Afrika Matematika

, Volume 29, Issue 5–6, pp 793–802 | Cite as

Unpredictability of initial coefficient bounds for m-fold symmetric bi-univalent starlike and convex functions defined by subordinations

  • Murat Çağlar
  • Palpandy Gurusamy
  • Erhan Deniz


In this paper, we introduce certain new subclasses of the bi-univalent function class \(\sigma \) in which both f and \(f^{-1}\) are m-fold symmetric analytic with their derivatives in the class \({\mathcal {P}}\) of analytic functions. Furthermore, we obtain coefficient bounds of \(|a_{m+1}|\) and \(|a_{2m+1}|\) for these new subclasses.


Analytic functions Univalent functions Bi-univalent functions Taylor–Maclaurin coefficients m-fold symmetric functions Subordination 

Mathematics Subject Classification

Primary 30C45 33C50 Secondary 30C80 



The authors Murat Çağlar and Erhan Deniz were supported by Kafkas University BAP-2016-FM-67 (The Scientific Research Projects Coordination Unit).


  1. 1.
    Ali, R.M., Lee, S.K., Ravichandran, V., Subramanian, S.: Coefficient estimates for bi-univalent Ma–Minda starlike and convex functions. Appl. Math. Lett. 25(3), 344–351 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Brannan, B.A., Clunie, J.G.: Aspects of Contemporary Complex Analysis. Academic Press, London (1980)zbMATHGoogle Scholar
  3. 3.
    Brannan, B.A., Taha, T.S.: On some classes of bi-univalent functions. Stud. Univ. Babes-Bolyai Math. 31(2), 70–77 (1986)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Bulut, S.: Coefficient estimates for a class of analytic and bi-univalent functions. Novi Sad J. Math. 43(2), 59–65 (2013)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Çağlar, M., Orhan, H., Yağmur, N.: Coefficient bounds for new subclasses of bi-univalent functions. Filomat 27, 1165–1171 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Crişan, O.: Coefficient estimates for certain subclasses of bi-univalent functions. Gen. Math. Notes 16, 93–102 (2013)MathSciNetGoogle Scholar
  7. 7.
    Deniz, E.: Certain subclasses of bi-univalent functions satisfying subordinate conditions. J. Class. Anal. 2, 49–60 (2013)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Duren, P.L.: Univalent Functions. Springer, New York (1983)zbMATHGoogle Scholar
  9. 9.
    Frasin, B.A., Aouf, M.K.: New subclasses of bi-univalent functions. Appl. Math. Lett. 24, 1569–1573 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Goyal, S.P., Goswami, P.: Estimate for initial Maclaurin coefficients of bi-univalent functions for a class defined by fractional derivatives. J. Egypt. Math. Soc. 20, 179–182 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Hayami, T., Owa, S.: Coefficient bounds for bi-univalent functions. Panam. Math. J. 22(4), 15–26 (2012)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Lewin, M.: On a coefficient problem for bi-univalent functions. Proc. Am. Math. Soc. 18, 63–68 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Ma, W.C., Minda, D.: A unified treatment of some special classes of univalent functions. In: Proceedings of the Conference on Complex Analysis, Tianjin, pp. 157–169. Internat. Press, Cambridge (1992)Google Scholar
  14. 14.
    Magesh, N., Rosy, T., Varma, S.: Coefficient estimate problem for a new subclass of bi-univalent functions. J. Complex Anal. 2013, Article ID 474231, 1–3 (2013)Google Scholar
  15. 15.
    Magesh, N., Yamini, J.: Coefficient bounds for certainsubclasses of bi-univalent functions. Int. Math. Forum 8, 1337–1344 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Murugusundaramoorthy, G., Magesh, N., Prameela. V.: Coefficient bounds for certain subclasses of bi-univalent functions. Abstr. Appl. Anal. 2013, Article ID 573017, 1–3 (2013)Google Scholar
  17. 17.
    Pommerenke, C.H.: On the coefficients of close-to-convex functions. Mich. Math. J. 9, 259–269 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Srivastava, H.M., Gaboury, S., Ghanim, F.: Coefficient estimates for some subclasses of \(m-\)fold symmetric bi-univalent functions, Acta Univ. Apulensis Math. Inform. 41, 153–164 (2015)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Srivastava, H.M., Sivasubramanian, S., Sivakumar, R.: Initial coefficient bounds for a subclass of m-fold symmetric bi-univalent functions. Tbil. Math. J. 7(2), 1–10 (2014)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Srivastava, H.M., Mishra, A.K., Gochhayat, P.: Certain subclasses of analytic and bi-univalent functions. Appl. Math. Lett. 23(10), 1188–1192 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Srivastava, H.M., Murugusundaramoorthy, G., Magesh, N.: Certain subclasses of bi-univalent functions associated with Hoholov operator. Glob. J. Math. Anal. 1(2), 67–73 (2013)Google Scholar
  22. 22.
    Srivastava, H.M., Bulut, S., Çağlar, M., Yağmur, N.: Coefficient estimates for a general subclass of analytic and bi-univalent functions. Filomat 27, 831–842 (2013)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© African Mathematical Union and Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2018

Authors and Affiliations

  • Murat Çağlar
    • 1
  • Palpandy Gurusamy
    • 2
  • Erhan Deniz
    • 1
  1. 1.Department of Mathematics, Faculty of Science and LettersKafkas UniversityKarsTurkey
  2. 2.Department of MathematicsVelammal Engineering CollegeChennaiIndia

Personalised recommendations