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

Open Access
Research Article

## Abstract

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

### Keywords

Analytic Function Differential Operator Integral Operator Unit Disk Schwarz Lemma

## 1. Introduction

Let denote the class of all functions of the form

which are analytic in the open unit disk , and .

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

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 ,

where and is the Al-Oboudi differential operator.

Remark 1.2.
1. (i)
For , , , , and , we have Alexander integral operator

which was introduced in [3].
1. (ii)
For , , , , and , we have the integral operator

that was studied in [4].
1. (iii)
For , , , , , we have the integral operator

which was studied in [5].
1. (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

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

The equality holds if and only if and .

Theorem 2.3.

then defined in Definition 1.1 is univalent in .

Proof. .

Since , , by (1.5), we have

for all .

On the other hand, we obtain
for . This equality implies that
or equivalently
By differentiating the above equality, we get
After some calculus, we obtain
By hypothesis, since , and since we have
(2.10)
So, we obtain
(2.11)

Remark 2.4.

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

Corollary 2.5.

Theorem 2.6.

(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.

(ii), and

(iii),

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

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