1. Introduction

Let be the class of analytic functions in the open unit disk with the usual normalization . 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 .

Let be the class of all functions which are analytic and univalent in and for which is convex with and for . We denote by and the subclasses of consisting of all analytic functions which are starlike and convex, respectively.

Let denote the class of functions of the form which are analytic in the punctured open unit disk . For , we denote by , and the subclasses of consisting of all meromorphic functions which are, respectively, starlike of order , convex of order and colse-to-convex of order and type in (see, for details, [1, 2]).

Making use of the principle of subordination between analytic functions, we introduce the subclasses , and of the class for and , which are defined by We note that the classes mentioned above are the familiar classes which have been used widely on the space of analytic and univalent functions in (see [3–5]) and for special choices for the functions and involved in these definitions, we can obtain the well-known subclasses of . For examples, we have

Now we define the function by

where is the Pochhammer symbol (or the shifted factorial) defined (in terms of the Gamma function) by Let . Denote by the operator defined by where the symbol stands for the Hadamard product (or convolution). The operator was introduced and studied by Liu and Srivastava [6]. Further, we remark in passing that this operator is closely related to the Carlson-Shaffer operator [7] defined on the space of analytic and univalent functions in .

Corresponding to the function , let be defined such that Analogous to , we now introduce a linear operator on as follows:

We note that We note that the operator is motivated essentially to the integral operator for analytic functions defined by Choi et al. [3], which extends the Noor integral operator studied by K. I. Noor and M. A. Noor [8] (also, see [9–13]).

Next, by using the operator , we introduce the following classes of meromprphic functions for , , and : We also note that

In particular, we set

In this paper, we investigate several inclusion properties of the classes , , and associated with the operator defined by (1.9).Some invariant properties under convolution are also considered for the classes mentioned above. 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. Let , , and be defined by (1.9). Then for , , where

Proof. From (1.8), we know that Therefore (2.1), (2.2) and (2.3) follow from (2.5) immediately.

Lemma 2.2 (see [14, pages 60-61]). Let . If or , then the function belongs to the class , where is defined by (2.4).

Lemma 2.3 (see [15]). Let and . Then for every analytic function in , where denote the closed convex hull of .
At first, the inclusion relationship involving the class is contained in Theorem 2.4 below.

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

Proof. Let . From the definition of , we have where is analytic in with and . By using (1.9), (2.1) and (2.8), we get Therefore by using (2.8), we obtain It follows from (2.9) and Lemma 2.2 that and , respectively. Let us put . Then by applying Lemma 2.3 to (2.10), we obtain since is convex univalent. Therefore from the definition of subordination and (2.12), we have or, equivalently, , which completes the proof of Theorem 2.4.
By using (1.13), (2.2) and (2.3), we have the following Theorem 2.5 and Theorem 2.6.

Theorem 2.5. Let , , and with . If or , then

Theorem 2.6. Let , , and with . If or , then
Next, we prove the inclusion theorem involving the class .

Theorem 2.7. Let , and with . If or , then

Proof. Applying (1.13) and Theorem 2.4, we observe that which evidently proves Theorem 2.7.
By using a similar method as in the proof of Theorem 2.7, we obtain the following two theorems below.

Theorem 2.8. Let , , and with . If or , then

Theorem 2.9. Let , , and with . If or , then
Taking in Theorems 2.4–2.9, we have the following corollaries below.

Corollary 2.10. Let and . If and , and and , then

Corollary 2.11. Let and . If and , and and , then

Corollary 2.12. Let and . If and , and and , then
To prove theorems below, we need the following lemma.

Lemma 2.13. Let with . If with and , then .

Proof. Let . Then where is an analytic function in with and . Thus we have
By using the similar arguments to those used in the proof of Theorem 2.4, we conclude that (2.24) is subordinated to in and so .
Finally, we give the inclusion properties involving the class .

Theorem 2.14. Let and with . If and , and and , then

Proof. We begin by proving that Let . Then there exists a function such that From (2.27), we obtain where is an analytic function in with and . By virtue of (2.3), Lemmas 2.2 and 2.13, we see that belongs to . Then, making use of (2.1), we have Therefore we prove that .
For the second part, by using arguments similar to those detailed above with (2.2), we obtain Thus the proof of Theorem 2.14 is completed.
The following results can be obtained by using the same techniques as in the proof of Theorem 2.14 and so we omit the detailed proofs involved.

Theorem 2.15. Let and with . If and , and