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A New General Integral Operator Defined by Al-Oboudi Differential Operator
Journal of Inequalities and Applications volume 2009, Article number: 158408 (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
Let denote the class of functions of the form
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F158408/MediaObjects/13660_2008_Article_1900_Equ1_HTML.gif)
which are analytic in the open unit disk , and
.
For , Al-Oboudi [1] introduced the following operator:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F158408/MediaObjects/13660_2008_Article_1900_Equ2_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F158408/MediaObjects/13660_2008_Article_1900_Equ3_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F158408/MediaObjects/13660_2008_Article_1900_Equ4_HTML.gif)
If is given by (1.1), then from (1.3) and (1.4) we see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F158408/MediaObjects/13660_2008_Article_1900_Equ5_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F158408/MediaObjects/13660_2008_Article_1900_Equ6_HTML.gif)
or equivalently
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F158408/MediaObjects/13660_2008_Article_1900_Equ7_HTML.gif)
for all .
A function is in the classs
, where
,
,
,
, if and only if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F158408/MediaObjects/13660_2008_Article_1900_Equ8_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F158408/MediaObjects/13660_2008_Article_1900_Equ14_HTML.gif)
or equivalently
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F158408/MediaObjects/13660_2008_Article_1900_Equ15_HTML.gif)
where ,
,
,
,
,
.
A function is in the class
if and only if
satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F158408/MediaObjects/13660_2008_Article_1900_Equ16_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F158408/MediaObjects/13660_2008_Article_1900_IEq70_HTML.gif)
and if and only if
and
, respectively, take all the values in the conic domain
which is included in the right-half plane such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F158408/MediaObjects/13660_2008_Article_1900_Equ21_HTML.gif)
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
.
A function is in the class
if and only if
satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F158408/MediaObjects/13660_2008_Article_1900_Equ22_HTML.gif)
where ,
,
,
.
A function is in the class
if and only if
satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F158408/MediaObjects/13660_2008_Article_1900_Equ23_HTML.gif)
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 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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F158408/MediaObjects/13660_2008_Article_1900_Equ27_HTML.gif)
where and
is the Al-Oboudi 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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F158408/MediaObjects/13660_2008_Article_1900_Equ28_HTML.gif)
Proof.
By (1.27), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F158408/MediaObjects/13660_2008_Article_1900_Equ29_HTML.gif)
Also, using (1.3) and (1.4), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F158408/MediaObjects/13660_2008_Article_1900_Equ30_HTML.gif)
On the other hand, from (2.2) and (2.3), we find
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F158408/MediaObjects/13660_2008_Article_1900_Equ31_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F158408/MediaObjects/13660_2008_Article_1900_Equ32_HTML.gif)
Thus by (2.2) and (2.4), we can write
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F158408/MediaObjects/13660_2008_Article_1900_Equ33_HTML.gif)
Finally, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F158408/MediaObjects/13660_2008_Article_1900_Equ34_HTML.gif)
which is the desired result.
Theorem 2.2.
Let,
,
, and
,
. Also suppose that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F158408/MediaObjects/13660_2008_Article_1900_Equ35_HTML.gif)
If , then the integral operator
, defined by (1.27), is in the class
, where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F158408/MediaObjects/13660_2008_Article_1900_Equ36_HTML.gif)
Proof.
Since , by (1.14) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F158408/MediaObjects/13660_2008_Article_1900_Equ37_HTML.gif)
for all . By (2.1), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F158408/MediaObjects/13660_2008_Article_1900_Equ38_HTML.gif)
So, (2.10) and (2.11) give us
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F158408/MediaObjects/13660_2008_Article_1900_Equ39_HTML.gif)
for all . Hence, we obtain
, where
.
Corollary 2.3.
Let ,
,
, and
. Also suppose that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F158408/MediaObjects/13660_2008_Article_1900_Equ40_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F158408/MediaObjects/13660_2008_Article_1900_Equ41_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F158408/MediaObjects/13660_2008_Article_1900_Equ42_HTML.gif)
If , then the integral operator
, defined by (1.27), is in the class
, where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F158408/MediaObjects/13660_2008_Article_1900_Equ43_HTML.gif)
Proof.
In Theorem 2.2, we consider .
Corollary 2.7.
Let and
. Also suppose that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F158408/MediaObjects/13660_2008_Article_1900_Equ44_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F158408/MediaObjects/13660_2008_Article_1900_Equ45_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F158408/MediaObjects/13660_2008_Article_1900_Equ46_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F158408/MediaObjects/13660_2008_Article_1900_Equ47_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F158408/MediaObjects/13660_2008_Article_1900_Equ48_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F158408/MediaObjects/13660_2008_Article_1900_Equ49_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F158408/MediaObjects/13660_2008_Article_1900_Equ50_HTML.gif)
If , then the integral operator
, defined by (1.27), is in the class
.
Proof.
Since , by (1.14) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F158408/MediaObjects/13660_2008_Article_1900_Equ51_HTML.gif)
for all .
On the other hand, from (2.1), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F158408/MediaObjects/13660_2008_Article_1900_Equ52_HTML.gif)
Considering (1.16) with the above equality, we find
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F158408/MediaObjects/13660_2008_Article_1900_Equ53_HTML.gif)
for all . This completes proof.
Corollary 2.17.
Let,
,
, and
. Also suppose that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F158408/MediaObjects/13660_2008_Article_1900_Equ54_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F158408/MediaObjects/13660_2008_Article_1900_Equ55_HTML.gif)
If , then the integral operator
, defined by (1.27), is in the class
.
Proof.
Since , by (1.22) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F158408/MediaObjects/13660_2008_Article_1900_Equ56_HTML.gif)
for all . Considering this inequality and (2.1), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F158408/MediaObjects/13660_2008_Article_1900_Equ57_HTML.gif)
for all . Hence by (1.23), we have
.
Corollary 2.20.
Let and
. Also suppose that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F158408/MediaObjects/13660_2008_Article_1900_Equ58_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F158408/MediaObjects/13660_2008_Article_1900_Equ59_HTML.gif)
If , then the integral operator
, defined by (1.27), is in the class
.
Proof.
Since , by (1.22) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F158408/MediaObjects/13660_2008_Article_1900_Equ60_HTML.gif)
for all . Considering this inequality and (2.1), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F158408/MediaObjects/13660_2008_Article_1900_Equ61_HTML.gif)
for all . Hence, by (1.8), we have
.
Corollary 2.23.
Let and
. Also suppose that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F158408/MediaObjects/13660_2008_Article_1900_Equ62_HTML.gif)
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].
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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):
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Bulut, S. A New General Integral Operator Defined by Al-Oboudi 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