Advertisement

Afrika Matematika

, Volume 28, Issue 5–6, pp 693–706 | Cite as

Coefficient estimates for some general subclasses of analytic and bi-univalent functions

  • H. M. Srivastava
  • S. GabouryEmail author
  • F. Ghanim
Article

Abstract

In the present investigation, we consider two new general subclasses \(\mathcal {H}_{\Sigma }(\tau , \mu , \lambda , \gamma ; \alpha )\) and \(\mathcal {H}_{\Sigma }(\tau , \mu , \lambda , \gamma ; \beta )\) of the class \(\Sigma \) consisting of analytic and bi-univalent functions in the open unit disk \(\mathbb {U}\). For functions belonging to the two classes introduced here, we find estimates on the Taylor–Maclaurin coeffcients \(|a_{2}|\) and \(|a_{3}|\). Several connections to some of the earlier known results are also pointed out.

Keywords

Analytic functions Bi-starlike functions Bi-univalent functions Coefficient estimates 

Mathematics Subject Classification

Primary 30C45 Secondary 30C10 

References

  1. 1.
    Brannan, D.A., Clunie, J.G. (Eds.) Aspects of Contemporary Complex Analysis. In: Proceedings of the NATO Advanced Study Institute held at the University of Durham, Durham; July 1–20,1979, Academic Press, New York and London, 1980Google Scholar
  2. 2.
    Brannan, D.A., Taha, T.S.: On some classes of bi-univalent functions. Studia Univ. Babeş-Bolyai Math. 31(2), 70–77 (1986)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Duren, P.L.: Univalent Functions, Grundlehren der MathematischenWissenschaften, vol. 259. Springer-Verlag, Berlin (1983)Google Scholar
  4. 4.
    Frasin, B.A.: Coefficcient bounds for certain classes of bi-univalent functions. Hacet. J. Math. Stat. 43(3), 383–389 (2014)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Frasin, B.A., Aouf, M.K.: New subclasses of bi-univalent functions. Appl. Math. Lett. 24, 1569–1573 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Hayami, T., Owa, S.: Coefficient bounds for bi-univalent functions. Pan Am. Math. J. 22(4), 15–26 (2012)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Keerthi, B.S., Raja, B.: Coefficient inequality for certain new subclasses of analytic bi-univalent functions. Theor. Math. Appl. 3(1), 1–10 (2013)zbMATHGoogle Scholar
  8. 8.
    Lewin, M.: On a coefficient problem for bi-univalent functions. Proc. Am. Math. Soc. 18, 63–68 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Li, X.-F., Wang, A.-P.: Two new subclasses of bi-univalent functions. Int. Math. Forum 7, 1495–1504 (2012)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Löwner, K.: Untersuchungen über schlichte konforme Abbildungen des Einheitskreises. Math. Ann. 89, 103–121 (1923)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Murugusundaramoorthy, G., Magesh, N., Prameela, V.: Coefficient bounds for certain subclasses of bi-univalent function. Abstr. Appl. Anal. 2013, 1–3 (2013) (Article ID 573017) Google Scholar
  12. 12.
    Netanyahu, E.: 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 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Peng, Z., Murugusundaramoorthy, G., Janani, T., Coefficient estimates of bi-univalent functions of complex order associated with the Hohlov operator. J. Complex Anal., 2014, 1–6 (2014) (Article ID 693908)Google Scholar
  14. 14.
    Pommerenke, C.: Univalent Functions (with a Chapter on Quadratic Differentials by Gerd Jensen). Vandenhoeck and Ruprecht, Göttingen (1975)zbMATHGoogle Scholar
  15. 15.
    Srivastava, H.M., Bansal, D.: Coefficient estimates for a subclass of analytic and bi-univalent functions. J. Egypt. Math. Soc. 23, 242–246 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Srivastava, H.M., Murugusundaramoorthy, G., Magesh, N.: Certain subclasses of bi-univalent functions associated with the Hohlov operator. Glob. J. Math. Anal. 1(2), 67–73 (2013)Google Scholar
  17. 17.
    Srivastava, H.M., Mishra, A.K., Das, M.K.: The Fekete–Szegö problem for a subclass of close-to-convex functions. Complex Var. Theory Appl. 44, 145–163 (2001)CrossRefzbMATHGoogle Scholar
  18. 18.
    Srivastava, H.M., Mishra, A.K., Gochhayat, P.: Certain subclasses of analytic and bi-univalent functions. Appl. Math. Lett. 23, 1188–1192 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Srivastava, H.M., Owa, S. (eds.): Current Topics in Analytic Function Theory. World Scientific Publishing Company, Singapore (1992)zbMATHGoogle Scholar
  20. 20.
    Srivastava, H.M., Sivasubramanian, S., Sivakumar, R.: Initial coefficient bounds for a subclass of m-fold symmetric bi-univalent functions. Tbilisi Math. J. 7(2), 1–10 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Styer, D., Wright, J.: Result on bi-univalent functions. Proc. Am. Math. Soc. 82, 243–248 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Tan, D.-L.: Coefficicent estimates for bi-univalent functions. Chin. Ann. Math. Ser. A 5, 559–568 (1984)zbMATHGoogle Scholar
  23. 23.
    Xu, Q.-H., Gui, Y.-C., Srivastava, H.M.: Coefficient estimates for a certain subclass of analytic and bi-univalent functions. Appl. Math. Lett. 25, 990–994 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Xu, Q.-H., Xiao, H.-G., Srivastava, H.M.: A certain general subclass of analytic and bi-univalent functions and associated coefficient estimates problems. Appl. Math. Comput. 218, 11461–11465 (2012)MathSciNetzbMATHGoogle Scholar

Copyright information

© African Mathematical Union and Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of VictoriaVictoriaCanada
  2. 2.China Medical UniversityTaichungPeople’s Republic of China
  3. 3.Department of Mathematics and Computer ScienceUniversity of Québec at ChicoutimiChicoutimiCanada
  4. 4.Department of Mathematics, College of SciencesUniversity of SharjahSharjahUnited Arab Emirates

Personalised recommendations