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

  • Serap Bulut
Open Access
Research Article

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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1. Introduction

Let Open image in new window denote the class of functions of the form

which are analytic in the open unit disk Open image in new window , and Open image in new window .

For Open image in new window , Al-Oboudi [1] introduced the following operator:
If Open image in new window is given by (1.1), then from (1.3) and (1.4) we see that

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.

or equivalently

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 .

Remark 1.2.
introduced by Frasin [3].
of Open image in new window -starlike functions of order Open image in new window defined by Sălăgean [2].
  1. (iii)
    In particular, the classes
     
are the classes of starlike functions of order Open image in new window and convex functions of order Open image in new window in Open image in new window , respectively.
  1. (iv)
    Furthermore, the classes
     
are familiar classes of starlike and convex functions in Open image in new window , respectively.

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.

or equivalently

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 .

Remark 1.3.
of Open image in new window -uniform starlike functions of order Open image in new window and type Open image in new window , [4].

Geometric Interpretation

and Open image in new window if and only if Open image in new window and Open image in new window , respectively, take all the values in the conic domain Open image in new window which is included in the right-half plane such that

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 .

Remark 1.4.
defined in [5].
  1. D.
    Breaz and N. Breaz [6] introduced and studied the integral operator
     

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.

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

Remark 1.6.

In Definition 1.5, if we set
  1. (i)
    , then we have [7, Definition 1].
     
  2. (ii)
    , Open image in new window and Open image in new window , then we have the integral operator defined by (1.26).
     
  3. (iii)
    , Open image in new window , then we have [8, Definition 1.1].
     

2. Main Results

The following lemma will be required in our investigation.

Lemma 2.1.

For the integral operator Open image in new window , 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.

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

Proof.

for all Open image in new window . By (2.1), we get
So, (2.10) and (2.11) give us

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.

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

Proof.

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

Corollary 2.7.

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.

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.

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.

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.

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.

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 .

On the other hand, from (2.1), we obtain
Considering (1.16) with the above equality, we find

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

Corollary 2.17.

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.

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.

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 . Considering this inequality and (2.1), we obtain

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

Corollary 2.20.

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.

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.

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 . Considering this inequality and (2.1), we obtain

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

Corollary 2.23.

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.

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

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Copyright information

© Serap Bulut. 2009

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.

Authors and Affiliations

  1. 1.Civil Aviation CollegeKocaeli Universityİzmit-KocaeliTurkey

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