Abstract and Applied Analysis

VolumeΒ 2011Β (2011), Article IDΒ 456729, 16 pages

http://dx.doi.org/10.1155/2011/456729

## A Class of Analytic Functions with Missing Coefficients

^{1}Department of Mathematics, Soochow University, Jiangsu, Suzhou 215006, China^{2}Department of Mathematics, Yangzhou University, Jiangsu, Yangzhou 225002, China

Received 1 March 2011; Accepted 9 May 2011

Academic Editor: PaulΒ Eloe

Copyright Β© 2011 Ding-Gong Yang and Jin-Lin Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Let and denote the class of functions of the form which are analytic in the open unit disk and satisfy the following subordination condition , for, for. 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 considered.

#### 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

Let us now calculate the minimum value of on the closed interval . Noting that and (4.7), we deduce from (4.12) that where Also and . Suppose that . Then, Hence, by virtue of the mathematical induction, we have for all and . This implies that In view of (4.14) and (4.18), we see that Further it follows from (4.9), (4.12), and (4.19) that where and Note that and . If we let denote the root in of the equation , then (4.20) yields the desired result (4.2).

To see that the bound is the best possible, we consider the function It is clear that for , which shows that the bound cannot be increased.

Setting , Theorem 4.1 reduces to the following result.

Corollary 4.2. *Let and . Then, is convex of order in
**
The result is sharp.*

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