SpringerLink
Log in

Certain Subclasses of Bi-Univalent Functions Associated with the Horadam Polynomials

  • Research paper
  • Published:
Iranian Journal of Science and Technology, Transactions A: Science Aims and scope Submit manuscript

Abstract

In Geometric Function Theory, there have been many interesting and fruitful usages of a wide variety of special functions and special polynomials. Here, in this article, we propose to make use of the Horadam polynomials which are known to include, as their particular cases, such potentially useful polynomials as (for example) the Fibonacci polynomials, the Lucas polynomials, the Pell polynomials, the Pell–Lucas polynomials, and the Chebyshev polynomials of the second kind. We aim first at introducing a new class of bi-univalent functions defined by means of the Horadam polynomials. For functions belonging to this new bi-univalent function class, we then derive coefficient inequalities and consider the celebrated Fekete–Szegö problem. We also provide relevant connections of our results with those considered in earlier investigations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Altınkaya Ş, Yalçın S (2015) Faber polynomial coefficient bounds for a subclass of bi-univalent functions. C R Acad Sci Paris Sér I(353):1075–1080

    Article  MathSciNet  MATH  Google Scholar 

  • Altınkaya Ş, Yalçın S (2017) On the Chebyshev polynomial coefficient problem of some subclasses of bi-univalent functions. Gulf J Math 5(3):34–40

    MathSciNet  MATH  Google Scholar 

  • Altınkaya Ş, Yalçın S (2017) Faber polynomial coefficient estimates for certain classes of bi-univalent functions defined by using the Jackson \((p, q)\)-derivative operator. J Nonlinear Sci Appl 10:3067–3074

    Article  MathSciNet  MATH  Google Scholar 

  • Brannan DA, Clunie JG (eds) (1980) Aspects of contemporary complex analysis. In: Proceedings of the NATO Advanced Study Institute (University of Durham, Durham; July 1–20, 1979), Academic Press, New York and London

  • Brannan DA, Taha TS (1985) On some classes of bi-unvalent functions, In: Mazhar SM, Hamoui A, Faour NS (eds) Mathematical analysis and its applications. Kuwait, pp 53–60, KFAS Proceedings Series, Vol. 3, Pergamon Press (Elsevier Science Limited), Oxford, 1988; see also Studia Univ. Babeş-Bolyai Math 31(2) (1986) 70–77

  • Bulut S (2014) Faber polynomial coefficient estimates for a comprehensive subclass of analytic bi-univalent functions. C R Acad Sci Paris Sér I(352):479–484

    Article  MathSciNet  MATH  Google Scholar 

  • Çağlar M, Deniz E, Srivastava HM (2017) Second Hankel determinant for certain subclasses of bi-univalent functions. Turk J Math 41:694–706

    Article  MathSciNet  MATH  Google Scholar 

  • Duren PL (1983) Univalent functions, Grundlehren der Mathematischen Wissenschaften, Band 259. Springer-Verlag, New York

    Google Scholar 

  • Fekete M, Szegö G (1933) Eine Bemerkung Über Ungerade Schlichte Funktionen. J Lond Math Soc 89:85–89

    Article  MATH  Google Scholar 

  • Filipponi P, Horadam AF (1990) Derivative sequences of Fibonacci and Lucas polynomials. In: Bergum GE, Philippou AN, Horadam AF (eds) Applications of Fibonacci Numbers, vol 4, pp 99–108, Proceedings of the fourth international conference on Fibonacci numbers and their applications, Wake Forest University, Winston-Salem, North Carolina; Springer (Kluwer Academic Publishers), Dordrecht, Boston and London, 1991. 4 (1991) 99–108

  • Filipponi P, Horadam AF (1993) Second derivative sequences of Fibonacci and Lucas polynomials. Fibonacci Quart. 31:194–204

    MathSciNet  MATH  Google Scholar 

  • Hamidi SG, Jahangiri JM (2016) Faber polynomial coefficients of bi-subordinate functions. C R Acad Sci Paris Sér I(354):365–370

    Article  MathSciNet  MATH  Google Scholar 

  • Horadam AF, Mahon JM (1985) Pell and Pell–Lucas polynomials. Fibonacci Quart 23:7–20

    MathSciNet  MATH  Google Scholar 

  • Hörçum T, Gökçen Koçer E (2009) On some properties of Horadam polynomials. Int Math Forum 4:1243–1252

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

    Article  MathSciNet  MATH  Google Scholar 

  • Lupaş AI (1999) A guide of Fibonacci and Lucas polynomials. Octagon Math Mag 7:2–12

    MathSciNet  Google Scholar 

  • Netanyahu E (1969) The minimal distance of the image boundary from the origin and the second coefficient of a univalent function in \(|z|<1\). Arch Ration Mech Anal 32:100–112

    Article  MathSciNet  MATH  Google Scholar 

  • Srivastava HM, Bansal D (2015) Coefficient estimates for a subclass of analytic and bi-univalent functions. J Egypt Math Soc 23:242–246

    Article  MathSciNet  MATH  Google Scholar 

  • Srivastava HM, Bulut S, Çağlar M, Yağmur N (2013) Coefficient estimates for a general subclass of analytic and bi-univalent functions. Filomat 27:831–842

    Article  MathSciNet  MATH  Google Scholar 

  • Srivastava HM, Gaboury S, Ghanim F (2017) Coefficient estimates for some general subclasses of analytic and bi-univalent functions. Afrika Mat 28:693–706

    Article  MathSciNet  MATH  Google Scholar 

  • Srivastava HM, Murugusundaramoorthy G, Magesh N (2013) Certain subclasses of bi-univalent functions associated with the Hohlov operator. Global J Math Anal 1(2):67–73

    Google Scholar 

  • Srivastava HM, Mishra AK, Gochhayat P (2010) Certain subclasses of analytic and bi-univalent functions. Appl Math Lett 23:1188–1192

    Article  MathSciNet  MATH  Google Scholar 

  • Srivastava HM, Sümer Eker S, Ali RM (2015) Coefficient bounds for a certain class of analytic and bi-univalent functions. Filomat 29:1839–1845

    Article  MathSciNet  MATH  Google Scholar 

  • Srivastava HM, Sümer Eker S, Hamidi SG, Jahangiri JM (2018) Faber polynomial coefficient estimates for bi-univalent functions defined by the Tremblay fractional derivative operator, Bull. Iranian Math. Soc. 44: 149–157

  • Szegö G (1975) Orthogonal Polynomials, Fourth edition, American Mathematical Society Colloquium Publications, vol 23. American Mathematical Society, Providence, Rhode Island

  • Vellucci P, Bersani AM (2016) The class of Lucas-Lehmer polynomials, Rend. Mat. Appl. (Ser. 7) 37: 43–62

  • Wang T-T, Zhang W-P (2012) Some identities involving Fibonacci, Lucas polynomials and their applications. Bull. Math. Soc. Sci. Math. Roumanie (New Ser.) 55(103):95–103

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia, V8W 3R4, Canada

    H. M. Srivastava

  2. Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, 40402, Taiwan, ROC

    H. M. Srivastava

  3. Department of Mathematics, Faculty of Arts and Science, Bursa Uludag University, TR-16059, Bursa, Turkey

    Şahsene Altınkaya & Sibel Yalçın

Authors
  1. H. M. Srivastava
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Şahsene Altınkaya
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Sibel Yalçın
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to H. M. Srivastava.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Srivastava, H.M., Altınkaya, Ş. & Yalçın, 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

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40995-018-0647-0

Keywords

Mathematics Subject Classification

Search

Navigation