# A New General Integral Operator Defined by Al-Oboudi Differential Operator

- Serap Bulut
^{1}Email author

**2009**:158408

**DOI: **10.1155/2009/158408

© Serap Bulut. 2009

**Received: **8 December 2008

**Accepted: **22 January 2009

**Published: **28 January 2009

## Abstract

We define a new general integral operator using Al-Oboudi differential operator. Also we introduce new subclasses of analytic functions. Our results generalize the results of Breaz, Güney, and Sălăgean.

## 1. Introduction

which are analytic in the open unit disk , and .

with .

Remark 1.1.

When , we get Sălăgean's differential operator [2].

Now we introduce new classes and as follows.

for all .

for all .

We note that if and only if .

- (i)For and , we have the classes(1.9)

- (ii)For and , we have the class(1.10)

- (iv)Furthermore, the classes(1.12)

- (v)For , we get(1.13)

Let us introduce the new subclasses , and , as follows.

where , , , , , .

where , , , , , .

We note that if and only if .

- (i)For , we have(1.17)

- (ii)For and , we have the class(1.18)

- (iv)For and , we have(1.20)

Geometric Interpretation

From elementary computations we see that represents the conic sections symmetric about the real axis. Thus is an elliptic domain for , a parabolic domain for , a hyperbolic domain for and a right-half plane for .

where , , , .

where , , , .

We note that if and only if .

- (i)For and , we have the classes(1.24)

- (ii)For , we have(1.25)

- D.Breaz and N. Breaz [6] introduced and studied the integral operator(1.26)

where and for all .

By using the Al-Oboudi differential operator, we introduce the following integral operator. So we generalize the integral operator .

Definition 1.5.

*Let*, ,

*and*, .

*One defines the integral operator*

where and is the Al-Oboudi differential operator.

Remark 1.6.

## 2. Main Results

The following lemma will be required in our investigation.

Lemma 2.1.

Proof.

which is the desired result.

Theorem 2.2.

*Let*, , , and , . Also suppose that

Proof.

for all . Hence, we obtain , where .

Corollary 2.3.

If , then the integral operator , defined by (1.27), is in the class , where is defined as in (2.9).

Proof.

In Theorem 2.2, we consider .

From Corollary 2.3, we immediately get Corollary 2.4.

Corollary 2.4.

If , then the integral operator , defined by (1.27), is in the class .

Remark 2.5.

If we set in Corollary 2.4, then we have [7, Theorem 1]. So Corollary 2.4 is an extension of Theorem 1.

Corollary 2.6.

*Let*and , . Also suppose that

Proof.

In Theorem 2.2, we consider .

Corollary 2.7.

*Let*and . Also suppose that

If , then the integral operator , defined by (1.27), is in the class , where is defined as in (2.16).

Proof.

In Corollary 2.6, we consider .

Corollary 2.8 readily follows from Corollary 2.7.

Corollary 2.8.

*Let*, and . Also suppose that

If , then the integral operator , defined by (1.27), is in the class .

Remark 2.9.

If we set in Corollary 2.8, then we have [7, Corollary 1].

Theorem 2.10.

*Let*, , and , . Also suppose that

If , then the integral operator , defined by (1.27), is in the class , where is defined as in (2.9).

Proof.

The proof is similar to the proof of Theorem 2.2.

Corollary 2.11.

*Let*, , and . Also suppose that

If , then the integral operator , defined by (1.27), is in the class , where is defined as in (2.9).

Proof.

In Theorem 2.10, we consider .

Remark 2.12.

If we set in Corollary 2.11, then we have [7, Theorem 2].

Corollary 2.13.

*Let*and , . Also suppose that

If , then the integral operator , defined by (1.27), is in the class , where is defined as in (2.16).

Proof.

In Theorem 2.10, we consider .

Corollary 2.14.

*Let*and . Also suppose that

If , then the integral operator , defined by (1.27), is in the class , where is defined as in (2.16).

Proof.

In Corollary 2.13, we consider .

Remark 2.15.

If we set in Corollary 2.14, then we have [7, Corollary 2].

Theorem 2.16.

*Let*, , and , . Also suppose that

If , then the integral operator , defined by (1.27), is in the class .

Proof.

for all .

for all . This completes proof.

Corollary 2.17.

*Let*, , , and . Also suppose that

If , then the integral operator , defined by (1.27), is in the class .

Proof.

In Theorem 2.16, we consider .

Remark 2.18.

If we set in Corollary 2.17, then we have [7, Theorem 3].

Theorem 2.19.

*Let*, , and . Also suppose that

If , then the integral operator , defined by (1.27), is in the class .

Proof.

for all . Hence by (1.23), we have .

Corollary 2.20.

*Let*and . Also suppose that

If , then the integral operator , defined by (1.27), is in the class .

Proof.

In Theorem 2.19, we consider .

Remark 2.21.

If we set in Corollary 2.20, then we have [7, Theorem 4].

Theorem 2.22.

*Let*, and . Also suppose that

If , then the integral operator , defined by (1.27), is in the class .

Proof.

for all . Hence, by (1.8), we have .

Corollary 2.23.

*Let*and . Also suppose that

If , then the integral operator , defined by (1.27), is in the class .

Proof.

In Theorem 2.22, we consider .

Remark 2.24.

If we set in Corollary 2.23, then we have [7, Theorem 5].

## Authors’ Affiliations

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## Copyright

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