Abstract

We consider the Fekete-Szegö problem with complex parameter for the class () of analytic functions.

1. Introduction and Preliminaries

Let denote the class of functions of the form which are analytic in the open unit disk and denote the subclass of that are univalent in . A function in is said to be in class of starlike functions of order zero in , if for . Let denote the class of all functions that are convex. Further, is convex if and only if is star-like. A function is said to be close-to-convex with respect to a fixed star-like function if and only if for . Let denote of all such close-to-convex functions [1].

Fekete and Szegö proved a noticeable result that the estimate holds for any normalized univalent function of the form (1.1) in the open unit disk and for . This inequality is sharp for each (see [2]). The coefficient functional on normalized analytic functions in the unit disk represents various geometric quantities, for example, when , , becomes , where denote the Schwarzian derivative of locally univalent functions in . In literature, there exists a large number of results about inequalities for corresponding to various subclasses of . The problem of maximising the absolute value of the functional is called the Fekete-Szegö problem; see [2]. In [3], Koepf solved the Fekete-Szegö problem for close-to-convex functions and the largest real number for which is maximised by the Koebe function is , and later in [4] (see also [5]), this result was generalized for functions that are close-to-convex of order .

Let be an analytic function with positive real part on with , which maps the unit disk onto a star-like region with respect to 1 which is symmetric with respect to the real axis. Let be the class of functions in for which and be the class of functions in for which where denotes the subordination between analytic functions. These classes were introduced and studied by Ma and Minda [6]. They have obtained the Fekete-Szegö inequality for the functions in the class .

Motivated by the class in paper [7], we introduce the following class.

Definition 1.1. Let , . A function is in th class , if where is defined the same as above.
If we set in (1.6), we get which is again a new class. We list few particular cases of this class discussed in the literature for was discussed recently by Swaminathan [7].The class for , where is considered in [8] (see also [9]).The class with was considered in [10] with reference to the univalencey of partial sums. whenever , the class considered in [11].
For geometric aspects of these classes, see the corresponding references. The class is new as the author Swaminathan [7] has introduced class which is subclass of the class , in his recent paper. To prove our main result, we need the following lemma.

Lemma 1.2 (see [12, 13]). If is a function with positive real part, then for any complex number , and the result is sharp for the functions given by

2. Fekete-Szegö Problem

Our main result is the following theorem.

Theorem 2.1. Let , where with . If given by (1.1) belongs to , , then for any complex number The result is sharp.

Proof. If , then there exists a Schwarz function analytic in with and in such that Define the function by Since is a Schwarz function, we see that and . Define the function by In view of (2.2), (2.3), (2.4), we have Thus, From (2.4), we obtain Therefore, we have where Our result now is followed by an application of Lemma 1.2. Also, by the application of Lemma 1.2 equality in (2.1) is obtained when but Putting value of we get the desired results.

For class , Thus, putting and in Theorem 2.1, we get the following corollary.

Corollary 2.2. If given by (1.1) belongs to , then