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

- 516 Downloads
- 2 Citations

## 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 Open image in new window , and Open image in new window .

with Open image in new window .

Remark 1.1.

When Open image in new window , we get Sălăgean's differential operator [2].

Now we introduce new classes Open image in new window and Open image in new window as follows.

for all Open image in new window .

for all Open image in new window .

We note that Open image in new window if and only if Open image in new window .

- (i)

- (ii)

- (iii)In particular, the classes(1.11)

- (iv)Furthermore, the classes(1.12)

- (v)For Open image in new window , we get(1.13)

Let us introduce the new subclasses Open image in new window , Open image in new window and Open image in new window , Open image in new window as follows.

where Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window .

where Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window .

We note that Open image in new window if and only if Open image in new window .

- (i)For Open image in new window , we have(1.17)

- (ii)

- (iii)For Open image in new window , we have(1.19)

- (iv)

Geometric Interpretation

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

where Open image in new window , Open image in new window , Open image in new window , Open image in new window .

where Open image in new window , Open image in new window , Open image in new window , Open image in new window .

We note that Open image in new window if and only if Open image in new window .

- (i)

- (ii)For Open image in new window , we have(1.25)

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

where Open image in new window and Open image in new window for all Open image in new window .

By using the Al-Oboudi differential operator, we introduce the following integral operator. So we generalize the integral operator Open image in new window .

Definition 1.5.

*Let*Open image in new window , Open image in new window ,

*and*Open image in new window , Open image in new window .

*One defines the integral operator*Open image in new window

where Open image in new window and Open image in new window is the Al-Oboudi differential operator.

Remark 1.6.

- (i), then we have [7, Definition 1].
- (ii), Open image in new window and Open image in new window , then we have the integral operator defined by (1.26).
- (iii)

## 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*Open image in new window , Open image in new window , Open image in new window Open image in new window , and Open image in new window , Open image in new window . Also suppose that

Proof.

for all Open image in new window . Hence, we obtain Open image in new window , where Open image in new window .

Corollary 2.3.

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

Proof.

In Theorem 2.2, we consider Open image in new window .

From Corollary 2.3, we immediately get Corollary 2.4.

Corollary 2.4.

If Open image in new window Open image in new window , then the integral operator Open image in new window , defined by (1.27), is in the class Open image in new window .

Remark 2.5.

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

Corollary 2.6.

*Let*Open image in new window Open image in new window and Open image in new window , Open image in new window . Also suppose that

Proof.

In Theorem 2.2, we consider Open image in new window .

Corollary 2.7.

*Let*Open image in new window Open image in new window and Open image in new window . Also suppose that

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

Proof.

In Corollary 2.6, we consider Open image in new window .

Corollary 2.8 readily follows from Corollary 2.7.

Corollary 2.8.

*Let*Open image in new window Open image in new window , and Open image in new window . Also suppose that

If Open image in new window Open image in new window , then the integral operator Open image in new window , defined by (1.27), is in the class Open image in new window .

Remark 2.9.

If we set Open image in new window in Corollary 2.8, then we have [7, Corollary 1].

Theorem 2.10.

*Let*Open image in new window , Open image in new window , Open image in new window Open image in new window and Open image in new window , Open image in new window . Also suppose that

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

Proof.

The proof is similar to the proof of Theorem 2.2.

Corollary 2.11.

*Let*Open image in new window , Open image in new window , Open image in new window Open image in new window and Open image in new window . Also suppose that

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

Proof.

In Theorem 2.10, we consider Open image in new window .

Remark 2.12.

If we set Open image in new window in Corollary 2.11, then we have [7, Theorem 2].

Corollary 2.13.

*Let*Open image in new window and Open image in new window , Open image in new window . Also suppose that

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

Proof.

In Theorem 2.10, we consider Open image in new window .

Corollary 2.14.

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

Proof.

In Corollary 2.13, we consider Open image in new window .

Remark 2.15.

If we set Open image in new window in Corollary 2.14, then we have [7, Corollary 2].

Theorem 2.16.

*Let*Open image in new window , Open image in new window , Open image in new window and Open image in new window , Open image in new window . Also suppose that

If Open image in new window Open image in new window , then the integral operator Open image in new window , defined by (1.27), is in the class Open image in new window .

Proof.

for all Open image in new window .

for all Open image in new window . This completes proof.

Corollary 2.17.

*Let*Open image in new window , Open image in new window , Open image in new window , and Open image in new window . Also suppose that

Proof.

In Theorem 2.16, we consider Open image in new window .

Remark 2.18.

If we set Open image in new window in Corollary 2.17, then we have [7, Theorem 3].

Theorem 2.19.

*Let*Open image in new window , Open image in new window , and Open image in new window . Also suppose that

Proof.

for all Open image in new window . Hence by (1.23), we have Open image in new window .

Corollary 2.20.

Proof.

In Theorem 2.19, we consider Open image in new window .

Remark 2.21.

If we set Open image in new window in Corollary 2.20, then we have [7, Theorem 4].

Theorem 2.22.

*Let*Open image in new window , Open image in new window and Open image in new window . Also suppose that

Proof.

for all Open image in new window . Hence, by (1.8), we have Open image in new window .

Corollary 2.23.

Proof.

In Theorem 2.22, we consider Open image in new window .

Remark 2.24.

If we set Open image in new window in Corollary 2.23, then we have [7, Theorem 5].

## References

- 1.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/S0161171204108090MathSciNetCrossRefMATHGoogle Scholar - 2.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/BFb0066543CrossRefGoogle Scholar - 3.Frasin BA:
**Family of analytic functions of complex order.***Acta Mathematica. Academiae Paedagogicae Nyíregyháziensis*2006,**22**(2):179–191.MathSciNetMATHGoogle Scholar - 4.Magdaş I:
*, Doctoral thesis*. University "Babeş-Bolyai", Cluj-Napoca, Romania; 1999.Google Scholar - 5.Acu M:
**Subclasses of convex functions associated with some hyperbola.***Acta Universitatis Apulensis*2006, (12):3–12.MathSciNetMATHGoogle Scholar - 6.Breaz D, Breaz N:
**Two integral operators.***Studia Universitatis Babeş-Bolyai. Mathematica*2002,**47**(3):13–19.MathSciNetMATHGoogle Scholar - 7.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 - 8.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 information

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.