On Certain Subclasses of Analytic Functions Involving Carlson-Shaffer Operator and Related to Lemniscate of Bernoulli

The object of the present investigation is to solve the Fekete-Szeg？ problem and determine the sharp upper bound to the second Hankel determinant for a new class of analytic functions involving the Carlson-Shaffer operator in the unit disk. We also obtain a sufficient condition for normalized analytic functions in the unit disk to be in this class. 1. Introduction and Preliminaries Let be the class of functions of the form which are analytic in the open unit disk . A function is said to be starlike of order , if Similarly, a function is said to be convex of order , if By usual notations, we write these classes of functions by and , respectively. We denote and , the familiar subclasses of starlike, convex functions in . Furthermore, let denote the class of analytic functions normalized by such that in . For functions and , analytic in , we say that is subordinate to , written as or , if there exists a Schwarz function , which (by definition) is analytic in with , , and , . Furthermore, if the function is univalent in , then we have the following equivalence relation (cf., e.g., [1]; see also [2]): For functions analytic in , we define the Hadamard product (or convolution) of and by Note that is also analytic in . Carlson and Shaffer [3] defined the linear operator in terms of the incomplete beta function by where and denotes the Pochhammer symbol (or shifted factorial) given, in terms of the Gamma function , by If is given by (1), then it follows from (7) that We note that for (i);(ii);(iii);(iv), the well-known Ruscheweyh derivative [4] of ;(v), the well-known Owa-Srivastava fractional differential operator [5]. We also observe that and . With the aid of the linear operator , we introduce a subclass of as follows. Definition 1. A function is said to be in the class , if it satisfies the condition It follows from (12) and the definition of subordination that a function satisfies the following subordination relation: We further note that if , then the function lies in the region bounded by the right half of the lemniscate of Bernoulli given by Noonan and Thomas [6] defined the th Hankel determinant of a sequence of real or complex numbers by This determinant has been studied by several authors including Noor [7] with the subject of inquiry ranging from the rate of growth of (as ) to the determination of precise bounds with specific values of and for certain subclasses of analytic functions in the unit disc . For , , , and , the Hankel determinant simplifies to The Hankel determinant was considered by Fekete and Szeg？ [8] and we refer to as the second