Sufficient Conditions for Univalence of an Integral Operator Defined by Al-Oboudi Differential Operator

Abstract

We investigate the univalence of an integral operator defined by Al-Oboudi differential operator.

1. Introduction

Let denote the class of all 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 .

When , we get Sălăgean's differential operator [2].

By using the Al-Oboudi differential operator, we introduce the following integral operator.

Definition 1.1.

Let and , . We define the integral operator ,

(1.6)

where and is the Al-Oboudi differential operator.

Remark 1.2.

• (i) For , , , , and , we have Alexander integral operator

  

  (1.7)

which was introduced in [3].

• (ii) For , , , , and , we have the integral operator

  

  (1.8)

that was studied in [4].

• (iii) For , , , , , we have the integral operator

  

  (1.9)

which was studied in [5].

• (iv) For , , , and , we have the integral operator

  

  (1.10)

which was studied in [6, 7].

2. Main Results

The following lemmas will be required in our investigation.

Lemma 2.1 (see [8]).

If the function is regular in the unit disk , , and

(2.1)

for all , then the function is univalent in .

Lemma 2.2 (Schwarz Lemma 2.2) (see [9, page 166]).

Let the analytic function be regular in and let . If, in , , then

(2.2)

and .

The equality holds if and only if and .

Theorem 2.3.

Let , , and , . If

(2.3)

then defined in Definition 1.1 is univalent in .

Proof. .

Since , , by (1.5), we have

(2.4)

for all .

On the other hand, we obtain

(2.5)

for . This equality implies that

(2.6)

or equivalently

(2.7)

By differentiating the above equality, we get

(2.8)

After some calculus, we obtain

(2.9)

By hypothesis, since , and since we have

(2.10)

So, we obtain

(2.11)

Thus .

Remark 2.4.

For , , , we have [5, Theorem 1].

Corollary 2.5.

Let , and , . If

(2.12)

and , then .

Theorem 2.6.

Let , and , . If

(i),

(ii), and

(iii),

then defined in Definition 1.1 is univalent in .

Proof. .

By (2.9), we get

(2.13)

This inequality implies that

(2.14)

By Schwarz lemma (Lemma 2.2), we have

(2.15)

or

(2.16)

for all .

So, by Lemma 2.1, .

Remark 2.7.

For , , , , , we have [7, Theorem 1].

Corollary 2.8.

Let , and , . If

(i),

(ii), and

(iii),

then .

In [10], similar results are given by using the Ruscheweyh differential operator.