- Research Article
- Open Access

# Some Properties of Certain Class of Integral Operators

- Jian-Rong Zhou
^{1}, - Zhi-Hong Liu
^{2}and - Zhi-Gang Wang
^{3}Email author

**2011**:531540

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

© Jian-Rong Zhou et al. 2011

**Received:**17 October 2010**Accepted:**10 January 2011**Published:**23 January 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.

## Keywords

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

## 1. Introduction and Preliminaries

*.*[1], Lashin [2] recently introduced and investigated the integral operator

Clearly, we know 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]).

and is the best dominant of (1.20).

Lemma 1.2 (see [6]).

The result is the best possible.

Lemma 1.3 (see [7]).

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.

The result is sharp.

Proof.

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

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.

The result is sharp.

Theorem 2.3.

The result is sharp.

Proof.

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.

The result is sharp when .

Proof.

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.

The result is sharp when .

## Declarations

### 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.

## Authors’ Affiliations

## References

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## Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.