Format: Online  ISSN 1943-2380

 Coefficient Inequality for Uniformly Convex Functions of Order $\alpha$ D. Vamshee Krishna, T. Ram Reddy Volume 5, Issue 1, 2013    Pages  25 - 41 Received   18 August 2011,   Accepted   23 May 2012 Abstract.  The objective of this paper is to obtain an upper bound to the second Hankel determinant $|a_{2}a_{4}-a_{3}^{2}|$ for the function $f$ and its inverse belonging to the uniformly convex functions of order $\alpha(0\leq\alpha<1)$, using Toeplitz determinants. Keywords.  Analytic function; Uniformly convex function; Parabolic starlike function; Upper bound; Second Hankel functional; Positive real function; Subordination; Toeplitz determinants. DOI.  10.5373/jarpm.1080.081811 Full Text:  PDF References : [1] A. Abubaker, M. Darus. Hankel Determinant for a class of analytic functions involving a generalized linear differential operator. Int. J. Pure Appl. Math., 2011, 69(4): 429 - 435. [2] R.M. Ali. Star likeness associated with parabolic regions. Int. J. Math. Math. Sci., 2005, 4: 561 - 570. [3] R.M. Ali. Coefficients of the inverse of strongly star like functions. Bull. Malays. Math. Sci. Soc., 2003, 26: 63 -71. [4] R.M. Ali, V. Singh. Coefficients of parabolic star like functions of order. Comput. Methods Funct. Theory 1994(Penang), Series in Approximations and Decompositions, World Scientific Publishing, New Jersey, 1995, 5: 23 - 36. [5] O. Al-Refai, M. Darus. Second Hankel determinant for a class of analytic functions defined by a fractional operator.European J. Sci. Res., 2009, 28(2): 234 - 241. [6] R. Ehrenborg. The Hankel determinant of exponential polynomials. Amer. Math. Monthly, 2000, 107: 557- 560. [7] A.W. Goodman. On uniformly convex functions. Ann. Polon. Math., 1991, 56(1): 87 - 92. [8] U. Grenander, G. Szego. Toeplitz forms and their application. Univ. of California Press, Berkeley and Los Angeles, 1958. [9] A. Janteng, S.A. Halim, M. Darus.Estimate on the second Hankel functional for functions whose derivative has apositive real part. J. Qual. Meas. Anal.(JQMA), 2008, 1: 189 - 195. [10] A. Janteng, S.A. Halim, M. Darus.Hankel determinant for starlike and convex functions. Int. J. Math. Anal.,2007, 1(13): 619 - 625. [11] A. Janteng, S.A. Halim, M. Darus.Coefficient inequality for a function whose derivative has a positive realpart. J. Inequal. Pure Appl. Math, 2006, 7(2): 1 - 5. [12] J. W. Layman. The Hankel transform and some of its properties. J. Integer Seq., 2001, 4:1 - 11. [13] S. K. Lee. Characterizations of parabolic starlike functions and the generalized uniformly convex functions.Master’s thesis, Universiti of Sains, Malaysia, Penang, Malaysia, 2000. [14] W.C. Ma, D. Minda. Uniformly convex functions. Ann. Polon. Math., 1992, 57(2): 165 - 175. [15] T.H. Mac Gregor. Functions whose derivative have a positive real part. Trans. Amer. Math. Soc., 1962, 104: 532 -537. [16] A.K. Mishra, P. Gochhayat. Second Hankel determinant for a class of analytic functions defined by fractionalderivative. Int. J. Math. Math. Sci., 2008, Article ID 153280, View Full Text via CrossRef. [17] G. Murugusundaramoorthy, N. Magesh. Coefficient inequalities for certain classes of analytic functions associated  with Hankel determinant. Bull Math Anal. Appl., 2009, 1(3): 85- 89. [18] J.W. Noonan, D.K. Thomas. On these cond Hankel determinant of a really mean p-Valent functions. Trans.Amer. Math. Soc., 1976, 223(2): 337 - 346. [19] K.I. Noor. Hankel determinant problem for the class of functions with bounded boundary rotation. Rev. Roum.Math. Pures Et Appl., 1983, 28(8): 731 - 739. [20] S. Owa, H.M. Srivastava. Univalentand star like generalised hypergeometric functions. Canad. J. Math., 1987,39(5): 1057 - 1077. [21] Ch. Pommerenke. Univalent functions. Vandenhoeck and Ruprecht, Gottingen, 1975. [22] F. Ronning. Uniformly convex functions and corresponding class of sta rlike functions. Proc. Amer. Math. Soc., 1993,118(1): 189 - 196. [23] F. Ronning. A survey on uniformlyconvex and uniformly starlike functions. Ann. Univ. Mariae Curie - SklodowskaSect. A, 1993, 47: 123 - 134.