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

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

(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].