#### Abstract

The main object of the present paper is to discuss some extensions of certain integral operators and to obtain their order of convexity. Several other closely related results are also considered.

#### 1. Introduction

Let be the class of analytic functions defined in the open unit disk of the complex plane .

We denote by the subclass of consisting of all univalent functions in . A function is *starlike function of order * if it satisfies
for some . We denote by the subclass of consisting of the functions which are starlike of order in . For , we obtain the class of starlike functions, denoted by .

A function is *convex of order * if it satisfies
for some (). We denote by the subclass of consisting of the functions which are convex of order in . For , we obtain the class of convex functions, denoted by .

A function is in the class if

Frasin and Jahangiri introduced in [1] the family consisting of functions satisfying the condition For we have , and for we have .

In this paper, we will obtain the order of convexity of the following general integral operators: where the functions are in for all .

In order to prove our main results, we recall the following lemma.

Lemma 1.1 (see [2, General Schwarz Lemma]). * Let the function be regular in the disk , with for fixed . If has one zero with multiplicity order bigger than for , then
**
The equality can hold only if
**
where is constant.*

#### 2. Main Results

Theorem 2.1. *Let be in the class for all . If () for all , then the integral operator
**
is in , where
**
and .*

*Proof. *Let be in the class . We have from (1.5) that
Also
Then
and, hence,
Applying the General Schwarz lemma, we have , for all . Therefore, from (2.6), we obtain
From (1.4) and (2.7), we see that

Letting and for all in Theorem 2.1, we have the following corollary.

Corollary 2.2. *Let be in the class for all . Then the integral operator defined in (1.5) is in , where
**
and .*

Letting and for all in Theorem 2.1, we have the following corollary.

Corollary 2.3. *Let be in the class for all . If () for all , then the integral operator defined in (1.5) is in , where
**
and .*

Letting , , and for all in Theorem 2.1, we have the following corollary.

Corollary 2.4. *Let be starlike functions in for all . If () for all , then the integral operator defined in (1.5) is convex in , where .*

Theorem 2.5. *Let be in the class for all . If () for all , then the integral operator
**
is in , where
**
and for all .*

*Proof. *Let be in the class . It follows from (1.6) that
and, hence,
Applying the General Schwarz lemma, we have for all . Therefore, from (2.14), we obtain
From (1.4) and (2.15), we see that
This completes the proof.

Letting and for all in Theorem 2.5, we have the following corollary.

Corollary 2.6. *Let be in the class for all . If () for all , then the integral operator defined in (1.6) is convex function in , where
*

Letting and for all in Theorem 2.5, we have the following corollary.

Corollary 2.7. *Let be in the class for all . If () for all , then the integral operator defined in (1.6) is in , where
**
and for all .*

Letting , and for all in Theorem 2.5, we have the following corollary.

Corollary 2.8. *Let be a starlike function in . If (), then the integral operator is convex in , where .*

Theorem 2.9. *Let be in the class for all . If () for all , then the integral operator
**
is in , where
**
and .*

*Proof. *Let be in the class . It follows from (1.7) that
So, from (2.21), we have
Applying the General Schwarz lemma, we have for all . Therefore, from (2.22), we obtain
From (1.4) and (2.23), we see that
This completes the proof.

Letting and for all in Theorem 2.9, we have the following corollary.

Corollary 2.10. *Let be in the class for all . If () for all , then the integral operator defined in (1.7) is convex function in , where
*

Letting and for all in Theorem 2.9, we have the following corollary.

Corollary 2.11. *Let be in the class for all . If () for all , then the integral operator defined in (1.7) is in , where
**
and .*

Letting , , and for all in Theorem 2.9, we have the following corollary.

Corollary 2.12. *Let be a starlike function in . If (), then the integral operator is convex in , where .*