# 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.

## 1. Introduction and Preliminaries

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

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.

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.

## 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

- Jung IB, Kim YC, Srivastava HM:
**The Hardy space of analytic functions associated with certain one-parameter families of integral operators.***Journal of Mathematical Analysis and Applications*1993,**176**(1):138–147. 10.1006/jmaa.1993.1204MATHMathSciNetView ArticleGoogle Scholar - Lashin AY:
**On certain subclasses of meromorphic functions associated with certain integral operators.***Computers & Mathematics with Applications*2010,**59**(1):524–531. 10.1016/j.camwa.2009.06.015MATHMathSciNetView ArticleGoogle Scholar - Wang Z-G, Liu Z-H, Sun Y:
**Some subclasses of meromorphic functions associated with a family of integral operators.***Journal of Inequalities and Applications*2009,**2009:**-18.Google Scholar - Sun Y, Kuang W-P, Liu Z-H:
**Subordination and superordination results for the family of Jung-Kim- Srivastava integral operators.***Filomat*2010,**24:**69–85. 10.2298/FIL1001069SMATHMathSciNetView ArticleGoogle Scholar - Miller SS, Mocanu PT:
**Differential subordinations and univalent functions.***The Michigan Mathematical Journal*1981,**28**(2):157–172.MATHMathSciNetView ArticleGoogle Scholar - Stankiewicz J, Stankiewicz Z:
**Some applications of the Hadamard convolution in the theory of functions.***Annales Universitatis Mariae Curie-Skłodowska Sectio A*1986,**40:**251–265.MATHMathSciNetGoogle Scholar - Srivastava HM, Owa S (Eds):
*Current Topics in Analytic Function Theory*. World Scientific, River Edge, NJ, USA; 1992:xiv+456.MATHGoogle Scholar

## Copyright

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