Hankel Determinant for a Class of Analytic Functions Related with Lemniscate of Bernoulli

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Ashok Kumar Sahoo, Jagannath Patel

Abstract

The object of the present investigation is to solve Fekete-Szegö problem and determine the sharp upper bound to the second Hankel determinant for a new class R of analytic functions in the unit disk.

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References

  1. P.L. Duren, Univalent Functions, Grundlehren der Mathematischen Wissenschaften, 259, Springer-Verlag, New York, USA (1983).
  2. M. Fekete and G. Szeg ¨o, Eine Bemerkung ¨uber ungerede schlichte funktionen, J. London Math. Soc., 8 (1933), 85-89.
  3. A. Janteng, S.A. Halim and M. Darus, Coefficient inequality for a function whose derivative has a positive real part, J. Inequal. Pure Appl. Math., 7 (2006), Article ID 50.
  4. A. Janteng, S.A. Halim and M. Darus, Estimate on the second Hankel functional for functions whose derivative has a positive real part, J. Quality Measurement and Analysis, 4 (2008), 189-195.
  5. F.R. Keogh and E.P. Merkes, A coefficient inequality for certain classes of analytic functions, Proc. Amer. Math. Soc., 20 (1969), 8-12.
  6. W. Koepf, On the Fekete-Szeg ¨o problem for close-to-convex functions-II, Arch. Math.(Basel), 49 (1987), 420-433.
  7. W. Koepf, On the Fekete-Szeg ¨o problem for close-to-convex functions, Proc. Amer. Math. Soc., 101 (1987), 89-95.
  8. R.J. Libera and E.J. Zlotkiewicz, Early coefficient of the inverse of a regular convex function, Proc. Amer. Math. Soc. 85 (2) (1982), 225-230.
  9. 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), 251-257.
  10. W. C. Ma and D. Minda, A unified treatment of some special classes of univalent functions, Proceedings of the Conference on Complex Analysis (Tianjin, 1992), Z. Li, F. Ren, L. Yang and S. Zhang (Eds.), Int. Press, Cambridge, MA,(1994) 157-169.
  11. T.H. MacGregor, Functions whose derivative have a positive real part. Trans. Amer. Math. Soc. 104(3) (1962), 532-537.
  12. J.W. Noonan and D.K. Thomas, On the second Hankel determinant of areally mean p-valent functions, Trans. Amer. Math. Soc., 223 (1976), 337-346.
  13. K. I. Noor, Hankel determinant problem for the class of functions with bounded boundary rotation, Rev. Roum. Math. Pures Et Appl., 28 (1983), no. 8, 731 - 739.