Abstract
In the present paper, we will define the bi-univalent function class \( \mathcal {S}_{\varSigma }^{\eta ,\mu }\left( p,q\right) \) related to the (p, q)-Chebyshev polynomials. Then we will derive the (p, q)-Chebyshev polynomial bounds for the initial coefficients and determine Fekete–Szegö functional for \(f\in \mathcal {S}_{\varSigma }^{\eta ,\mu }\left( p,q\right) \).
Similar content being viewed by others
References
Altınkaya, Ş., Yalçın, S.: Coefficient estimates for two new subclasses of bi-univalent functions with respect to symmetric points. J. Funct. Spaces 2015, 1–5 (2015)
Altınkaya, Ş., Yalçın, S.: On the Chebyshev polynomial coefficient problem of some subclasses of bi-univalent functions. Gulf J. Math. 5, 34–40 (2017)
Brannan, D.A., Taha, T.S.: On some classes of bi-univalent functions. Stud. Univ. Babeş Bolyai Math. 31, 70–77 (1986)
Brannan, D.A., Clunie, J.G.: Aspects of Contemporary Complex Analysis, (Proceedings of the NATO Advanced Study Institute Held at University of Durham: July 1–20, 1979). Academic Press, New York (1980)
Duren, P.L.: Univalent Functions, Grundlehren der Mathematischen Wissenschaften, vol. 259. Springer, New York (1983)
Doha, E.H.: The first and second kind Chebyshev coefficients of the moments of the general-order derivative of an infinitely differentiable function. Int. J. Comput. Math. 51, 21–35 (1994)
Frasin, B.A., Aouf, M.K.: New subclasses of bi-univalent functions. Appl. Math. Lett. 24, 1569–1573 (2011)
Filipponi, P., Horadam, A.F.: Derivative sequences of Fibonacci and Lucas polynomials. Appl. Fibonacci Numbers 4, 99–108 (1991)
Filipponi, P., Horadam, A.F.: Second derivative sequences of Fibonacci and Lucas polynomials. Fibonacci Q. 31, 194–204 (1993)
Hussain, S., Khan, S., Zaighum, M.A., Darus, M., Shareef, Z.: Coefficients bounds for certain subclass of bi-univalent functions associated with Ruscheweyh q-differential operator. J. Complex Anal. 1–9 (2017) (2826514)
Khan, S., Khan, N., Hussain, S., Ahmad, Q.Z., Zaighum, M.A.: Some subclasses of bi-univalent functions associated with Srivastava–Attiya operator. Bull. Math. Anal. Appl. 9, 37–44 (2017)
Kızılateş, C., Tuğlu, N., Çekim, B.: On the $(p, q)$-Chebyshev polynomials and related polynomials. Mathematics 7, 1–12 (2019)
Lewin, M.: On a coefficient problem for bi-univalent functions. Proc. Am. Math. Soc. 18, 63–68 (1967)
Lupas, A.: A guide of Fibonacci and Lucas polynomials. Octag. Math. Mag. 7, 2–12 (1999)
Ma, R., Zhang, W.: Several identities involving the Fibonacci numbers and Lucas numbers. Fibonacci Q. 45, 164–170 (2007)
Mason, J.C.: Chebyshev polynomials approximations for the L-membrane eigenvalue problem. SIAM J. Appl. Math. 15, 172–186 (1967)
Mason, J.C., Handscomb, D.C.: Chebyshev Polynomials. Chapman & Hall, Boca Raton (2003)
Netanyahu, E.: The minimal distance of the image boundary from the origin and the second coefficient of a univalent function in $\left|z\right|<1,$ Arch. Ration. Mech. Anal. 32, 100–112 (1969)
Özkoç, A., Porsuk, A.: A note for the $(p, q)$-Fibonacci and Lucas quarternion polynomials. Konuralp J. Math. 5, 36–46 (2017)
Srivastava, H.M., Mishra, A.K., Gochhayat, P.: Certain subclasses of analytic and bi-univalent functions. Appl. Math. Lett. 23, 1188–1192 (2010)
Wang, T., Zhang, W.: Some identities involving Fibonacci, Lucas polynomials and their applications. Bull. Math. Soc. Sci. Math. Roum. 55, 95–103 (2012)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Altınkaya, Ş., Yalçın, S. The (p, q)-Chebyshev polynomial bounds of a general bi-univalent function class. Bol. Soc. Mat. Mex. 26, 341–348 (2020). https://doi.org/10.1007/s40590-019-00246-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40590-019-00246-2