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# Some Subclasses of Meromorphic Functions Associated with a Family of Integral Operators

*Journal of Inequalities and Applications*
**volume 2009**, Article number: 931230 (2009)

## Abstract

Making use of the principle of subordination between analytic functions and a family of integral operators defined on the space of meromorphic functions, we introduce and investigate some new subclasses of meromorphic functions. Such results as inclusion relationships and integral-preserving properties associated with these subclasses are proved. Several subordination and superordination results involving this family of integral operators are also derived.

## 1. Introduction and Preliminaries

Let denote the class of functions of the form

which are *analytic* in the *punctured* open unit disk

Let , where is given by (1.1) and is defined by

Then the Hadamard product (or convolution) of the functions and is defined by

Let denote the class of functions of the form

which are analytic and convex in and satisfy the condition

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:

Analogous to the integral operator defined by Jung et al*.* [1], Lashin [2] introduced and investigated the following integral operator:

defined, in terms of the familiar Gamma function, by

By setting

we define a new function in terms of the Hadamard product (or convolution):

Then, motivated essentially by the operator , we now introduce the operator

which is defined as

where (and throughout this paper unless otherwise mentioned) the parameters and are constrained as follows:

We can easily find from (1.14), (1.15), and (1.17) that

where is the Pochhammer symbol defined by

Clearly, we know that .

It is readily verified from (1.19) that

By making use of the principle of subordination between analytic functions, we introduce the subclasses , , and of the class which are defined by

Indeed, the above mentioned function classes are generalizations of the general meromorphic starlike, meromorphic convex, meromorphic close-to-convex and meromorphic quasi-convex functions in analytic function theory (see, for details, [3–12]).

Next, by using the operator defined by (1.19), we define the following subclasses , , and of the class :

Obviously, we know that

In order to prove our main results, we need the following lemmas.

Lemma 1.1 (see [13]).

Let . Suppose also that is convex and univalent in with

If is analytic in with , then the subordination

implies that

Lemma 1.2 (see [14]).

Let be convex univalent in and let be analytic in with

If is analytic in and , then the subordination

implies that

The main purpose of the present paper is to investigate some inclusion relationships and integral-preserving properties of the subclasses

of meromorphic functions involving the operator . Several subordination and superordination results involving this operator are also derived.

## 2. The Main Inclusion Relationships

We begin by presenting our first inclusion relationship given by Theorem 2.1.

Theorem 2.1.

Let and with

Then

Proof.

We first prove that

Let and suppose that

where is analytic in with Combining (1.21) and (2.4), we find that

Taking the logarithmical differentiation on both sides of (2.5) and multiplying the resulting equation by , we get

By virtue of (2.1), an application of Lemma 1.1 to (2.6) yields , that is . Thus, the assertion (2.3) of Theorem 2.1 holds.

To prove the second part of Theorem 2.1, we assume that and set

where is analytic in with . Combining (1.22), (2.1), and (2.7) and applying the similar method of proof of the first part, we get , that is Therefore, the second part of Theorem 2.1 also holds. The proof of Theorem 2.1 is evidently completed.

Theorem 2.2.

Let and with (2.1) holds. Then

Proof.

In view of (1.25) and Theorem 2.1, we find that

Combining (2.9) and (2.10), we deduce that the assertion of Theorem 2.2 holds.

Theorem 2.3.

Let , and with (2.1) holds. Then

Proof.

We begin by proving that

Let . Then, by definition, we know that

with , Moreover, by Theorem 2.1, we know that , which implies that

We now suppose that

where is analytic in with Combining (1.21) and (2.15), we find that

Differentiating both sides of (2.16) with respect to and multiplying the resulting equation by , we get

In view of (1.21), (2.14), and (2.17), we conclude that

By noting that (2.1) holds and

we know that

Thus, an application of Lemma 1.2 to (2.18) yields

that is , which implies that the assertion (2.12) of Theorem 2.3 holds.

By virtue of (1.22) and (2.1), making use of the similar arguments of the details above, we deduce that

The proof of Theorem 2.3 is thus completed.

Theorem 2.4.

Let , and with (2.1) holds. Then

Proof.

In view of (1.26) and Theorem 2.3, and by similarly applying the method of proof of Theorem 2.2, we conclude that the assertion of Theorem 2.4 holds.

## 3. A Set of Integral-Preserving Properties

In this section, we derive some integral-preserving properties involving two families of integral operators.

Theorem 3.1.

Let with and

Then the integral operator defined by

belongs to the class .

Proof.

Let . Then, from (3.2), we find that

By setting

we observe that is analytic in with . It follows from (3.3) and (3.4) that

Differentiating both sides of (3.5) with respect to logarithmically and multiplying the resulting equation by , we get

Since (3.1) holds, an application of Lemma 1.1 to (3.6) yields

which implies that the assertion of Theorem 3.1 holds.

Theorem 3.2.

Let with and (3.1) holds. Then the integral operator defined by (3.2) belongs to the class .

Proof.

By virtue of (1.25) and Theorem 3.1, we easily find that

The proof of Theorem 3.2 is evidently completed.

Theorem 3.3.

Let with and (3.1) holds. Then the integral operator defined by (3.2) belongs to the class .

Proof.

Let . Then, by definition, we know that there exists a function such that

Since , by Theorem 3.1, we easily find that , which implies that

We now set

where is analytic in with . From (3.3), and (3.11), we get

Combining (3.10), (3.11), and (3.12), we find that

By virtue of (1.21), (3.10), and (3.13), we deduce that

The remainder of the proof of Theorem 3.3 is much akin to that of Theorem 2.3. We, therefore, choose to omit the analogous details involved. We thus find that

which implies that . The proof of Theorem 3.3 is thus completed.

Theorem 3.4.

Let with and (3.1) holds. Then the integral operator defined by (3.2) belongs to the class .

Proof.

In view of (1.26) and Theorem 3.3, and by similarly applying the method of proof of Theorem 3.2, we deduce that the assertion of Theorem 3.4 holds.

Theorem 3.5.

Let with and

Then the function defined by

belongs to the class .

Proof.

Let and suppose that

Combining (3.17) and (3.18), we have

Now, in view of (3.17), (3.18), and (3.19), we get

Since (3.16) holds, an application of Lemma 1.1 to (3.20) yields

that is, . We thus complete the proof of Theorem 3.5.

Theorem 3.6.

Let with and (3.16) holds. Then the function defined by (3.17) belongs to the class .

Proof.

By virtue of (1.25) and Theorem 3.5, and by similarly applying the method of proof of Theorem 3.2, we conclude that the assertion of Theorem 3.6 holds.

Theorem 3.7.

Let with and (3.16) holds. Then the function defined by (3.17) belongs to the class .

Proof.

Let . Then, by definition, we know that there exists a function such that (3.9) holds. Since , by Theorem 3.5, we easily find that , which implies that

We now set

where is analytic in with . From (3.17) and (3.23), we get

Combining (3.22), (3.23), and (3.24), we find that

Furthermore, by virtue of (1.22), (3.22), and (3.25), we deduce that

The remainder of the proof of Theorem 3.7 is similar to that of Theorem 2.3. We, therefore, choose to omit the analogous details involved. We thus find that

which implies that . The proof of Theorem 3.7 is thus completed.

Theorem 3.8.

Let with and (3.16) holds. Then the function defined by (3.17) belongs to the class .

Proof.

By virtue of (1.26) and Theorem 3.7, and by similarly applying the method of proof of Theorem 3.2, we deduce that the assertion of Theorem 3.8 holds.

## 4. Subordination and Superordination Results

In this section, we derive some subordination and superordination results associated with the operator . By similarly applying the methods of proof of the results obtained by Cho et al*.* [15], we get the following subordination and superordination results. Here, we choose to omit the details involved. For some other recent sandwich-type results in analytic function theory, one can find in [16–30] and the references cited therein.

Corollary 4.1.

Let . If

where

then the subordination relationship

implies that

Furthermore, the function is the best dominant.

Corollary 4.2.

Let . If

where

then the subordination relationship

implies that

Furthermore, the function is the best dominant.

Denote by the set of all functions that are analytic and injective on , where

and such that for . If is subordinate to , then is superordinate to . We now derive the following superordination results.

Corollary 4.3.

Let . If

where is given by (4.2), also let the function be univalent in and , then the subordination relationship

implies that

Furthermore, the function is the best subordinant.

Corollary 4.4.

Let . If

where is given by (4.6), also let the function be univalent in and , then the subordination relationship

implies that

Furthermore, the function is the best subordinant.

Combining the above mentioned subordination and superordination results involving the operator , we get the following "sandwich-type results".

Corollary 4.5.

Let . If

where is given by (4.2), also let the function be univalent in and , then the subordination chain

implies that

Furthermore, the functions and are, respectively, the best subordinant and the best dominant.

Corollary 4.6.

Let . If

where is given by (4.6), also let the function be univalent in and , then the subordination chain

implies that

Furthermore, the functions and are, respectively, the best subordinant and the best dominant.

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## Acknowledgments

The present investigation was supported by the *Scientific Research Fund of Hunan Provincial Education Department* under Grant 08C118 of China. The authors would like to thank Professor R. M. Ali for sending several valuable papers to them.

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Wang, ZG., Liu, ZH. & Sun, Y. Some Subclasses of Meromorphic Functions Associated with a Family of Integral Operators.
*J Inequal Appl* **2009, **931230 (2009). https://doi.org/10.1155/2009/931230

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DOI: https://doi.org/10.1155/2009/931230

### Keywords

- Analytic Function
- Similar Argument
- Integral Operator
- Unit Disk
- Open Unit