Abstract and Applied Analysis

Volumeย 2008, Article IDย 246876, 8 pages

http://dx.doi.org/10.1155/2008/246876

## Inclusion Properties for Certain Subclasses of Analytic Functions Defined by a Linear Operator

Department of Applied Mathematics, Pukyong National University, Pusan 608-737, Korea

Received 8 August 2007; Revised 29 October 2007; Accepted 23 November 2007

Academic Editor: John Michaelย Rassias

Copyright ยฉ 2008 Nak Eun Cho. 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 purpose of the present paper is to investigate some inclusion properties of certain subclasses of analytic functions associated with a family of linear operators, which are defined by means of the Hadamard product (or convolution). Some integral preserving properties are also considered.

#### 1. Introduction

Let denote the class of functions of the form which are analytic in the open unit disk . If and are analytic in , we say that is subordinate to , written or if there exists an analytic function in with and for such that . We denote by , , and the subclasses of consisting of all analytic functions which are, respectively, starlike, convex, and close-to-convex in .

Let be the class of all functions which are analytic and univalent in and for which is convex with and for .

Making use of the principle of subordination between analytic functions, many authors investigated the subclasses , , and of the class for (cf. [1, 2]), which are defined by For in the definitions defined above, we have the well-known classes , , and , respectively. Furthermore, for the function classes and investigated by Janowski [3] (also see [4]), it is easily seen that

We now define the function by where is the Pochhammer symbol (or the shifted factorial) defined (in terms of the Gamma function) by

We also denote by : the operator defined by where the symbol () stands for the Hadamard product (or convolution). Then it is easily observed from definitions (1.4) and (1.6) that and Furthermore, we note that where the symbol denotes the familiar Ruscheweyh derivative [5] (also, see [6]) for . The operator was introduced and studied by Carlson and Shaffer [7] which has been used widely on the space of analytic and univalent functions in (see also [8]).

By using the operator , we introduce the following classes of analytic functions for , and : We also note that In particular, we set

In this paper, we investigate several inclusion properties of the classes , , and . The integral preserving properties in connection with the operator are also considered. Furthermore, relevant connections of the results presented here with those obtained in earlier works are pointed out.

#### 2. Inclusion Properties Involving the Operator

The following lemmas will be required in our investigation.

Lemma 2.1 (See [9, Pages 60-61]). *Let . If or , then the function defined by (1.4) belongs to the class . *

Lemma 2.2 (See [10]). *Let and . Then for every analytic function in , ** where denote the
closed convex hull of .*

Theorem 2.3. *Let , , and . If or , then *

*Proof. * Let . Then there exists an analytic function in with and such that By using (1.6)
and (2.3), we have It follows from
(2.3) and Lemma 2.1 that and , respectively. Then by applying Lemma 2.2 to (2.4),
we obtain since is convex
univalent. Therefore, from the definition of subordination and (2.5), we have or,
equivalently, , which completes the proof of Theorem 2.3.

Theorem 2.4. * Let , and . If or , then *

*Proof. *(). Using a similar argument as in the proof of Theorem
2.3, we obtain where is an analytic
function in with and . Applying Lemma 2.1 and the fact that , we see that since is convex
univalent. Thus the proof of Theorem 2.3 is completed.

Corollary 2.5. *Let , , and . If and , then *

Theorem 2.6. *Let , and . If and , then *

*Proof. *Applying (1.9) and Corollary 2.5,
we observe that which evidently
proves Theorem 2.6.

Taking in Corollary 2.5 and Theorem 2.6, we have the following corollary.

Corollary 2.7. *Let and . If and , then *

To prove the theorems below, we need the following lemma.

Lemma 2.8. *Let . If and , then . *

*Proof. * Let . Then where is an analytic
function in with and . Thus we have By using
similar arguments to those used in the proof of Theorem 2.3, we conclude that
(2.15) is subordinated to in and so .

Theorem 2.9. *Let , and . If and , then *

*Proof. *
First of all, we show that Let . Then there exists a function such that From (2.18), we
obtain where is an analytic
function in with and . By virtue of Lemmas 2.1 and 2.8, we see that belongs to . Then we have which implies
that .

Moreover, the proof of the second part is similar to that
of the first part and so we omit the details involved.

#### 3. Inclusion Properties Involving Various Operators

The next theorem shows that the classes , , and are invariant under convolution with convex functions.

Theorem 3.1. *Let , , and let . Then *(i)(ii)(iii)

*Proof. * (i) Let . Then we have By using the
same techniques as in the proof of Theorem 2.3, we obtain (i).

(ii) Let . Then, by (1.9), and hence from
(i), . Since we have (ii)
applying (1.9) once again.

(iii) Let . Then there exists a function such that where is an analytic
function in with and . From Lemma 2.8, we have that . Since we obtain
(iii).

Now we consider the following operators [5, 11] defined by It is well known ([12], see also [5]) that the operators and are convex univalent in . Therefore, we have the following result, which can be obtained from Theorem 3.1 immediately.

Corollary 3.2. *Let , , and let be defined by (3.5). Then *(i)(ii)(iii)

#### Acknowledgments

The author would like to express his gratitude to the referees for their valuable suggestions. This work was supported by Pukyong National University Research Fund in 2007 (PK-2007-013).

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