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A New General Integral Operator Defined by AlOboudi Differential Operator
Journal of Inequalities and Applications volume 2009, Article number: 158408 (2009)
Abstract
We define a new general integral operator using AlOboudi differential operator. Also we introduce new subclasses of analytic functions. Our results generalize the results of Breaz, Güney, and Sălăgean.
1. Introduction
Let denote the class of functions of the form
which are analytic in the open unit disk , and .
For , AlOboudi [1] introduced the following operator:
If is given by (1.1), then from (1.3) and (1.4) we see that
with .
Remark 1.1.
When , we get Sălăgean's differential operator [2].
Now we introduce new classes and as follows.
A function is in the classes , where , , , , if and only if
or equivalently
for all .
A function is in the classs , where , , , , if and only if
for all .
We note that if and only if .
Remark 1.2.

(i)
For and , we have the classes
(1.9)
introduced by Frasin [3].

(ii)
For and , we have the class
(1.10)
of starlike functions of order defined by Sălăgean [2].

(iii)
In particular, the classes
(1.11)
are the classes of starlike functions of order and convex functions of order in , respectively.

(iv)
Furthermore, the classes
(1.12)
are familiar classes of starlike and convex functions in , respectively.

(v)
For , we get
(1.13)
Let us introduce the new subclasses , and , as follows.
A function is in the class if and only if satisfies
or equivalently
where , , , , , .
A function is in the class if and only if satisfies
where , , , , , .
We note that if and only if .
Remark 1.3.

(i)
For , we have
(1.17)

(ii)
For and , we have the class
(1.18)
of uniform starlike functions of order and type , [4].

(iii)
For , we have
(1.19)

(iv)
For and , we have
(1.20)
Geometric Interpretation
and if and only if and , respectively, take all the values in the conic domain which is included in the righthalf plane such that
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 righthalf plane for .
A function is in the class if and only if satisfies
where , , , .
A function is in the class if and only if satisfies
where , , , .
We note that if and only if .
Remark 1.4.

(i)
For and , we have the classes
(1.24)
defined in [5].

(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 AlOboudi 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 AlOboudi differential operator.
Remark 1.6.
In Definition 1.5, if we set
2. Main Results
The following lemma will be required in our investigation.
Lemma 2.1.
For the integral operator , defined by (1.27), one has
Proof.
By (1.27), we get
Also, using (1.3) and (1.4), we obtain
On the other hand, from (2.2) and (2.3), we find
Thus by (2.2) and (2.4), we can write
Finally, we obtain
which is the desired result.
Theorem 2.2.
Let, , , and , . Also suppose that
If , then the integral operator , defined by (1.27), is in the class , where
Proof.
Since , by (1.14) we have
for all . By (2.1), we get
So, (2.10) and (2.11) give us
for all . Hence, we obtain , where .
Corollary 2.3.
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 2.2, we consider .
From Corollary 2.3, we immediately get Corollary 2.4.
Corollary 2.4.
Let , , , and . Also suppose that
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
If , then the integral operator , defined by (1.27), is in the class , where
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.
Since , by (1.14) we have
for all .
On the other hand, from (2.1), we obtain
Considering (1.16) with the above equality, we find
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.
Since , by (1.22) we have
for all . Considering this inequality and (2.1), we obtain
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.
Since , by (1.22) we have
for all . Considering this inequality and (2.1), we obtain
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].
References
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Bulut S: Some properties for an integral operator defined by AlOboudi differential operator. Journal of Inequalities in Pure and Applied Mathematics 2008.,9(4, article 115):
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Bulut, S. A New General Integral Operator Defined by AlOboudi Differential Operator. J Inequal Appl 2009, 158408 (2009). https://doi.org/10.1155/2009/158408
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DOI: https://doi.org/10.1155/2009/158408
Keywords
 Analytic Function
 Differential Operator
 Convex Function
 Integral Operator
 Geometric Interpretation