On the Fekete-Szeg？ Problem for a Class of Analytic Functions

Let denote the class of functions which are analytic in the unit disk and given by the power series . Let be the class of convex functions. In this paper, we give the upper bounds of for all real number and for any in the family , Re for？？some . 1. Introduction Let denote the class of functions which are analytic in the unit disk and satisfy . The set of all functions that are univalent will be denoted by . Let and be the classes of convex, starlike of order and close-to-convex functions, respectively. Fekete and Szeg？ [1] proved that holds for any and that this inequality is sharp. The coefficient functional on in plays an important role in function theory. For example, , where is the Schwarzian derivative. The problem of maximizing the absolute value of the functional is called the Fekete-Szeg？ problem. In the literature, there exist a large number of results about the Fekete-Szeg？ problem (see, for instance, [2–11]). For and , let denote the class of functions satisfying and for some . Al-Abbadi and Darus [7] investigated the Fekete-Szeg？ problem on the class . Let be the class of functions in satisfying the inequality for some function . In [11], Srivastara et al. studied the Fekete-Szeg？ problem on the class for by proving that Srivastara et al. held that the inequality (5) was sharp. However, the extremal function given in [11] did not exist in the case of . In this paper, we solve the Fekete-Szeg？ problem for the family As a corollary of the main result, we find the sharp upper bounds for absolute value of the Fekete-Szeg？ functional for the class defined by Clearly, is a subclass of . In the case of , we get sharp estimation of the absolute value of the Fekete-Szeg？ functional for the class and for all real number , which prove that the inequality (5) is not sharp actually when . 2. Main Result Let be the class of functions that are analytic in and satisfy for all . The following two lemmas can be found in [2]. Lemma 1 (see [2]). If is in the class , then, for any complex number , one has . The inequality is sharp. Lemma 2 (see [2]). If is in the class and is a complex number, then . The inequality is sharp. Theorem 3. If is in the class and is a real number, then Proof. By definition, is in the class if and only if there exists a function such that . A simple computation shows . Thus, So, by Lemmas 1 and 2, we have Putting and , we get from (10) that , where Since and , we will calculate the maximum value of for . Case 1. Suppose . Then it follows from (11) that Since does not have a local maximum at any point of the open rectangle . Hence,