Bonfring International Journal of Data Mining, Vol. 4, No. 3, August 2014 16 Hankel Determinant for a Subclass of Alpha Convex Functions Gagandeep Singh and Gurcharanjit Singh Abstract--- In the present investigation, the upper bound of second Hankel determinant 2 3 4 2 a a a − for functions belonging to the subclass of analytic functions is studied. Results presented in this paper would extend the corresponding results of various authors. ( ) B A M , ; α Keywords--- Analytic Functions, Starlike Functions, Convex Functions, Alpha Convex Functions, Subordination, Schwarz Function, Second Hankel Determinant I. INTRODUCTION LET A be the class of analytic functions of the form () ∑ ∞ = + = 2 k k k z a z z f (1.1) In the unit disc { } : 1 E z z = < . By S we denote the class of functions () A z f ∈ and univalent in E. Let U be the class of Schwarzian functions () ∞ k ∑ = = 1 k k z d z w Which are analytic in the unit disc { } : 1 E z z = < 0 ) 0 ( = w and satisfying the conditions and () . 1 < z w Let f and g be two analytic functions in E. Then f is said to be subordinate to g (symbolically if there exists a Schwarz function w , such that ) g f p () U z ∈ () ( ) ( ) . z w g z f = In 1976, Noonan and Thomas [9] stated the qth Hankel determinant of for and as () z 1 ≥ q f 1 ≥ n () . ... ... 1 1 + − + n q n n a a a H ... ... ... ... ... ... 2 2 1 − + − + q n q n q a a ... ... 1 + n = a n Gagandeep Singh, Department of Mathematics, M.S.K. Girls College, Bharowal(Tarn-Taran), Punjab, India. E-mail: kamboj.gagandeep@yahoo.in Gurcharanjit Singh, Department of Mathematics, Guru Nanak Dev University College, Chungh(Tarn-Taran), Punjab, India. E-mail: dhillongs82@yahoo.com The Hankel determinant plays an important role in the study of singularities; for instance, see[2]. This is also important in the study of power series with integral coefficients, see [2] and Cantor[1]. Hankel determinants are useful in showing that a function of bounded characterstic in the unit disc i.e. a function which is a ratio of two bounded analytic functions with its Laurent series around the origin having integral coefficients, is rational. It is well known that the Fekete-Szegö functional () . 1 2 2 2 3 H a a = − This functional is further generalized as 2 2 3 a a μ − for μ (real as well as complex). It is a very great combination of the two coefficients which describes the area problems posted earlier by Gron wall in 1914-15. Moreover, we also know that the functional 2 3 4 2 a a a − is equivalent to ( ) . 2 2 H The functional 2 3 4 2 a a a − has been studied by many authors see ([4-6],[10],[14-16]). Janteng et al. [4] have considered the functional 2 3 4 2 a a a − and found a sharp bound for the function in the subclass R of S, consisting of functions whose derivative has a positive real part studied by Mac- Gregor[9]. In their work, they have shown that if f R f ∈ , then . 9 4 2 3 4 − 2 ≤ a a a The same authors [5] also obtained the second Hankel determinant and sharp upper bounds for the familier subclasses namely, starlike and convex functions denoted by and ∗ S K of and have shown that S 1 ≤ 2 3 4 2 − a a a and 8 1 ≤ 2 3 − a a 4 2 a respectively. Motivated by the above mentioned results obtained by different authors in this direction, in this paper, we introduce certain subclass of analytic functions and obtain an upper bound to the functional 2 3 4 2 a a a − for the function f belonging to this class, defined as follows: ( ) B A M , ; α denote the subclass of functions ( ) A z f ∈ and satisfying the condition ISSN 2277 - 5048 | © 2014 Bonfring