Keywords:Analytic Functions, Integral Operators, General Schwarz Lemma
![](https://www.scirp.org/html/htmlimages\9-5300736x\4f871df7-7ac9-44f9-9e33-26c880ba693a.png)
1. Introduction
Let
be the unit disk and A be the class of all functions of the form
(1)
which are analytic in U and satisfy the conditions
.
We denote by S the class of univalent and regular functions.
In order to derive our main results, we have to recall here the following univalence conditions.
Theorem 1.1. [1] (Becker’s univalence criterion).
If the function f is regular in unit disk U,
and
, (2)
then the function f is univalent in U.
Theorem 1.2. [2] If the function g is regular in U and
in U, then for all
the following inequalities hold
(3)
and
![](https://www.scirp.org/html/htmlimages\9-5300736x\dcac621c-b499-4678-89f6-50dc60592094.png)
the equalities hold in case
where
and
.
Remark 1.3. [2] For
, from inequality (3) we obtain for every ![](https://www.scirp.org/html/htmlimages\9-5300736x\7a6b6bd0-fe8c-4c7a-95d2-426b0511b905.png)
(4)
and, hence
(5)
Considering
and
, then
for all
.
2. Main Results
In this paper we study the univalence of the following general integral operators:
(6)
where
and
,
(7)
where
and
.
Theorem 2.1. Let
,
,
,
,
,
,
If
(8)
for all
, for all
and
![](https://www.scirp.org/html/htmlimages\9-5300736x\3014014d-f42c-4cfa-8713-401b4662aacb.png)
(9)
(10)
where
![](https://www.scirp.org/html/htmlimages\9-5300736x\767b192a-1d92-4615-9af8-10fad0b7617a.png)
then the function
(11)
is in the class S.
Proof. We have
,
, for all
and
, when
.
Let us consider the function:
(12)
From (6), we have:
(13)
and
(14)
From (13) and (14), we have:
![](https://www.scirp.org/html/htmlimages\9-5300736x\d3e045b0-52dc-4576-ae79-86924da3ae31.png)
Using relations before the function h has the form:
(15)
We have:
![](https://www.scirp.org/html/htmlimages\9-5300736x\43d1d7dd-d53d-47e3-9948-c06304ab69d1.png)
By using the relations (15), (8) and (9), we obtain:
(16)
(17)
Applying Remark 1.3 for the function h, we obtain:
(18)
From (18), we get:
(19)
for all
.
Let us consider the function: ![](https://www.scirp.org/html/htmlimages\9-5300736x\246a8e7a-0272-4de8-a840-4e70f253d01f.png)
![](https://www.scirp.org/html/htmlimages\9-5300736x\1e288198-39ef-498d-81d8-87646fb004f1.png)
Since
, it results:
![](https://www.scirp.org/html/htmlimages\9-5300736x\23fb7afe-c1c0-416b-83c7-bf57a7b852dc.png)
Using this result and the form (19), we have:
(20)
for all
.
Applying the condition (10) in relation (20), we obtain:
![](https://www.scirp.org/html/htmlimages\9-5300736x\576a5ba3-2a84-4a6a-9987-59c8fc1872db.png)
for all
and from Theorem 1.1, we have
.
Corollary 2.2. Let
be a complex number and the functions
,
,
,
.
If
(21)
for all
and the constant
satisfies the condition:
(22)
then the function
(23)
is in the class S.
Proof. We consider
in Theorem 2.1.
Remark 2.3. For
,
,
and
in relation (11), we obtain the integral operator
, introduced by J. W. Alexander in [3] .
Remark 2.4. For
,
,
,
in relation (6), we obtain the integral operator
, defined and studied by V. Pescar in [4] [5] .
Remark 2.5. For
, for all
, we get the integral operator
,
studied by D. Breaz, N. Breaz in [6] and D. Breaz in [7] .
Theorem 2.6.
Let
,
,
,
,
,
,
.
If
(24)
for all
, for all
and ![](https://www.scirp.org/html/htmlimages\9-5300736x\2ea03703-5be3-4152-aa4f-f12af08334ff.png)
(25)
(26)
where
![](https://www.scirp.org/html/htmlimages\9-5300736x\a8a6d74a-c95b-454f-93d0-a8255ac099ed.png)
then the function
(27)
is in the class S.
Proof. We have
, for all
and
, when
.
Let us consider the function:
(28)
From (27), we have:
(29)
and
(30)
From (29) and (30), we get:
(31)
Using relation (31) the function p has the form:
![](https://www.scirp.org/html/htmlimages\9-5300736x\f9602e43-3b9f-4ab1-94cd-e8e94dc7670b.png)
We have:
![](https://www.scirp.org/html/htmlimages\9-5300736x\d6057901-9772-4c55-873b-d28826e7a8f0.png)
By using the relations (24), (25) and (28), we obtain:
(32)
and
(33)
Applying Remark 1.3 for the function p, we obtain:
(34)
From (34), we get:
(35)
for all
.
Let us consider the function ![](https://www.scirp.org/html/htmlimages\9-5300736x\48ce2376-9b1e-4079-9859-78bada43b442.png)
![](https://www.scirp.org/html/htmlimages\9-5300736x\c384e5bd-fa26-4c7f-a881-af2bb0b0779c.png)
Since
, it results:
![](https://www.scirp.org/html/htmlimages\9-5300736x\57e5f063-c0c5-4788-96ec-e9971fe96d0c.png)
Using this result and the form (35), we have:
(36)
for all
.
Applying the condition (26) in relation (36), we obtain:
![](https://www.scirp.org/html/htmlimages\9-5300736x\32b0b0f1-32e0-4be2-a987-a6a8f9f0d1bc.png)
for all
and from Theorem 1.1, we have
.
Corollary 2.7. Let
be a complex number and the functions
,
,
,
.
If
(37)
for all
and the constant
satisfies the condition:
(38)
then the function
(39)
is in the class S.
Proof. We consider
in Theorem 2.6.
Remark 2.8. For
,
,
,
in relation (27), we obtain the integral operator
, defined and studied by V. Pescar in [8] [9] .
Remark 2.9. For
and
in relation (27), we obtain the integral operator
, introduced and studied by N. Ularu and D. Breaz in [10] and [11] .
Acknowledgements
This work was supported by the strategic project PERFORM, POSDRU 159/1.5/S/138963, inside POSDRU Romania 2014, co-financed by the European Social Fund-Investing in People.