Hankel Determinant for p-Valent Alpha-Convex Functions

The objective of the present paper is to obtain the sharp upper bound of for p-valent α-convex functions of the form in the unit disc .


Introduction
Let   be the class of analytic functions of the form in the unit disc  = { : || < 1} with  ∈  = {1, 2, 3, . ..}.Let  be the subclass of  1 = , consisting of univalent functions. *  is the class consisting of functions of the form (1) and satisfying the condition Re {   ()  () } > 0,  ∈ . ( The functions of the class  *  are called p-valent starlike functions.In particular,  * 1 ≡  * , the class of starlike functions. is the class of functions of the form (1), satisfying the condition The functions of the class   are known as p-valent convex functions.Particularly,  1 ≡ , the class of convex functions.
Let   () ( ≥ 0) be the class of functions of the form (1), satisfying the condition Functions in the class   () are known as -valent alpha-convex functions.For  = 1, the class   () reduces to the class () of alpha-convex functions introduced by Mocanu [1].Also   (0) ≡  *  and   (1) ≡   .In 1976, Noonan and Thomas [2] stated the th Hankel determinant for  ≥ 1 and  ≥ 1 as This determinant has also been considered by several authors.For example, Noor [3] determined the rate of growth of   () as  → ∞ for functions given by (1) with bounded boundary.Ehrenborg [4] studied the Hankel determinant of exponential polynomials.Also Hankel determinant was studied by various authors including Hayman [5] and Pommerenke [6].In [7], Janteng et al. studied the Hankel determinant for the classes of starlike and convex functions.Again Janteng et al. discussed the Hankel determinant problem for the classes of starlike functions with respect to symmetric points and convex functions with respect to symmetric Geometry points in [8] and for the functions whose derivative has a positive real part in [9].Also Hankel determinant for various subclasses of -valent functions was investigated by various authors including Krishna and Ramreddy [10] and Hayami and Owa [11].
Easily, one can observe that the Fekete and Szegö functional is  2 (1).Fekete and Szegö [12] then further generalised the estimate | 3 −  2  2 |, where  is real and  ∈ .For our discussion in this paper, we consider the Hankel determinant in the case of  = 2 and  = 2: In this paper, we seek sharp upper bound of the functional +2 | for functions belonging to the class   ().The results due to Janteng et al. [7] follow as special cases.

Preliminary Results
Let  be the family of all functions  analytic in  for which Re(()) > 0 and for  ∈ .
Putting  = 1 and  = 1 in Theorem 3, we obtain the following result due to Janteng et al. [7].