Second Hankel Determinant for Bi-Univalent Analytic Functions Associated with Hohlov Operator

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G. Murugusundaramoorthy, K. Vijaya, Second Hankel Determinant for Bi-Univalent Analytic Functions Associated with Hohlov Operator, Int. J. Anal. Appl., 8 (1) (2015), 22-29.

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G. Murugusundaramoorthy, K. Vijaya

Abstract

In the present paper, we consider asubclass of the function class $\Sigma$ of bi-univalent  analytic functions in the open unit disk $\Delta$ associated with Hohlovoperator and we obtain the functional $|a_2a_4 - a_3^2|$ for the  function class $\Sigma$. Our result gives corresponding $|a_2a_4 - a_3^2|$ for the  subclasses of $\Sigma$ defined in the literature.

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References

  1. D.A. Brannan, J.G. Clunie(Eds.), Aspects of Contemporary Complex Analysis (Proceeding of the NATO Advanced study Institute held at the University of Durham, Durham: July 1-20, 1979), Academic Press, New York and London, 1980.
  2. D.A. Brannan, J. Clunie, W.E. Kirwan, Coefficient estimates for a class of star-like functions, Canad. J. Math. 22(1970), 476-485.
  3. D.A. Brannan, T.S. Taha, On some classes of bi-univalent functions, Studia Univ. BabesBolyai Math. 31(2)(1986), 70-77.
  4. P.L. Duren, Univalent Functions, in: Grundlehren der Mathematischen Wissenschaften, Band 259, Springer-Verlag, New York, Berlin, Heidelberg and Tokyo, 1983.
  5. E.Deniz, M.C ¸a`glar, and H. Orhan,Second hankel determinant for bi-starlike and bi-convex functions of order β,arxiv:1303.2504v2.
  6. M.Fekete and G.Szeg ¨o, Eine Bemerkung ¨uber ungerade schlichte Funktionen, J. London. Math. Soc., 8(1933), 85-89.
  7. B.A. Frasin, M.K. Aouf, New subclasses of bi-univalent functions, Appl. Math. Lett. 24(2011), 1569-1973.
  8. U.Grenander and G.Szeg ¨o, Toeplitz forms and their applications, Univ. of California Press, Berkeley and Los Angeles, 1958.
  9. J.Hummel, The coefficient regions of starlike functions, Pacific. J. Math., 7 (1957), 1381- 1389.
  10. J.Hummel, Extremal problems in the class of starlike functions, Proc. Amer. Math. Soc., 11 (1960), 741-749.
  11. Yu.E. Hohlov, Operators and operations in the class of univalent functions (in Russian), Izv. VysËœs. UËœcebn. Zaved. Matematika 10(1978) 83-89.
  12. A.Janteng, S.A.Halim and M.Darus, Coefficient inequality for a function whose derivative has a positive real part, J.Ineq. Pure and Appl. Math., Vol.7, 2 (50) (2006), 1-5.
  13. F.R.Keogh and E.P.Merkes, A coefficient inequality for certain classes of analytic functions, Proc. Amer. Math. Soc., 20 (1969), 8-12.
  14. M. Lewin, On a coefficient problem for bi-univalent functions, Proc. Amer. Math. Soc. 18(1967) 63-68.
  15. R.J.Libera and E.J.Zlotkiewicz, Early coefficients of the inverse of a regular convex function, Proc. Amer. Math. Soc., 85(2) (1982), 225-230.
  16. R.J.Libera and E.J.Zlotkiewicz, Coefficient bounds for the inverse of a function with derivative in P, Proc. Amer. Math. Soc., 87(2) (1983), 251-289.
  17. T.H.MacGregor, Functions whose derivative has a positive real part, Trans. Amer. Math. Soc., 104 (1962), 532-537.
  18. E. Netanyahu, The minimal distance of the image boundary from the origin and the second coefficient of a univalent function in |z| < 1, Arch. Rational Mech. Anal. 32(1969) 100-112.
  19. J.W.Noonan and D.K.Thomas, On the second Hankel determinant of areally mean p-valent functions, Trans. Amer. Math. Soc., 223 (2) (1976), 337-346
  20. T.Panigarhi and G. Murugusundaramoorthy, Coefficient bounds for Bi- univalent functions analytic functions associated with Hohlov operator, Proc. Jangjeon Math. Soc,16 (1) (2013) 91-100.
  21. Ch. Pommerenke, Univalent functions, Vandenhoeck and Ruprecht, G ¨ottingen, 1975.
  22. H.M. Srivastava, A.K. Mishra, P. Gochhayat, Certain subclasses of analytic and bi-univalent functions, Appl. Math. Lett. 23(2010), 1188-1192.
  23. T.S. Taha, Topics in Univalent Function Theory, Ph.D Thesis, University of London, 1981.