Horadam Polynomial coefficient estimates for the classes of \(\lambda \)–bi-pseudo-starlike and Bi-Bazilevič Functions


In this investigation, we propose to make use of the Horadam polynomials and introduce two classes of bi-univalent functions. For functions belonging to these classes, the coefficient inequalities and the Fekete–Szegö inequalities are discussed. Some interesting remarks of the results presented here are also investigated.

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  1. 1.

    Ali, R.M., S.K. Lee, V. Ravichandran, and S. Supramanian. 2012. Coefficient estimates for bi-univalent Ma-Minda starlike and convex functions. Applied Mathematics Letters 25 (3): 344–351.

    MathSciNet  Article  Google Scholar 

  2. 2.

    Altınkaya, Ş., and S. Yalçin. 2017. On the Chebyshev polynomial coefficient problem of some subclasses of bi-univalent functions. Gulf Journal of Mathematics 5 (3): 34–40.

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Altınkaya, Ş., and S. Yalçin. 2018. On the (p, q)-Lucas polynomial coefficient bounds of the bi-univalent function class. Boletin de la Sociedad Matematica Mexicana 1–9.

  4. 4.

    Bulut, S. 2018. Coefficient estimates for a class of analytic bi-univalent functions related to pseudo-starlike functions. Miskolc Mathematical Notes 19 (1): 149–156.

    MathSciNet  Article  Google Scholar 

  5. 5.

    Bulut, S., N. Magesh, and V.K. Balaji. 2017. Initial bounds for analytic and bi-univalent functions by means of Chebyshev polynomials. J. Class. Anal. 11 (1): 83–89.

    MathSciNet  Article  Google Scholar 

  6. 6.

    Deniz, E., J.M. Jahangiri, S.G. Hamidi, and S.K. Kina. 2018. Faber polynomial coefficients for generalized bi-subordinate functions of complex order. Journal of Mathematical Inequalities 12 (3): 645–653.

    MathSciNet  Article  Google Scholar 

  7. 7.

    Eker, S.S., and B. Şeker. 2018. On \(\lambda \)-pseudo bi-starlike and \(\lambda \)-pseudo bi-convex functions with respect to symmetrical points. Tbilisi Mathematical Journal 11 (1): 49–57.

    MathSciNet  Article  Google Scholar 

  8. 8.

    Girgaonkar, V.B., and S.B. Joshi. 2018. Coefficient estimates for certain subclass of bi-univalent functions associated with Chebyshev polynomial. Ganita 68 (1): 79–85.

    MathSciNet  Google Scholar 

  9. 9.

    Horadam, A.F., and J.M. Mahon. 1985. Pell and Pell–Lucas polynomials. Fibonacci Quarterly 23 (1): 7–20.

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Horzum, T., and E.G. Kocer. 2009. On some properties of Horadam polynomials. International Mathematical Forum 4 (25–28): 1243–1252.

    MathSciNet  MATH  Google Scholar 

  11. 11.

    Joshi, S., Ş. Altınkaya, and S. Yalçin. 2017. Coefficient estimates for Sãlãgean type \(\lambda \)-bi-pseudo-starlike functions. Kyungpook Mathematical Journal 57 (4): 613–621.

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Joshi, S., S. Joshi, and H. Pawar. 2016. On some subclasses of bi-univalent functions associated with pseudo-starlike functions. Journal of the Egyptian Mathematical Society 24 (4): 522–525.

    MathSciNet  Article  Google Scholar 

  13. 13.

    Magesh, N., and S. Bulut. 2018. Chebyshev polynomial coefficient estimates for a class of analytic bi-univalent functions related to pseudo-starlike functions. Afrika Matematika 29 (1–2): 203–209.

    MathSciNet  Article  Google Scholar 

  14. 14.

    Srivastava, H.M., Ş. Altınkaya, and S. Yalçin. 2018. Certain subclasses of bi-univalent functions associated with the Horadam polynomials. Iran J Sci Technol Trans Sci 1–7.

  15. 15.

    Srivastava, H.M., S. Sümer Eker, S.G. Hamidi, and J.M. Ahangiri. 2018. Faber polynomial coefficient estimates for bi-univalent functions defined by the Tremblay fractional derivative operator. The Bulletin of the Iranian Mathematical Society 44: 149–157.

    MathSciNet  Article  Google Scholar 

  16. 16.

    Srivastava, H.M., A.K. Mishra, and P. Gochhayat. 2010. Certain subclasses of analytic and bi-univalent functions. Applied Mathematics Letters 23 (10): 1188–1192.

    MathSciNet  Article  Google Scholar 

  17. 17.

    Zireh, A., E.A. Adegani, and M. Bidkham. 2018. Faber polynomial coefficient estimates for subclass of bi-univalent functions defined by quasi-subordinate. Mathematica Slovaca 68 (2): 369–378.

    MathSciNet  Article  Google Scholar 

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The authors are grateful to the referees for their valuable suggestions.

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Correspondence to N. Magesh.

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Abirami, C., Magesh, N., Yamini, J. et al. Horadam Polynomial coefficient estimates for the classes of \(\lambda \)–bi-pseudo-starlike and Bi-Bazilevič Functions. J Anal 28, 951–960 (2020). https://doi.org/10.1007/s41478-020-00224-2

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  • Analytic functions
  • Bi-univalent functions
  • Bi-pseudo-starlike functions
  • Bi-Bazilevič
  • Horadam polynomials
  • Fekete-Szegö inequality

Mathematics Subject Classification

  • Primary 11B 39
  • 30C45
  • 33C45
  • Secondary 30C50
  • 33C05