- Research Article
- Open Access

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

- Serap Bulut
^{1}Email author

**2009**:158408

https://doi.org/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.

## Keywords

- Analytic Function
- Differential Operator
- Convex Function
- Integral Operator
- Geometric Interpretation

## 1. Introduction

which are analytic in the open unit disk , and .

Remark 1.1.

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

Now we introduce new classes and as follows.

- (ii)

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

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 .

- (ii)

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

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

Definition 1.5.

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.

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.

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.

Proof.

Corollary 2.7.

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.

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.

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.

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.

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.

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.

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

Proof.

for all . This completes proof.

Corollary 2.17.

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.

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.

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.

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.

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

## References

- Al-Oboudi FM:
**On univalent functions defined by a generalized Sălăgean operator.***International Journal of Mathematics and Mathematical Sciences*2004,**2004**(27):1429–1436. 10.1155/S0161171204108090MathSciNetView ArticleMATHGoogle Scholar - Sălăgean GŞ:
**Subclasses of univalent functions.**In*Complex Analysis-Fifth Romanian-Finnish Seminar, Part 1 (Bucharest, 1981), Lecture Notes in Mathematics*.*Volume 1013*. Springer, Berlin, Germany; 1983:362–372. 10.1007/BFb0066543View ArticleGoogle Scholar - Frasin BA:
**Family of analytic functions of complex order.***Acta Mathematica. Academiae Paedagogicae Nyíregyháziensis*2006,**22**(2):179–191.MathSciNetMATHGoogle Scholar - Magdaş I:
*, Doctoral thesis*. University "Babeş-Bolyai", Cluj-Napoca, Romania; 1999.Google Scholar - Acu M:
**Subclasses of convex functions associated with some hyperbola.***Acta Universitatis Apulensis*2006, (12):3–12.MathSciNetMATHGoogle Scholar - Breaz D, Breaz N:
**Two integral operators.***Studia Universitatis Babeş-Bolyai. Mathematica*2002,**47**(3):13–19.MathSciNetMATHGoogle Scholar - Breaz D, Güney HÖ, Sălăgean GŞ: A new general integral operator. Tamsui Oxford Journal of Mathematical Sciences. Accepted Tamsui Oxford Journal of Mathematical Sciences. AcceptedGoogle Scholar
- Bulut S:
**Some properties for an integral operator defined by Al-Oboudi differential operator.***Journal of Inequalities in Pure and Applied Mathematics*2008.,**9**(4, article 115):Google Scholar

## 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.