D. Minda, A unified treatment of some special classes of univalent functions, Proceedings of the Conference on Complex Analysis (Tianjin, 1992), 1994 (1994), 157-169.
• [20] C. Pommerenke, Univalent functions , Vandenhoeck & Ruprecht, Gottingen,(1975).
• [21] M. I. S. Robertson, On the theory of univalent functions, Ann. of Math., 37 (1936), 374-408.
• [22] R. M. Ali, S. K. Lee, V. Ravichandran, S. Supramaniam, Coefficient estimates for bi-univalent Ma-Minda starlike and convex functions, Appl. Math. Lett., 25 (2012), 344-351.
• [23] H. M. Srivastava, A. K. Mishra, P. Gochhayat, Certain subclasses of analytic and bi-univalent functions, Appl. Math.Lett., 23(2010), 1188-1192.
• [24] R. Ma, W. Zhang, Several identities involving the Fibonacci numbers and Lucas numbers Fibonacci Q. 45 (2007), 164-170 .
• [25] P Vellucci, A.M. Bersani, The class of Lucas–Lehmer polynomials Rend. Mat. Appl. serie VII 37 (2016), 43-62.
• [26] P. Vellucci, A.M. Bersani, Orthogonal polynomials and Riesz bases applied to the solution of Love’s equation Math. Mech. Complex Syst. 4(2016), 55-66.
• [27] P. Vellucci, A.M. Bersani, Ordering of nested square roots of 2 according to the Gray code Ramanujan J. 45(2018), 197-210.
• [28] T. Wang, W. Zhang, Some identities involving Fibonacci, Lucas polynomials and their applications Bull. Math. Soc. Sci. Math. Roum. 55 (2012), 95-103.
• [29] S. Ruscheweyh,New criteria for univalent functions, Proc. Amer. Math. Soc., (49), (1975), 109-115.