Second Hankel Determinant for a Certain Subclass of Bi-univalent Functions

  • Nizami Mustafa
  • Gangadharan Mrugusundaramoorthy
  • Thambidurai Janani


In this paper, we introduce a subclass of analytic and bi-univalent functions in the open unit disk. Here, we give upper bound estimates for the second Hankel determinant of the functions that belong to this class. Some interesting applications and conclusions of the results obtained in this paper are also discussed.


Univalent function analytic function bi-univalent function Hankel determinant 

Mathematics Subject Classification

Primary 30C45 30C50 Secondary 30C55 


  1. 1.
    Brannan, D.A., Taha, T.S.: On some classes of bi-univalent functions. Stud. Univ. Babes-Bolyai Math. 31, 70–77 (1986)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Deniz, E., Çağlar, M., Orhan, H.: Second Hankel determinant for bi-stalike and bi-convex functions of order \(\beta \). Appl. Math. Comput. 271, 301–307 (2015)MathSciNetGoogle Scholar
  3. 3.
    Duren, P.L.: Univalent Functions (Grundlehren der Mathematischen Wissenschaften, 259, New York). Springer, New York (1983)Google Scholar
  4. 4.
    Fekete, M., Szegö, G.: Eine Bemerkung über ungerade schichte Funktionen. J. Lond. Math. Soc. 8(8), 85–89 (1933)CrossRefzbMATHGoogle Scholar
  5. 5.
    Goodman, A.W.: Univalent Functions. Volume I. Polygonal, Washington (1983)zbMATHGoogle Scholar
  6. 6.
    Grenander, U., Szegö, G.: Toeplitz Forms and Their Applications. California Monographs in Mathematical Sciences. University of California Press, Berkeley, CA, USA (1958)Google Scholar
  7. 7.
    Hummel, J.: The coefficient regions of starlike functions. Pacific J. Math. 7, 1381–1389 (1957)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Hummel, J.: Extremal problems in the class of starlike functions. Proc. Am. Math. Soc. 11, 741–749 (1960)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Kanas, S.: An unified approach to the FeketeSzeg problem. Appl. Math. Comput. 218(17), 8453–8461 (2012)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Kanas, S., Lecko, A., Stankiewicz, J.: Differential subordinations and geometric means. Complex Var Theory Appl Int J 28(3), 201–209 (1996)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Kanas, S., Kim, S.-A., Sivasubramanian, S.: Verification of Brannan and Clunies conjecture for certain subclasses of bi-univalent function. Ann. Polon. Math. 113, 295–304 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Keogh, F.R., Merkes, E.P.: A coefficient inequality for certain classes of analytic functions. Proc. Am. Math. Soc. 20, 8–12 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Lewin, M.: On a coefficient problem for bi-univalent functions. Proc. Am. Math. Soc. 18(1), 63–68 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Libera, R.J., Złotkiewicz, E.J.: Early coefficients of the inverse of a regular convex function. Proc. Am. Math. Soc. 85(2), 225–230 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Libera, R.J., Złotkiewicz, E.J.: Coefficient bounds for the inverse of a function with derivative in \({\cal{P}}\). Proc. Am. Math. Soc. 87(2), 251–257 (1983)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Ma, W. C., Minda, D.: A unified treatment of some special classes of functions. In: Proceedings of the Conference on Complex Analysis, Tianjin, 1992, pp. 157–169, Conf. Proc. Lecture Notes Anal. 1, Int. Press, Cambridge, MA, (1994)Google Scholar
  17. 17.
    Netanyahu, E.: The minimal distance of the image boundary from the origin and the second coefficient of a univalent function in \( \left|z\right|<1\). Arch. Ration. Mech. Anal. 32, 100–112 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Noonan, J.W., Thomas, D.K.: On the second Hankel determinant of a really mean p-valent functions. Trans. Am. Math. Soc. 223, 337–346 (1976)zbMATHGoogle Scholar
  19. 19.
    Lewin, M.: On a coefficient problem for bi-univalent functions. Proc. Am. Math. Soc. 18, 63–68 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Orhan, H., Magesh, N., Yamini, J.: Bounds for the second Hankel determinant of certain bi-univalent functions. Turkish J. Math. 40, 665–678 (2016)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Srivastava, H.M., Mishra, A.K., Das, M.K.: The Fekete-Szegö-problem for a subclass of close-to-convex functions. Complex Var. Elliptic Equ. 44(2), 145–163 (2001)zbMATHGoogle Scholar
  22. 22.
    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
  23. 23.
    Srivastava, H.M., Owa, S.: Current Topics in Analytic Function Theory. World Scientific, Singapore (1992)CrossRefzbMATHGoogle Scholar
  24. 24.
    Taha, T.S.: Topics in Uivalent Function Theory. Ph.D, University of London (1981)Google Scholar
  25. 25.
    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
  26. 26.
    Xu, Q.H., Xiao, H.G., Srivastava, H.M.: A certain general subclass of analytic and bi-univalent functions and associated coefficient estimate problems. Appl. Math. Comput. 218, 11461–11465 (2012)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Zaprawa, P.: On the Fekete-Szegö problem for classes of bi-univalent functions. Bull. Belg. Math. Soc. Simon Stevin 1, 169–178 (2014)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Nizami Mustafa
    • 1
  • Gangadharan Mrugusundaramoorthy
    • 2
  • Thambidurai Janani
    • 2
  1. 1.Department of Mathematics, Faculty of Science and LettersKafkas UniversityKarsTurkey
  2. 2.School of Advanced SciencesVIT UniversityVelloreIndia

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