Coefficient Inequality for Transforms of Starlike and Convex Functions with Respect to Symmetric Points


  • 투고 : 2014.01.02
  • 심사 : 2014.10.30
  • 발행 : 2015.06.23


The objective of this paper is to obtain sharp upper bound for the second Hankel functional associated with the $k^{th}$ root transform $[f(z^k)]^{\frac{1}{k}}$ of normalized analytic function f(z) when it belongs to the class of starlike and convex functions with respect to symmetric points, defined on the open unit disc in the complex plane, using Toeplitz determinants.


starlike and convex functions with respect to symmetric points;upper bound;second Hankel functional;positive real function;Toeplitz determinants


  1. A. Abubaker and M. Darus, Hankel Determinant for a class of analytic functions involving a generalized linear differential operator, Int. J. Pure Appl. Math., 69(4)(2011), 429 - 435.
  2. R. M. Ali, S. K. Lee, V. Ravichandran and S. Supramaniam, The Fekete-Szego coefficient functional for transforms of analytic functions, Bull. Iran. Math. Soc., 35(2)(2009), 119-142.
  3. R. M. Ali, Coefficients of the inverse of strongly starlike functions, Bull. Malays. Math. Sci. Soc., (second series), 26(1)(2003), 63 - 71.
  4. P. L. Duren, Univalent functions, 259 Grundlehren der Mathematischen Wissenschaften, New York, Springer-verlag XIV, 328, 1983.
  5. R. Ehrenborg, The Hankel determinant of exponential polynomials, Amer. Math. Monthly, 107(6)(2000), 557 - 560.
  6. U. Grenander and G. Szego, Toeplitz forms and their applications. 2nd ed. New York (NY): Chelsea Publishing Co., 1984.
  7. A. Janteng, S. A. Halim and M. Darus, Hankel determinant for starlike and convex functions, Int. J. Math. Anal., (Ruse) 4(13-16)(2007), 619 - 625.
  8. R. J. Libera and E. J. Zlotkiewicz, Coefficient bounds for the inverse of a function with derivative in P, Proc. Amer. Math. Soc., 87(1983), 251-257.
  9. A. K. Mishara and S. N. Kund, The second Hankel determinant for a class of analytic functions associated with the carlson-shaffer operator, Tamkang J. Math., 44(1)(2013), 73 - 82.
  10. N. Mohamed, D. Mohamad and S. Cik Soh, Second Hankel determinant for certain generalized classes of analytic functions, Int. J. Math. Anal., (Ruse) 6(17-20)(2012), 807 - 812.
  11. K. I. Noor, Hankel determinant problem for the class of functions with bounded boundary rotation, Rev. Roum. Math. Pures Et Appl., 28(8)(1983), 731 - 739.
  12. Ch. Pommerenke, Univalent functions, Gottingen: Vandenhoeck and Ruprecht, 1975.
  13. Ch. Pommerenke, On the coefficients and Hankel determinants of univalent functions, J. Lond. Math. Soc., 41(1966), 111 - 122.
  14. Prithvipal Singh, A study of some subclasses of analytic functions in the unit disc, Ph. D Thesis, 1979, IIT Kanpur.
  15. T. RamReddy and D. Vamshee Krishna, Hankel Determinant for starlike and convex functions with respect to symmetric points, J. Ind. Math. Soc., 79(1-4)(2012), 161 - 171.
  16. Ratanchand, Some aspects of functions analytic in the unit disc, Ph. D Thesis 1978, IIT Kanpur.
  17. K. Sakaguchi, On a certain univalent mapping, J. Math. Soc. Japan, 11(1959), 72 - 75.
  18. B. Simon, Orthogonal polynomials on the unit circle, part 1. Classical theory. Vol.54, American mathematical society colloquium publications. Providence (RI): American Mathematical Society, 2005.