#### Abstract

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 Szegö [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-Szegö problem. In the literature, there exist a large number of results about the Fekete-Szegö problem (see, for instance, [211]).

For and , let denote the class of functions satisfying and for some . Al-Abbadi and Darus [7] investigated the Fekete-Szegö 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-Szegö 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-Szegö problem for the family As a corollary of the main result, we find the sharp upper bounds for absolute value of the Fekete-Szegö 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-Szegö 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, must attain its maximum at a boundary point. Since for and we have
Case 2. Suppose . Then, we get from (11) that Since must attain its maximum at a boundary point of the rectangle .
Since we get
Case 3. Suppose . Then, (11) gives
Since must attain its maximum at a boundary point of the rectangle . Since it follows that
Combining (15), (19) with (23), we get (8). Since inequalities in Lemmas 1 and 2 are sharp, it follows that inequality (8) is also sharp. The proof is completed.

Since if and only if , by a simple calculation, we have the following.

Corollary 4. If , then

Letting in (24), we get the following.

Corollary 5. If , then

Remark 6. In [11], Srivastava et al. gave a function satisfying , where and Srivastava held that and when . But implies that . So satisfying the above conditions does not exist.

#### Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgment

The paper is supported by Educational Commission of Hubei Province of China (D2011006).