Keywords:Analytic Functions, Integral Operators, General Schwarz Lemma
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
the equalities hold in case 
where 
and
.
Remark 1.3. [2] For
, from inequality (3) we obtain for every 
(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
(9)
(10)
where
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:
Using relations before the function h has the form:
(15)
We have:
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: 
Since
, it results:
Using this result and the form (19), we have:
(20)
for all
.
Applying the condition (10) in relation (20), we obtain:
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 
(25)
(26)
where
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:
We have:
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 
Since
, it results:
Using this result and the form (35), we have:
(36)
for all
.
Applying the condition (26) in relation (36), we obtain:
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.