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

Serap Bulut

doi:10.1155/2009/158408

Research Article
Open Access

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

Serap Bulut¹Email author

Journal of Inequalities and Applications20092009:158408

https://doi.org/10.1155/2009/158408

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

Analytic Function
Differential Operator
Convex Function
Integral Operator
Geometric Interpretation

1. Introduction

Let

denote the class of functions of the form

(1.1)

which are analytic in the open unit disk , and .

For

, Al-Oboudi [1] introduced the following operator:

(1.2)

(1.3)

(1.4)

If

is given by (1.1), then from (1.3) and (1.4) we see that

(1.5)

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

(1.6)

or equivalently

(1.7)

for all .

A function

is in the classs

, where

,

, if and only if

(1.8)

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

(1.14)

or equivalently

(1.15)

where , , , , , .

A function

is in the class

if and only if

satisfies

(1.16)

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 right-half plane such that

(1.21)

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

(1.22)

where , , , .

A function

is in the class

if and only if

satisfies

(1.23)

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

(1.27)

where and is the Al-Oboudi differential operator.

Remark 1.6.

In Definition 1.5, if we set

(i)

, then we have [7, Definition 1].
(ii)

, and , then we have the integral operator defined by (1.26).
(iii)

, , 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

, defined by (1.27), one has

(2.1)

Proof.

By (1.27), we get

(2.2)

Also, using (1.3) and (1.4), we obtain

(2.3)

On the other hand, from (2.2) and (2.3), we find

(2.4)

(2.5)

Thus by (2.2) and (2.4), we can write

(2.6)

Finally, we obtain

(2.7)

which is the desired result.

Theorem 2.2.

Let

,

, and

,

. Also suppose that

(2.8)

If

, then the integral operator

, defined by (1.27), is in the class

, where

(2.9)

Proof.

Since

, by (1.14) we have

(2.10)

for all

. By (2.1), we get

(2.11)

So, (2.10) and (2.11) give us

(2.12)

for all . Hence, we obtain , where .

Corollary 2.3.

Let

,

, and

. Also suppose that

(2.13)

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

(2.14)

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

(2.15)

If

, then the integral operator

, defined by (1.27), is in the class

, where

(2.16)

Proof.

In Theorem 2.2, we consider .

Corollary 2.7.

Let

and

. Also suppose that

(2.17)

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

(2.18)

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

(2.19)

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

(2.20)

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

(2.21)

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

(2.22)

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

(2.23)

If , then the integral operator , defined by (1.27), is in the class .

Proof.

Since

, by (1.14) we have

(2.24)

for all .

On the other hand, from (2.1), we obtain

(2.25)

Considering (1.16) with the above equality, we find

(2.26)

for all . This completes proof.

Corollary 2.17.

Let

,

, and

. Also suppose that

(2.27)

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

(2.28)

If , then the integral operator , defined by (1.27), is in the class .

Proof.

Since

, by (1.22) we have

(2.29)

for all

. Considering this inequality and (2.1), we obtain

(2.30)

for all . Hence by (1.23), we have .

Corollary 2.20.

Let

and

. Also suppose that

(2.31)

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

(2.32)

If , then the integral operator , defined by (1.27), is in the class .

Proof.

Since

, by (1.22) we have

(2.33)

for all

. Considering this inequality and (2.1), we obtain

(2.34)

for all . Hence, by (1.8), we have .

Corollary 2.23.

Let

and

. Also suppose that

(2.35)

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

(1)

Civil Aviation College, Kocaeli University, Arslanbey Campus, 41285 İzmit-Kocaeli, Turkey

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