Abstract and Applied Analysis

Volume 2013 (2013), Article ID 890404, 8 pages

http://dx.doi.org/10.1155/2013/890404

## Certain Subclasses of Multivalent Analytic Functions

^{1}Department of Mathematics, Suqian College, Suqian 223800, China^{2}Department of Mathematics, Yangzhou University, Yangzhou 225002, China

Received 26 February 2013; Revised 17 May 2013; Accepted 29 June 2013

Academic Editor: Pedro M. Lima

Copyright © 2013 Yi-Ling Cang 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

Two new subclasses and of multivalent analytic functions are introduced. Distortion inequalities and inclusion relation for and are obtained. Some results of the partial sums of functions in these classes are also given.

#### 1. Introduction

Throughout this paper, we assume that Let denote the class of functions of the form which are analytic in the unit disk .

For functions and analytic in , we say that is subordinate to in and write , if there exists a Schwarz function in such that

Let Then the Hadamard product (or convolution) of and is defined by The following lemma will be required in our investigation.

Lemma 1. *Let defined by (2) satisfy
**
Then
**
where
*

*Proof. *For defined by (2), the function in (8) can be expressed as
with
In view of (1) and (9), we see that

Let inequality (6) hold. Then from (10) and (12) we deduce that
Hence, by the maximum modulus theorem, we arrive at (7).

We now consider the following two subclasses of .

*Definition 2. *A function defined by (2) is said to be in the class if and only if it satisfies the coefficient inequality (6).

*Definition 3. *A function defined by (2) is said to be in the class if and only if it satisfies

It is obvious from Definitions 2 and 3 that

If we write
then it is easy to verify that
Thus we obtain the following inclusion relations:
Therefore, by Lemma 1, we see that each function in the classes and is starlike with respect to -symmetric points. Analytic functions which are starlike with respect to symmetric points and related functions have been extensively investigated in [1–6]. Recently, several authors have obtained many important properties and characteristics of multivalent analytic functions (see, e.g., [7–11]).

The main object of this paper is to present some distortion inequalities of functions in the classes and which we have introduced here. In particular some results of inclusion relation and convolution of functions in these classes are also given. Further we derive several interesting results of the partial sums of functions in these classes.

#### 2. Main Results

Theorem 4. *Let and suppose that either*(a)* and , or*(b)* and .*

(i)* If , then, for , **
The bounds in (19) are best possible for the function defined by
* (ii)* If , then, for , **
The bounds in (21) are best possible for the function defined by
*

*Proof. *Let . For and , we have , and so
For and , we have and
If either (a) or (b) is satisfied, then
(i) If
then it follows from (23) to (25) that
Hence we have
for . (ii) If
then (23) to (25) yield
This leads to (21). The proof of the theorem is complete.

Theorem 5. *Let
* (i)* If , then, for , **
Equalities in (33) are attained, for example, by
* (ii)* If , then, for , **
Equalities in (36) are attained, for example, by
*

*Proof. *Note that implies that
(i) For , it follows from (23), (24), and (38) that
From this we can get (33). (ii) For , from (23), (24), and (38) we deduce that
Hence we have (36). The proof of the theorem is complete.

Theorem 6. *Let .*(i)*If , then, for ,
**The bounds in (41) are sharp for the function defined by
* (ii)* If , then, for , **
The bounds in (43) are sharp for the function defined by
*

*Proof. *Let . For and , we have , , and
For and , we have , and so
(i) If
then it follows from (45) and (46) that
Hence we have
for .(ii) If
then (45) and (46) yield
This leads to (43). Thus we complete the proof.

Next, we generalize the inclusion relation which is mentioned in (18).

Theorem 7. *If , then
**
where
*

*Proof. *Since and , we see that

Let . In order to prove that , we need only to find the smallest such that
for all , that is, that

For and , (56) is equivalent to
Noting (1), a simple calculation shows that for all real and , and so the function is decreasing in . Therefore
For and , (56) becomes

Consequently, by taking
it follows from (55) to (60) that . The proof is complete.

*Remark 8. * If we take in Theorem 7, then from (1) we have . This shows that

Theorem 9. *Let . Then
**
where
*

*Proof. * For , from Lemma 1 we have (7), which is equivalent to
or to
Obviously
If we put
then, for ,
and, for ,

Now, making use of (65) to (69), we arrive at
for , and . This gives the desired result (62). The proof of the theorem is complete.

Corollary 10. *Let . Then
**
where is the same as in Theorem 9.*

*Proof. *Since if and only if
it follows from Theorem 9 that
Thus we complete the proof.

Finally, we derive certain results of the partial sums of functions in the classes and .

Let be given by (2) and define the partial sums and by For simplicity we use the notation defined by (16).

Theorem 11. *Let and let either*(a)* and , or*(b)* and .**Then, for , we have
**
The bounds in (75) and (76) are best possible for each .*

*Proof. *If either (a) or (b) is satisfied, then, for ,

Let . Then it follows from (77) that

If we put
for and , then and we deduce from (78) that
This implies that for , and so (75) holds true for .

Similarly, by setting
it follows from (78) that
Hence we have (76) for .

For , replacing (78) by
and proceeding as the above, we see that (75) and (76) are also true.

Furthermore, taking the function defined by
we have ,
Thus the proof of Theorem 11 is completed.

*Remark 12. *Replacing by , it follows from Theorem 11 that inequalities (75) and (76) are true. In Theorem 13 we improve the bounds in (75) and (76) for .

Theorem 13. *Let and let either (a) or (b) in Theorem 11 be satisfied. Then, for , one has
**
The bounds in (86) are sharp for the function defined by
*

*Proof. *In view of the assumptions of Theorem 13, it follows from (77) that
If we put
then (88) leads to . Hence we have (86). Sharpness can be verified easily.

#### Acknowledgment

The authors would like to express sincere thanks to the referees for careful reading and suggestions which helped them improve the paper.

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