Abstract and Applied Analysis

1. Introduction

Let denote the class of functions of the form which are analytic in the open unit disk . Let and denote the subclasses of whose members are univalent and starlike, respectively.

For functions and analytic in , we say that is subordinate to in and we write , if there exists an analytic function in such that Furthermore, if the function is univalent in , then Throughout our present discussion, we assume that We introduce the following subclass of .

Definition 1.1. A function is said to be in the class if it satisfies where The classes have been studied by several authors (see [1–5]). Recently, Gao and Zhou [6] showed some mapping properties of the following subclass of : Note that For further information of the above classes (with ) and related analytic function classes, see Srivastava et al. [7], Yang and Liu [8], Kim [9], and Kim and Srivastava [10].

In this paper, we obtain sharp bounds on , and coefficient estimates for functions belonging to the class . Conditions for univalency and starlikeness, convolution properties, and the radius of convexity are also presented. One can see that the methods used in [6] do not work for the more general class than .

2. The bounds on , , and in

In this section, we let where and With (2.1), it is easily seen that the function given by (1.6) can be expressed as

Theorem 2.1. Let . Then, for , The bounds in (2.4) are sharp for the function defined by

Proof. The analytic function given by (1.6) is convex (univalent) in (cf. [11]) and satisfies . Thus, for ,
Let . Then, we can write where is analytic and for . By the Schwarz lemma, we know that . It follows from (2.7) that which leads to or to Since we deduce from (2.6) and (2.10) that Now, by using (2.3) and (2.12), we can obtain (2.4).
Furthermore, for the function defined by (2.5), we find that Hence, and from (2.13), we see that the bounds in (2.4) are the best possible.
Hereafter, we write

Corollary 2.2. Let . Then, for , The results are sharp.

Proof. For , it follows from (2.12) (used in the proof of Theorem 2.1) that for . From these, we have the desired results.

The bounds in (2.16) and (2.17) are sharp for the function

Theorem 2.3. Let . Then, for , The results are sharp.

Proof. Noting that an application of Theorem 2.1 yields (2.20). Furthermore, the results are sharp for the function defined by (2.5).

Corollary 2.4. Let . Then, for , The results are sharp for the function defined by (2.19).

Proof. For , it follows from (2.6) and (2.10) (with ) that for and . Making use of (2.21) and (2.23), we can obtain (2.22).

Theorem 2.5. Let and and ). If then , where the symbol stands for the familiar Hadamard product (or convolution) of two analytic functions in .

Proof. Since and ), it follows from Corollary 2.4 (with ) and (2.24) that Thus, has the Herglotz representation where is a probability measure on the unit circle and .
For , we have where In view of the function is convex (univalent) in , we deduce from (2.26) to (2.28) that This shows that .

Corollary 2.6. Let , and Then, .

Proof. By taking , and , (2.24) in Theorem 2.5 becomes that is, Hence, the desired result follows as a special case from Theorem 2.5.

Remark 2.7. R. Singh and S. Singh [4, Theorem 3] proved that, if and belong to , then . Obviously, for Corollary 2.6 generalizes and improves Theorem 3 in [4].

Theorem 2.8. Let and . Then, for , The result is sharp, with the extremal function defined by (2.5).

Proof. It is well known that for and , Since , we have and so (2.35) leads to By virtue of (1.6), (2.10), and (2.36), we have for and . Now, by using (2.3), (2.21) and (2.37), we can obtain (2.34).

Theorem 2.9. Let Then, The result is sharp for each .

Proof. It is known (cf. [12]) that, if where is analytic in and is analytic and convex univalent in , then .
By (2.38), we have where and is given by (1.6). Since the function is analytic and convex univalent in , it follows from (2.41) that which gives (2.39).
Next, we consider the function It is easy to verify that The proof of Theorem 2.9 is completed.

3. The Univalency and Starlikeness of

Theorem 3.1. if and only if

Proof. Let and (3.1) be satisfied. Then, by (2.16) in Corollary 2.2, we see that . Thus, is close-to-convex and univalent in .
On the other hand, if then the function defined by (2.19) satisfies and as . Hence, there exists a point such that . This implies that is not univalent in and so the theorem is proved.

Theorem 3.2. Let (3.1) in Theorem 3.1 be satisfied. If and then .

Proof. We first show that Equation (3.5) is obvious when . For , we have where Since and is a monotonically decreasing sequence. Therefore, the inequality (3.5) follows from (3.6).
Let . Then, Define in by In view of (3.1) in Theorem 3.1 is satisfied, the function is univalent in , and so is analytic in . Also it follows from (3.10) that

We want to prove now that for . Suppose that there exists a point such that Then, applying a result of Miller and Mocanu [13, Theorem 4], we have For , we deduce from Corollaries 2.2 and 2.4, (3.1), (3.5), (3.11), (3.13), and (3.4) that But this contradicts (3.9) at . Therefore, we must have and the proof of Theorem 3.2 is completed.

Remark 3.3. In [6, Theorem 4(ii)], the authors gave the following: if and is the solution of the equation then for . However, this result is not true because the series in (3.15) diverges.

4. The Radius of Convexity

Theorem 4.1. Let belong to the class defined by and . Then, where is the root in of the equation The result is sharp.

Proof. For , we can write where is analytic and in . Differentiating both sides of (4.4) logarithmically, we arrive at Put and . Then, (4.4) implies that With the help of the Carathéodory inequality it follows from (4.5) and (4.6) that where and because of (4.6) and (4.7). In view of (4.10) and (4.11), we see that