A New General Integral Operator Defined by Al-Oboudi Differential Operator
- Serap Bulut1Email author
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
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.