Rocky Mountain Journal of Mathematics

Coefficient inequality for certain $p$-valent analytic functions

D. Vamshee Krishna

Full-text: Open access

Abstract

The objective of this paper is to obtain an upper bound to the second Hankel determinant $|a_{p+1}a_{p+3}-a_{p+2}^{2}|$ for certain $p$-valent analytic functions, using Toeplitz determinants.

Article information

Source
Rocky Mountain J. Math., Volume 44, Number 6 (2014), 1941-1959.

Dates
First available in Project Euclid: 2 February 2015

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1422885102

Digital Object Identifier
doi:10.1216/RMJ-2014-44-6-1941

Mathematical Reviews number (MathSciNet)
MR3310956

Zentralblatt MATH identifier
1309.30013

Subjects
Primary: 30C45: Special classes of univalent and multivalent functions (starlike, convex, bounded rotation, etc.) 30C50: Coefficient problems for univalent and multivalent functions

Keywords
Analytic function $p$-valent function starlike and convex functions upper bound second Hankel determinant positive real function Toeplitz determinants

Citation

Krishna, D. Vamshee. Coefficient inequality for certain $p$-valent analytic functions. Rocky Mountain J. Math. 44 (2014), no. 6, 1941--1959. doi:10.1216/RMJ-2014-44-6-1941. https://projecteuclid.org/euclid.rmjm/1422885102


Export citation

References

  • 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 (2011), 429–435.
  • R.M. Ali, Coefficients of the inverse of strongly starlike functions, Bull. Malay. Math. Sci. Soc. 26 (2003), 63–71.
  • O. Al-Refai and M. Darus, Second Hankel determinant for a class of analytic functions defined by a fractional operator, Europ. J. Sci. Res. 28 (2009), 234–241.
  • P.L. Duren, Univalent functions, Grund. Math. Wiss. 259, Springer, New York, 1983.
  • R. Ehrenborg, The Hankel determinant of exponential polynomials, Amer. Math. Month. 107 (2000), 557–560.
  • A.W. Goodman, Univalent functions, Vol. I and Vol. II, Mariner Publishing Company, Inc., Tampa, FL, 1983.
  • U. Grenander and G. Szegö, Toeplitz forms and their applications, second edition, Chelsea Publishing Co., New York, 1984.
  • A. Janteng, S.A. Halim and M. Darus, Hankel determinant for starlike and convex functions, Int. J. Math. Anal. 1 (2007), 619–625.
  • ––––, Coefficient inequality for a function whose derivative has a positive real part, J. Inequal. Pure Appl. Math. 7 (2006), 1–5.
  • J.W. Layman, The Hankel transform and some of its properties, J. Int. Seq. 4 (2001), 1–11.
  • R.J. Libera and E.J. Zlotkiewicz, Coefficient bounds for the inverse of a function with derivative in $\mathscr{P}$, Proc. Amer. Math. Soc. 87 (1983), 251–257.
  • ––––, Early coefficients of the inverse of a regular convex function, Proc. Amer. Math. Soc. 85 (1982), 225–230.
  • T.H. Mac Gregor, Functions whose derivative have a positive real part, Trans. Amer. Math. Soc. 104 (1962), 532–537.
  • B.S. Mehrok and Gagandeep Singh, Estimate of the second Hankel determinant for certain classes of analytic functions, Sci. Magna 8 (2012), 85–94.
  • A.K. Mishra and P. Gochhayat, Second Hankel determinant for a class of analytic functions defined by fractional derivative, Int. J. Math. Math. Sci. 2008, Article ID 153280, 2008, 1–10.
  • A.K. Mishra and S.N. Kund, The second Hankel determinant for a class of analytic functions associated with the Carlson-Shaffer operator, Tamkang J. Math. 44 (2013), 73–82.
  • N. Mohamed, D. Mohamed and S. Cik Soh, Second Hankel determinant for certain generalized classes of analytic functions, Int. J. Math. Anal. 6 (2012), 807–812.
  • G. Murugusundaramoorthy and N. Magesh, Coefficient inequalities for certain classes of analytic functions associated with Hankel determinant, Bull. Math. Anal. Appl. 1 (2009), 85–89.
  • 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.
  • K.I. Noor, Hankel determinant problem for the class of functions with bounded boundary rotation, Rev. Roum. Math. Pures Appl. 28 (1983), 731–739.
  • S. Owa and H.M. Srivastava, Univalent and starlike generalised hypergeometric functions, Canad. J. Math. 39 (1987), 1057–1077.
  • Ch. Pommerenke, Univalent functions, Vandenhoeck and Ruprecht, Gottingen, 1975.
  • ––––, On the Hankel determinants of univalent functions, Mathematika 14 (1967), 108–112.
  • ––––, On the coefficients and Hankel determinants of univalent functions, J. Lond. Math. Soc. 41 (1966), 111–122.
  • T. RamReddy and D. Vamshee Krishna, Hankel Determinant for starlike and convex functions with respect to symmetric points, J. Indian Math. Soc. 79 (2012), 161–171.
  • B. Simon, Orthogonal polynomials on the unit circle, Part 1. Classical theory, Amer. Math. Soc. Colloq. Publ. 54, American Mathematical Society, Providence, RI, 2005.