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ISRN Mathematical Analysis

Volume 2012 (2012), Article ID 403028, 12 pages

http://dx.doi.org/10.5402/2012/403028

## Some Properties of Certain Subclasses of Analytic Functions with Complex Order

^{1}School of Mathematics and Statistics, Anyang Normal University, Anyang, Henan 455002, China^{2}School of Econometrics and Management, Changsha University of Science and Technology, Changsha, Hunan 410114, China^{3}Department of Mathematics, Huaihua University, Huaihua, Hunan 418008, China

Received 2 November 2011; Accepted 30 November 2011

Academic Editor: G. Martin

Copyright © 2012 Zhi-Gang Wang et al. 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

The main purpose of this paper is to derive some coefficient inequalities and subordination properties for certain subclasses of analytic functions involving the Salagean operator. Relevant connections of the results presented here with those obtained in earlier works are also pointed out.

#### 1. Introduction

Let denote the class of functions of the form
which are * analytic* in the * open* unit disk

For , we denote by and the usual subclasses of consisting of functions which are, respectively, starlike of order and convex of order in . Clearly, we know that

A function is said to be in the class if it satisfies the inequality for some . Also, a function is said to be in the class if and only if . The classes and were introduced and investigated recently by Owa and Srivastava [1] (see also Nishiwaki and Owa [2], Owa and Nishiwaki [3], and Srivastava and Attiya [4]).

Sălăgean [5] introduced the operator We note that

Given two functions , where is given by (1.1) and is defined by the Hadamard product (or convolution) is defined by

For two functions and , analytic in , we say that the function is subordinate to in , and write if there exists a Schwarz function , which is analytic in with such that Indeed, it is known that Furthermore, if the function is univalent in , then we have the following equivalence:

In recent years, Deng [6] (see also Kamali [7], Altintaş et al*.* [8], Srivastava et al*.* [9], and Xu et al*.* [10]) introduced and investigated the following subclass of involving the S Sălăgean lagean operator and obtained the coefficient bounds for this function class.

*Definition 1.1. *A function is said to be in the class if it satisfies the inequality
where

It is easy to see that the class includes the classes and as its special cases.

Now, motivated essentially by the above-mentioned function classes, we introduce the following subclass of of analytic functions.

*Definition 1.2. *A function is said to be in the class if it satisfies the inequality:
where

It is also easy to see that the classes and are special cases of the class .

In this paper, we aim at proving some coefficient inequalities and subordination properties for the classes and . The results presented here would provide extensions of those given in earlier works. Several other new results are also obtained.

#### 2. Coefficient Inequalities

In this section, we derive some coefficient inequalities for the classes and .

Theorem 2.1. *Let
**
If satisfies the coefficient inequality
**
then .*

*Proof. *To prove , it is sufficient to show that
By noting that
it follows from (2.2) that the above last expression is bounded by . This completes the proof of Theorem 2.1.

Theorem 2.2. *Let
**
If satisfies the coefficient inequality
**
then .*

*Proof. *To prove , it suffices to show that
We consider defined by
Thus, for , we have
It follows from (2.6) that , which implies that (2.7) holds, that is, . The proof of Theorem 2.2 is evidently completed.

To prove our next result, we need the following lemma.

Lemma 2.3. *Let and . Suppose also that the sequence is defined by
**
then
*

*Proof. *We make use of the principle of mathematical induction to prove the assertion (2.11) of Lemma 2.3. Indeed, from (2.10), we know that
which implies that (2.11) holds for .

We now suppose that (2.11) holds for , then
Combining (2.10) and (2.13), we find that
which shows that (2.11) holds for . The proof of Lemma 2.3 is evidently completed.

*Theorem 2.4. Let , then
*

*Proof. *We first suppose that
where
Next, by setting
we easily find that . It follows from (2.18) that
We now find from (2.16), (2.18), and (2.19) that
By evaluating the coefficients of in both the sides of (2.20), we get
On the other hand, it is well known that
Combining (2.21) and (2.22), we easily get

Suppose that and . We define the sequence as follows:
In order to prove that
we use the principle of mathematical induction. By noting that
thus, assuming that
we find from (2.23) and (2.24) that
Therefore, by the principle of mathematical induction, we have
as desired.

By virtue of Lemma 2.3 and (2.24), we know that
Combining (2.17), (2.29), and (2.30), we readily arrive at the coefficient estimates (2.15) asserted by Theorem 2.4.

*Remark 2.5. *Setting , , and in Theorem 2.4, we get the corresponding results obtained by Owa and Nishiwaki [3].

*Remark 2.6. *We cannot show that the result of Theorem 2.4 is sharp. Indeed, if one can prove the sharpness of Theorem 2.4, the sharpness of the corresponding result obtained by Deng [6] follows easily.

*3. Subordination Properties*

*3. Subordination Properties*

*In view of Theorems 2.1 and 2.2, we now introduce the following subclasses:
which consist of functions whose Taylor-Maclaurin coefficients satisfy the inequalities (2.2) and (2.6), respectively.*

*A sequence of complex numbers is said to be a subordinating factor sequence if, whenever of the form (1.1) is analytic, univalent, and convex in , we have the subordination *

*To derive the subordination properties for the classes and , we need the following lemma.*

*Lemma 3.1 (see [11]). The sequence is a subordinating factor sequence if and only if
*

*Theorem 3.2. If and , then
for
where, for convenience,
The constant factor
in the subordination result (3.4) cannot be replaced by a larger one.*

*Proof. *Let and suppose that
then
where is defined by (3.7).

If
is a subordinating factor sequence with , then the subordination result (3.4) holds. By Lemma 3.1, we know that this is equivalent to the inequality
Since
is an increasing function of , and using Theorem 2.1, we have
This evidently proves the inequality (3.12), and hence also the subordination result (3.4), asserted by Theorem 3.2. The inequality (3.5) asserted by Theorem 3.2 follows from (3.4) by setting
Finally, we consider the function defined by
which belongs to the class . Thus, by (3.4), we know that
Furthermore, it can be easily verified for the function given by (3.16) that
We thus complete the proof of Theorem 3.2.

* The proof of the following subordination result is much akin to that of Theorem 3.2. We, therefore, choose to omit the analogous details involved.*

*Corollary 3.3. If and , then
for
where, for convenience,
The constant factor
in the subordination result (3.19) cannot be replaced by a larger one.*

*Remark 3.4. *Putting , , and in Corollary 3.3, we get the corresponding results obtained by Srivastava and Attiya [4].

*Acknowledgments*

*Acknowledgments*

*The present investigation was supported by the National Natural Science Foundation under grants 11101053, 70971013, and 71171024, the Natural Science Foundation of Hunan Province under grant 09JJ1010, the Key Project of Chinese Ministry of Education under grant 211118, the Excellent Youth Foundation of Educational Committee of Hunan Province under grant 10B002, the Open Fund Project of Key Research Institute of Philosophies and Social Sciences in Hunan Universities under grant 11FEFM02, and the Key Project of Natural Science Foundation of Educational Committee of Henan Province under grant 12A110002 of China.*

*References*

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