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# Some Properties of Certain Class of Integral Operators

*Journal of Inequalities and Applications*
**volumeÂ 2011**, ArticleÂ number:Â 531540 (2011)

## Abstract

The main object of this paper is to derive some inequality properties and convolution properties of certain class of integral operators defined on the space of meromorphic functions.

## 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

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] recently introduced and investigated the 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 , Wang et al*.* [3] introduced the operator

which is defined as

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

and is the Pochhammer symbol defined by

Clearly, we know that .

It is readily verified from (1.15) that

Recently, Wang et al*.* [3] obtained several inclusion relationships and integral-preserving properties associated with some subclasses involving the operator , some subordination and superordination results involving the operator are also derived. Furthermore, Sun et al*.* [4] investigated several other subordination and superordination results for the operator .

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

Lemma 1.1 (see [5]).

Let be analytic and convex univalent in with . Suppose also that is analytic in with . If

then

and is the best dominant of (1.20).

Let denote the class of functions of the form

which are analytic in and satisfy the condition

Lemma 1.2 (see [6]).

Let

Then

The result is the best possible.

Lemma 1.3 (see [7]).

Let

Then

In the present paper, we aim at proving some inequality properties and convolution properties of the integral operator .

## 2. Main Results

Our first main result is given by Theorem 2.1 below.

Theorem 2.1.

Let and . If satisfies the condition

then

The result is sharp.

Proof.

Suppose that

Then is analytic in with . Combining (1.18) and (2.3), we find that

From (2.1), (2.3), and (2.4), we get

By Lemma 1.1, we obtain

or equivalently,

where is analytic in with

Since and , we deduce from (2.7) that

By noting that

the assertion (2.2) of Theorem 2.1 follows immediately from (2.9) and (2.10).

To show the sharpness of (2.2), we consider the function defined by

For the function defined by (2.11), we easily find that

it follows from (2.12) that

This evidently completes the proof of Theorem 2.1.

In view of (1.19), by similarly applying the method of proof of Theorem 2.1, we get the following result.

Corollary 2.2.

Let and . If satisfies the condition

then

The result is sharp.

For the function given by (1.1), we here recall the integral operator

defined by

Theorem 2.3.

Let , and . Suppose also that is given by (2.17). If satisfies the condition

then

The result is sharp.

Proof.

We easily find from (2.17) that

Suppose that

It follows from (2.18), (2.20) and (2.21) that

The remainder of the proof of Theorem 2.3 is much akin to that of Theorem 2.1, we therefore choose to omit the analogous details involved.

Theorem 2.4.

Let and . If is defined by

and each of the functions satisfies the condition

then

The result is sharp when .

Proof.

Suppose that satisfy conditions (2.24). By setting

it follows from (2.24) and (2.26) that

Combining (1.18) and (2.26), we get

For the function given by (2.23), we find from (2.28) that

where

By noting that and , it follows from Lemma 1.2 that

Furthermore, by Lemma 1.3, we know that

In view of (2.24), (2.30), and (2.32), we deduce that

When , we consider the functions which satisfy conditions (2.24) and are given by

It follows from (2.26), (2.28), (2.30), and (2.34) that

Thus, we have

The proof of Theorem 2.4 is evidently completed.

With the aid of (1.19), by applying the similar method of the proof of Theorem 2.4, we obtain the following result.

Corollary 2.5.

Let and . If is defined by (2.23) and each of the functions satisfies the condition

then

The result is sharp when .

## References

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## Acknowledgments

This work was supported by the *National Natural Science Foundation under Grant* 11026205, the *Science Research Fund of Guangdong Provincial Education Department* under Grant LYM08101, the *Natural Science Foundation of Guangdong Province* under Grant 10452800001004255, and the *Excellent Youth Foundation of Educational Committee of Hunan Province* under Grant 10B002 of the People's Republic of China.

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### Cite this article

Zhou, JR., Liu, ZH. & Wang, ZG. Some Properties of Certain Class of Integral Operators.
*J Inequal Appl* **2011**, 531540 (2011). https://doi.org/10.1155/2011/531540

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DOI: https://doi.org/10.1155/2011/531540

### Keywords

- Integral Operator
- Unit Disk
- Open Unit
- Meromorphic Function
- Gamma Function