#### Abstract

We define new subclasses of meromorphic -valent functions by using certain differential operator. Combining the differential operator and certain integral operator, we introduce a general -valent meromorphic function. Then we prove the sufficient conditions for the function in order to be in the new subclasses.

#### 1. Introduction

Let denote the class of meromorphic functions of the form which are analytic and -valent in the punctured unit disc: A function is said to be in the class of meromorphic -valent starlike of order if it satisfies the following inequality:

For , Saif and Kılıçman [1] introduced the linear operator , as follows: and in general, for , we can write It is easy to see that, for , we have

Meromorphically multivalent functions have been extensively studied by several authors; see, for example, Uralegaddi and Somanatha [2, 3], Liu and Srivastava [4, 5], Mogra [6, 7], Srivastava et al. [8], Aouf et al. [9, 10], Joshi and Srivastava [11], Owa et al. [12], and Kulkarni et al. [13].

Now, for , we define the following new subclasses.

*Definition 1. *Let a function be analytic in . Then is in the class if, and only if, satisfies

where , , , .

From (6), one can see that (7) is equivalent to

*Remark 2. *In Definition 1, if we set(i) and , then we have [14, Definition 1.1];(ii) and , then we have , the class of meromorphic -valent starlike of order ;(iii) and , then we have [14, Definition 1.7].

*Definition 3. *Let a function be analytic in . Then is in the class if, and only if, satisfies
where , , , , .

Inequality (9) is equivalent to

*Remark 4. *In Definition 3, if we set(i) and , then we have [14, Definition 1.3];(ii)for and , then we have [14, Definition 1.8].

*Definition 5. *Let a function be analytic in . Then is in the class , if, and only if, satisfies
where , , , .

Inequality (11) is equivalent to

*Remark 6. *In Definition 5, if we set(i) and , then we have [14, Definition 1.5];(ii)for and , then we have [14, Definition 1.9].

Recently, Mohammed and Darus [15] introduced the following -valent meromorphic function: where is the integral operator introduced and studied by the authors [15, 16] and defined by where For we obtain [17]. It is clear that

By using the differential operator given by (4), we introduce the following -valent meromorphic function.

*Definition 7. *Let , and , . One defines the -valent meromorphic function :,
where and is the differential operator given by (4).

*Remark 8. *If we set , and , then we have the -valent meromorphic function given by (13).

#### 2. Main Results

To prove our main results, we need the following lemma.

Lemma 9. *For the -valent meromorphic function given by (18), one has
*

*Proof. *From (18), we have
Differentiating (20) logarithmically and then by simple computation, we get
From (6), we obtain
Then using (22) on the right-hand side of (21), one gets
Multiplying (23) by yields that
or, equivalently, we can write that
which is the desired result.

Our first theorem is as follows.

Theorem 10. *Let , , and , . Suppose that
**
If , then the function defined by (18) is in the class , where
*

*Proof. *Since , by (9), we have
By (19), we get
This is equivalent to
From (28) together with (30), we can get
Hence, we obtain , where .

Corollary 11. *Let , , , , and , . Suppose that
**
If , then the function , defined by (18), is in the class , where is defined as in (27).*

*Proof. *In Theorem 10, we consider .

By Corollary 11, we easily get the following.

Corollary 12. *Let , , , , and , . Suppose that
**
If , then the function , defined by (18), is in the class .*

Now, we prove a sufficient condition for the function defined by (18) to belong to the class .

Theorem 13. *Let , , , and , . Suppose that
**
If , then the function defined by (18) is in the class .*

*Proof. *Since , by (9), we have
On the other hand, from (19), we obtain the following:
Considering (10) with the above equality, we find
The proof is complete.

Corollary 14. *Let , , , and . Suppose that
**
If , then the function defined by (18) is in the class .*

*Proof. *In Theorem 13, we consider that

Next, for the function defined by (18) to belong to the class , we have the following result.

Theorem 15. *Let , , and . Suppose that
**
If , then the function .*

*Proof. *Since , by (11), we have
Combining (12), (30), and the above inequality, we obtain
which is
and finally
Hence, by (12), we have .

Corollary 16. *Let and . Suppose that
**
If , then the function defined by (18) is in the class .*

*Proof. *In Theorem 15, we consider .

Finally, we end this paper by the following theorem and its consequence.

Theorem 17. *Let , , and . Suppose that
**
If , then the function defined by (18) is in the class .*

*Proof. *Since , by (11), we have
Considering this inequality and (30), we obtain
Hence, we have .

Corollary 18. *Let and . Suppose that
**
If , then the function defined by (18) is in the class .*

*Proof. *In Theorem 17, we consider that .

For other work that we can look at regarding differential and integral operators, see [14, 18–24].

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgment

The work here is fully supported by UKM′s Grants: AP-2013-009 and DIP-2013-001.