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The (pq)-Chebyshev polynomial bounds of a general bi-univalent function class

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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 (pq)-Chebyshev polynomials. Then we will derive the (pq)-Chebyshev polynomial bounds for the initial coefficients and determine Fekete–Szegö functional for \(f\in \mathcal {S}_{\varSigma }^{\eta ,\mu }\left( p,q\right) \).

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References

  1. 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)

    Article  Google Scholar 

  2. Altınkaya, Ş., Yalçın, S.: On the Chebyshev polynomial coefficient problem of some subclasses of bi-univalent functions. Gulf J. Math. 5, 34–40 (2017)

    MathSciNet  MATH  Google Scholar 

  3. Brannan, D.A., Taha, T.S.: On some classes of bi-univalent functions. Stud. Univ. Babeş Bolyai Math. 31, 70–77 (1986)

    MathSciNet  MATH  Google Scholar 

  4. 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)

    MATH  Google Scholar 

  5. Duren, P.L.: Univalent Functions, Grundlehren der Mathematischen Wissenschaften, vol. 259. Springer, New York (1983)

    Google Scholar 

  6. 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)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  8. Filipponi, P., Horadam, A.F.: Derivative sequences of Fibonacci and Lucas polynomials. Appl. Fibonacci Numbers 4, 99–108 (1991)

    Article  MathSciNet  Google Scholar 

  9. Filipponi, P., Horadam, A.F.: Second derivative sequences of Fibonacci and Lucas polynomials. Fibonacci Q. 31, 194–204 (1993)

    MathSciNet  MATH  Google Scholar 

  10. 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)

  11. 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)

    MathSciNet  MATH  Google Scholar 

  12. Kızılateş, C., Tuğlu, N., Çekim, B.: On the $(p, q)$-Chebyshev polynomials and related polynomials. Mathematics 7, 1–12 (2019)

    MATH  Google Scholar 

  13. Lewin, M.: On a coefficient problem for bi-univalent functions. Proc. Am. Math. Soc. 18, 63–68 (1967)

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  15. Ma, R., Zhang, W.: Several identities involving the Fibonacci numbers and Lucas numbers. Fibonacci Q. 45, 164–170 (2007)

    MathSciNet  MATH  Google Scholar 

  16. Mason, J.C.: Chebyshev polynomials approximations for the L-membrane eigenvalue problem. SIAM J. Appl. Math. 15, 172–186 (1967)

    Article  MathSciNet  Google Scholar 

  17. Mason, J.C., Handscomb, D.C.: Chebyshev Polynomials. Chapman & Hall, Boca Raton (2003)

    MATH  Google Scholar 

  18. 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)

    Article  MathSciNet  Google Scholar 

  19. Özkoç, A., Porsuk, A.: A note for the $(p, q)$-Fibonacci and Lucas quarternion polynomials. Konuralp J. Math. 5, 36–46 (2017)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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Altınkaya, Ş., Yalçın, S. The (pq)-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

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  • DOI: https://doi.org/10.1007/s40590-019-00246-2

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