• Research Article
• Open Access

# Some Properties of Certain Class of Integral Operators

Journal of Inequalities and Applications20112011:531540

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

• Received: 17 October 2010
• Accepted: 10 January 2011
• Published:

## 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

Let denote the class of functions of the form
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
Furthermore, if the function is univalent in , then we have the following equivalence:
Analogous to the integral operator defined by Jung et al. , Lashin  recently introduced and investigated the integral operator
we define a new function in terms of the Hadamard product (or convolution)
Then, motivated essentially by the operator , Wang et al.  introduced the operator
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 .

Recently, Wang et al.  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.  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 ).

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

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 ).

The result is the best possible.

Lemma 1.3 (see ).

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

The result is sharp.

Proof.

Then is analytic in with . Combining (1.18) and (2.3), we find that
where is analytic in with
Since and , we deduce from (2.7) 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

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

The result is sharp.

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

Theorem 2.3.

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

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.

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

The result is sharp when .

Proof.

Suppose that satisfy conditions (2.24). By setting
For the function given by (2.23), we find from (2.28) that
By noting that and , it follows from Lemma 1.2 that
When , we consider the functions which satisfy conditions (2.24) and are given by

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

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

(1)
Department of Mathematics, Foshan University, Foshan, Guangdong, 528000, China
(2)
Department of Mathematics, Honghe University, Mengzi, Yunnan, 661100, China
(3)
School of Mathematics and Computing Science, Changsha University of Science and Technology, Yuntang Campus, Changsha, Hunan, 410114, China

## References 