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).