On New 𝑝-Valent Meromorphic Function Involving Certain Differential and Integral Operators

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.