#### 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, [2–11]).

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).