Estimation of Upper Bounds for Initial Coefficients and Fekete-Szegö Inequality for a Subclass of Analytic Bi-univalent Functions

  • G. SaravananEmail author
  • K. Muthunagai
Conference paper
Part of the Trends in Mathematics book series (TM)


In this article we have introduced a class \(\mathcal {\tilde {R}}_{\varSigma }(\eta ,q,\varsigma ),\eta \in \mathbb {C}-\{0\} \) of bi-univalent functions defined by symmetric q-derivative operator. We have estimated the upper bounds for the initial coefficients and Fekete- Szeg\(\ddot {o}\) inequality by making use of Chebyshev polynomials.


Bi-univalent Chebyshev polynomials Symmetric q-derivative operator 


  1. 1.
    Aldweby, H., Darus, M.: A subclass of harmonic univalent functions associated with q-analogue of Dziok- Srivastava operator. ISRN Math. Anal. (2013) doi:382312, 6 pages.Google Scholar
  2. 2.
    Altinkaya, Ş., Yalçn, S.: Estimates on coefficients of a general subclass of bi-univalent functions associated with symmetric q- derivative operator by means of the chebyshev polynomials. Asia Pacific Journal of Mathematices. 4, no. 2, 90–99 (2017).zbMATHGoogle Scholar
  3. 3.
    Aydoğan, M., Kahramaner, Y., Polatoğlu, Y.: Close-to-Convex Functions Defined by Fractional Operator. Appl. Math. Sci. 7, 2769–2775 (2013).MathSciNetCrossRefGoogle Scholar
  4. 4.
    Brahim, K. L., Sidomou, Y.: On some symmetric q-special functions. Le Mat. 68, 107–122 (2013).MathSciNetzbMATHGoogle Scholar
  5. 5.
    Brannan, D. A., Clunie, J. G.: Aspects of Contemporary Complex Analysis. Academic Press London and New York (1980).Google Scholar
  6. 6.
    Doha, E. H.: The first and second kind Chebyshev cofficients of the moments of the general-order derivative of an infinitely differentiable function. Int. J. Comput. Math. 51, 21–35 (1994).CrossRefGoogle Scholar
  7. 7.
    El-Ashwah, R. M.: Subclasses of bi-univalent functions defined by convolution. Journal of the Egyptian Mathematical Society. 22, 348–351 (2014).MathSciNetCrossRefGoogle Scholar
  8. 8.
    Frasin, B. A., Aouf, M. K.: New subclasses of bi-univalent functions. Applied Mathematics Letters. 24, 1569–1573 (2011).MathSciNetCrossRefGoogle Scholar
  9. 9.
    Gasper, G., Rahman, M.: Basic Hypergeometric Series. Cambridge Univ. Press Cambridge MA (1990).zbMATHGoogle Scholar
  10. 10.
    Hayami, T., Owa, S.: Coefficient bounds for bi-univalent functions. Panamerican Mathematical Journal. 22, 15–26 (2012).MathSciNetzbMATHGoogle Scholar
  11. 11.
    Jackson, F. H.: On q-functions and a certain difference operator. Trans. Royal Soc. Edinburgh. 46, 253–281 (1908).CrossRefGoogle Scholar
  12. 12.
    Lewin, M.: On a Coefficient problem for bi-univalent functions. Proc. Amer. Math.Soc. 18, 63–68 (1967).MathSciNetCrossRefGoogle Scholar
  13. 13.
    Mohammed, A., Darus, M.: A generalized operator involving the q-hypergeometric function. Mat. Vesnik, 65, 454–465 (2013).MathSciNetzbMATHGoogle Scholar
  14. 14.
    Mason, J. C.: Chebyshev polynomials approximations for the L-membrane eigenvalue problem. SIAM J. Appl. Math. 15, 172–186 (1967).MathSciNetCrossRefGoogle Scholar
  15. 15.
    Polatoğlu, Y.: Growth and distortion theorems for generalized q-starlike functions. Adv. Math., Sci. J. 5, 7–12 (2016).Google Scholar
  16. 16.
    Pommerenke, C.: Univalent Functions. Vandenhoeck & Ruprecht, G\(\ddot {o}\)ttingen (1975).Google Scholar
  17. 17.
    Purohit, S. D., Raina, R. K.: Fractional q -calculus and certain subclass of univalent analytic functions. Mathematica. 55, 62–74 (2013).MathSciNetzbMATHGoogle Scholar
  18. 18.
    \(\ddot {O}\)zkan Ucar, H. E.: Cofficient inequalties for q-starlike functions. Appl. Math. Comp. 276, 122–126 (2016).Google Scholar
  19. 19.
    Vijaya, R., Sudharsan, T. V., Sivasubramanian, S.: Coefficient Estimates for Certain Subclasses of Biunivalent Functions Defined by Convolution. International Journal of Analysis. (2016) doi: 6958098, 5 pages.Google Scholar
  20. 20.
    Xu, Q. H., Gui, Y. C., Srivastava, H. M.: Coefficient estimates for a certain subclass of analytic and bi-univalent functions. Applied Mathematics Letters. 25, no. 6, 990–994 (2012).MathSciNetCrossRefGoogle Scholar
  21. 21.
    Xu, Q. H., Xiao, H. G., Srivastava, H. M.: A certain general subclass of analytic and bi-univalent functions and associated coefficient estimate problems. Applied Mathematics and Computation. 218, no. 23, 11461–11465 (2012).MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.School of Advanced SciencesVIT ChennaiChennaiIndia
  2. 2.Department of MathematicsPatrician College of Arts and ScienceChennaiIndia
  3. 3.School of Advanced Sciences, VIT ChennaiChennaiIndia

Personalised recommendations