Skip to main content

A new characterization of (PQ)-Lucas polynomial coefficients of the bi-univalent function class associated with q-analogue of Noor integral operator

Abstract

In this study, by using Lucas polynomials, subordination and q-analogue of Noor integral operator, we will introduce an interesting new class \(Q^{q,\mu }(\tau ,\alpha ;x)\) of bi-univalent functions. Also we will obtain (PQ)-Lucas polynomial coefficient estimates and Fekete–Szegö inequalities for this new class.

This is a preview of subscription content, access via your institution.

References

  1. Ahmad, B., Khan, M.G., Frasin, B.A., Aouf, M.K., Abdeljawad, T., Mashwani, W.K., Arif, M.: On \(q\)-analogue of meromorphic multivalent functions in lemniscate of Bernoulli domain. AIMS Math. 6(4), 3037–3052 (2021). https://doi.org/10.3906/mat-1903-38

    MathSciNet  Article  MATH  Google Scholar 

  2. Akgül, A.: (P;Q)-Lucas polynomial coefficient inequalities of the bi-univalent function class. Turk. J. Math. 43(5), 2170–2176 (2019). https://doi.org/10.3906/mat-1905-17

    MathSciNet  Article  MATH  Google Scholar 

  3. Akgül, A., Sakar, F.M.: A certain subclass of bi-univalent analytic functions introduced by means of the \(q\)-analogue of Noor integral operator and Horadam polynomials. Turk. J. Math. 43(5), 2275–2286 (2019)

    MathSciNet  Article  Google Scholar 

  4. Akgül, A., Alçin, B.: Based on a family of bi-univalent functions introduced through the q-analogue of Noor integral operator. Int. J. Open Problems Compt. Math. 15(1), 20–33 (2022)

    Google Scholar 

  5. Akgül, A.: On a family of bi-univalent functions related to the Fibonacci numbers. Math. Moravica. 26(1), 103–112 (2022)

    MathSciNet  Article  Google Scholar 

  6. Alamoush A.G.: Coefficient estimates for certain subclass of bi functions associated the Horadam polynomials. (2018). arXiv:1812.10589 (arXiv preprint)

  7. Aldweby, H., Darus, M.: A subclass of harmonic univalent functions associated with-analogue of Dziok–Srivastava operator. ISRN Math. Anal. 2013, 6 (2013). https://doi.org/10.1155/2013/382312

    MathSciNet  Article  MATH  Google Scholar 

  8. Altinkaya, Ş: Inclusion properties of Lucas polynomials for bi-univalent functions introduced through the \(q\)-analogue of Noor integral operator. Turk. J. Math. 43(2), 620–629 (2019). https://doi.org/10.3906/mat-1805-86

    MathSciNet  Article  MATH  Google Scholar 

  9. Altinkaya, Ş, Yalçin, S.: On the \((p, q)\)-Lucas polynomial coefficient bounds of the bi-univalent function class. Bol. Soc. Mat. Mex. 20, 1–9 (2018)

    MATH  Google Scholar 

  10. Amourah, A., Frasin, B.A., Murugusundaramoorthy, G., Al-Hawary, T.: Bi-Bazilevič functions of order \(\vartheta +i\delta \) associated with \((p;q)\)- Lucas polynomials. AIMS Math. 6(5), 4296–4305 (2021). https://doi.org/10.3934/math.2021254

  11. Amourah, A., Alamoush, A., Al-Kaseasbeh, M.: Gegenbauer polynomials and bi-univalent functions. Palestine J. Math. 10(2), 625–632 (2021)

    MathSciNet  MATH  Google Scholar 

  12. Amourah, A., Al-Hawary, T., Frasin, B.A.: Application of Chebyshev polynomials to certain class of bi-Bazilevic functions of order \(\alpha +i\beta \). Afr. Mat. 32(5), 1059–66 (2021). https://doi.org/10.1007/s13370-021-00881-x

    MathSciNet  Article  MATH  Google Scholar 

  13. Amourah, A., Frasin, B.A., Ahmad, M., Yousef, F.: Exploiting the Pascal distribution series and gegenbauer polynomials to construct and study a new subclass of analytic bi-univalent functions. Symmetry 14(1), 147 (2021). https://doi.org/10.3390/sym14010147

    Article  Google Scholar 

  14. Arif, M., Haq, M.U., Liu, J.L.: A subfamily of univalent functions associated with-analogue of Noor integral operator. J. Funct. Sp. 2018, 1–5 (2018). https://doi.org/10.1155/2018/3818915

    MathSciNet  Article  MATH  Google Scholar 

  15. Çağlar M., Deniz E., Kazimoǧlu S.: Fekete-Szegö problem for a subclass of analytic functions defined by Chebyshev polynomials. In: 3rd International Conference on Mathematical and Related Sciences: Current Trend and Developments; Turkey, pp. 114–120 (2020). https://doi.org/10.5269/bspm.51024

  16. Deniz, E., Orhan, H.: The Fekete–Szegö problem for a generalized subclass of analytic functions. Kyungpook Math. J. 50(1), 37–47 (2010)

    MathSciNet  Article  Google Scholar 

  17. Duren, P.L.: Univalent Functions. Springer, New York (1983)

    MATH  Google Scholar 

  18. Fekete, M., Szegö, G.: Eine Bemerkung über ungerade schlichte Funktionen. J. Lond. Math. Soc. 1(2), 85–9 (1933)

    Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  20. Filipponi, P., Horadam, A.F.: Second derivative sequences of Fibonacci and Lucas polynomials. Fibonacci Quart. 31(3), 194–204 (1993)

    MathSciNet  MATH  Google Scholar 

  21. Frasin, B.A., Swamy, S.R., Aldawish, I.: A comprehensive family of bi-univalent functions defined by k-Fibonacci numbers. J. Funct. Sp. (2021). https://doi.org/10.1155/2021/4249509

    Article  MATH  Google Scholar 

  22. Horzum, T., Koçer, E.G.: On some properties of Horadam polynomials. Int. Math. Forum 4(25), 1243–1252 (2009)

    MathSciNet  MATH  Google Scholar 

  23. Keogh, F.R., Merkes, E.P.: A coefficient inequality for certain classes of analytic functions. Proc. Am. Math. Soc. 20(1), 8–12 (1969)

    MathSciNet  Article  Google Scholar 

  24. Khan, B., Srivastava, H.M., Tahir, M., Darus, M., Ahmed, Q.Z., Khan, N.: Applications of a certain \(q\)- integral operator to the subclasses of analytic and bi-univalent functions. AIMS Math. 6(1), 1024–1039 (2020). https://doi.org/10.3934/math.2021061

    MathSciNet  Article  MATH  Google Scholar 

  25. Koshy, T.: Fibonacci and Lucas Numbers with Applications. Wiley, New York (2019)

    MATH  Google Scholar 

  26. Lee, G., Asci, M.: Some Properties of the \((p, q)-\)Fibonacci and \((p, q)-\)Lucas Polynomials. J. Appl. Math. 2012, 1–18 (2012). https://doi.org/10.1155/2012/264842

    MathSciNet  Article  MATH  Google Scholar 

  27. Lewin, M.: On a coefficient problem for bi-univalent functions. Proc. Am. Math. Soc. 18(1), 63–68 (1967)

    MathSciNet  Article  Google Scholar 

  28. Lupas, A.: A guide of Fibonacci and Lucas polynomials. Octagon Math. Mag. 7(1), 2–12 (1999)

    MathSciNet  Google Scholar 

  29. Al-Hawary, T., Amourah, A., Frasin, B.A.: Fekete–Szegö inequality for bi-univalent functions by means of Horadam polynomials. Bol. Soc. Mat. Mex. 27(79), 1–12 (2021). https://doi.org/10.1007/s40590-021-00385-5(0123456789)

    Article  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  31. Noor, K.I.: On new classes of integral operators. J. Natur. Geom. 16(1–2), 71–80 (1999)

    MathSciNet  MATH  Google Scholar 

  32. Noor, K.I., Noor, M.A.: On integral operators. J. Math. Anal. Appl. 238(2), 341–52 (1999)

    MathSciNet  Article  Google Scholar 

  33. Orhan, H., Deniz, E., Çaǧlar, M.: Fekete–Szegö problem for certain subclasses of analytic functions. Demonstratio Math. 45(4), 835–846 (2012). https://doi.org/10.1515/dema-2013-0423

    MathSciNet  Article  MATH  Google Scholar 

  34. Özkoç, A., Porsuk, A.: A note for the \((p;q)\)-Fibonacci and Lucas quaternion polynomials. Konuralp J. Math. 5(2), 36–46 (2017)

    MathSciNet  MATH  Google Scholar 

  35. Sakar, F.M.: Estimate for initial Tschebyscheff polynomials coefficients on a certain subclass of bi-univalent functions defined by Salagean differential operator. Acta Univ. Apulensis Math. Inform. 54, 45–54 (2018)

    MathSciNet  MATH  Google Scholar 

  36. Sǎlǎgean, G.S.: Subclasses of Univalent Functions. Complex Analysis-Fifth Romanian-Finnish Seminar, Part-1(Bucharest, 1981). Lecture Notes in Math, vol. 1013, pp. 363–372. Springer, Berlin (1983)

  37. Srivastava, H.M., Altinkaya, Ş, Yalçin, S.: Certain subclasses of bi-univalent functions associated with the Horadam polynomials. Iran J. Sci. Technol. Trans. Sci. 43, 1873–1879 (2019). https://doi.org/10.1007/s40995-018-0647-0

  38. Srivastava, H.M., Mishra, A.K., Gochhayat, P.: Certain subclasses of analytic and bi-univalent functions. Appl. Math. Lett. 23, 1188–1192 (2010). https://doi.org/10.1016/j.aml.2010.05.009

  39. Srivastava, H.M.: Univalent Functions, Fractional Calculus, and Associated Generalized Hypergeometric Functions. Univalent Functions, Fractional Calculus, and Their Applications. Ellis Horwood, Chichester (1989)

    MATH  Google Scholar 

  40. Srivastava, H.M., Altinkaya, Ş, Yalçın, S.: Certain subclasses of bi-univalent functions associated with the Horadam polynomials. Iran J. Sci. Technol. Trans. Sci. 43(4), 1873–1879 (2019). https://doi.org/10.1007/s40995-018-0647-0(0123456789

    MathSciNet  Article  Google Scholar 

  41. Tingting, W., Wenpeng, Z.: Some identities involving Fibonacci, Lucas polynomials and their applications. Bull. Math. Soc. Sci. Math. Roumanie 20, 95–103 (2012)

    MathSciNet  MATH  Google Scholar 

  42. Vellucci P., Bersani A. M.: The class of Lucas–Lehmer polynomials. Rend. Mat. Appl. 7(37), 43–62 (2016). arXiv:1603.01989 (arXiv preprint) (2016)

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Arzu Akgül.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Akgül, A., Sakar, F.M. A new characterization of (PQ)-Lucas polynomial coefficients of the bi-univalent function class associated with q-analogue of Noor integral operator. Afr. Mat. 33, 87 (2022). https://doi.org/10.1007/s13370-022-01016-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s13370-022-01016-6

Keywords

  • (P, Q)-Lucas polynomials
  • q-analogue of Noor integral operator
  • Coefficient bounds
  • Bi-univalent functions

Mathematics Subject Classification

  • 30C45
  • 30C50