Ann. Funct. Anal. 2 (2011), no. 2, 4250 [612108]

Ann. Funct. Anal. 2 (2011), no. 2, 42{50
Annals of Functional Analysis
ISSN: 2008-8752 (electronic)
URL: www.emis.de/journals/AFA/
THE UNIVALENCE CONDITIONS FOR A FAMILY OF
INTEGRAL OPERATORS
LAURA STANCIU1AND DANIEL BREAZ2
Communicated by M. S. Moslehian
Abstract. The main object of the present paper is to discuss some univa-
lence conditions for a family of integral operators. Several other closely-related
results are also considered.
1.Introduction and preliminaries
LetU=fz2C:jzj<1gbe the open unit disk and Adenote the class of functions
of the form
f(z) =z+1X
n=2anzn;
which are analytic in Uand satisfy the following usual normalization condition
f(0) =f0(0)1 = 0:
Also, letSdenote the subclass of Aconsisting of functions f(z) which are uni-
valent inU:
We begin by recalling a theorem dealing with a univalence criterion, which will
be required in our present work.
In [1], Pascu gave the following univalence criterion for the functions f2A:
Theorem 1.1. (Pascu [1]). Letf2A and2C. IfRe()>0and
1jzj2Re()
Re()zf00(z)
f0(z)1 (z2U)
Date : Received: 12 June 2011; Accepted: 25 July 2011.
Corresponding author.
2010 Mathematics Subject Classication. Primary 30C55; Secondary 30C45.
Key words and phrases. Analytic functions; Univalence conditions; Integral Operators.
42

UNIVALENCE CONDITIONS FOR A FAMILY OF INTEGRAL OPERATORS 43
then the function F(z)given by
F(z) =
Zz
0t1f0(t)dt1

is in the univalent function class SinU:
In this paper, we consider three general families of integral operators.
The rst family of integral operators, studied by Breaz and Breaz [2], is dened
as follows:
F1;2;


;n;(z) =
Zz
0t1nY
i=1fi(t)
t1
idt!1

(1.1)
The second family of integral operators was introduced by Breaz and Breaz [3]
and it has the following form:
G1;2;


;n(z) = nX
i=1(i1) + 1!Zz
0nY
i=1(fi(t))i1dt! 1
(Pn
i=1(i1)+1)
(1.2)
Finally, Breaz and Breaz [4] considered the following family of integral operators
H1;2;


;n;(z) =
Zz
0t1nY
i=1fi(t)
ti
dt!1

(1.3)
In the present paper, we propose to investigate further univalence conditions
involving the general families of integral operators dened by (1.1), (1.2) and
(1.3).
2.Main results
Theorem 2.1. Let the functions fi2A;(i2f1;


;ng)andi,be complex
numbers with Re()0for alli2f1;2;


;ng. If
(a):4nX
i=11
jijRe();Re()2(0;1)
or
(b):nX
i=11
jij1
4;Re()2[1;1)
then the function F1;2;


;n;(z)dened by (1.1) is in the class S:
Proof. We dene the function
h(z) =Zz
0nY
i=1fi(t)
t1
idt: (2.1)
Becausefi2S;fori2f1;2;


;ng;we havezf0
i(z)
fi(z)1 +jzj
1jzj(2.2)

44 L. STANCIU, D. BREAZ
for allz2U:
Now, we calculate for h(z) the derivates of the rst and second order.
From (2.1) we obtain
h0(z) =nY
i=1fi(z)
z1
i
and
h00(z) =1
inX
i=1fi(z)
z1i
izf0
i(z)fi(z)
z2nY
k=1
k6=ifk(z)
z1
i:
After the calculus we obtain that
zh00(z)
h0(z)=nX
i=11
izf0
i(z)
fi(z)1
which readily shows that
1jzj2Re()
Re()zh00(z)
h0(z)=1jzj2Re()
Re()nX
i=11
izf0
i(z)
fi(z)1
1jzj2Re()
Re()nX
i=11
jijzf0
i(z)
fi(z)1
1jzj2Re()
Re()nX
i=11
jijzf0
i(z)
fi(z)+ 1
: (2.3)
From (2.2) and (2.3) we obtain
1jzj2Re()
Re()zh00(z)
h0(z)1jzj2Re()
Re()nX
i=11
jij1 +jzj
1jzj+ 1
1jzj2Re()
Re()nX
i=11
jij2
1jzj
1jzj2Re()
Re()2
1jzjnX
i=11
jij: (2.4)
Now, we consider the cases
i1):0<Re()<1:
We have
1jzj2Re()1jzj2(2.5)
for allz2U:
From (2.4) and (2.5), we have
1jzj2Re()
Re()zh00(z)
h0(z)1jzj2
Re()2
1jzjnX
i=11
jij

UNIVALENCE CONDITIONS FOR A FAMILY OF INTEGRAL OPERATORS 45
4
Re()nX
i=11
jij: (2.6)
Using the condition (a). and (2.6) we have
1jzj2Re()
Re()zh00(z)
h0(z)1 (2.7)
for allz2U:
i2):Re()1:
We have
1jzj2Re()
Re()1jzj2(2.8)
for allz2U:
From (2.8) and (2.4) we have
1jzj2Re()
Re()zh00(z)
h0(z)
1jzj2
2
(1jzj)nX
i=11
jij
4nX
i=11
jij: (2.9)
Using the condition (b). and (2.9) we obtain
1jzj2Re()
Re()zh00(z)
h0(z)1 (2.10)
for allz2U:
From (2:7) and (2:10) we obtain that the function F1;2;


;n;(z) dened by
(1.1) is in the class S:

Settingn= 1 in Theorem 2.1 we have
Corollary 2.2. Letf2A and; be complex numbers with Re()0:If
(a):4
jjRe();Re()2(0;1)
or
(b):1
jj1
4;Re()2[1;2]
then the function
F(z) =
Zz
0t1f(t)
t1

dt!1

is in the classS:

46 L. STANCIU, D. BREAZ
Theorem 2.3. Let the functions fi2 A (i2f1;2;


;ng); ;ibe complex
numbers for all i2f1;2;


;ng; = (Pn
i=1(i1) + 1) andRe()0:If
(a):4nX
i=1ji1jRe();Re()2(0;1)
or
(b):nX
i=1ji1j1
4;Re()2[1;1)
then the function G1;2;


;n(z)dened by 1:2is in the classS:
Proof. Dening the function h(z) by
h(z) =Zz
0nY
i=1fi(t)
ti1
dt
we take the same steps as in the proof. of Theorem 2.1.
Then, we obtain that
1jzj2Re()
Re()zh00(z)
h0(z)=1jzj2Re()
Re()nX
i=1(i1)zf0
i(z)
fi(z)1
1jzj2Re()
Re()nX
i=1ji1jzf0
i(z)
fi(z)1
1jzj2Re()
Re()nX
i=1ji1jzf0
i(z)
fi(z)+ 1
: (2.11)
From (2.2) and (2.11) we obtain
1jzj2Re()
Re()zh00(z)
h0(z)1jzj2Re()
Re()nX
i=1ji1j1 +jzj
1jzj+ 1
1jzj2Re()
Re()nX
i=1ji1j2
1jzj
1jzj2Re()
Re()2
1jzjnX
i=1ji1j: (2.12)
Now, we consider the cases
i1):0<Re()<1:
We have
1jzj2Re()1jzj2(2.13)
for allz2U:
From (2.12) and (2.13), we have
1jzj2Re()
Re()zh00(z)
h0(z)1jzj2
Re()2
1jzjnX
i=1ji1j

UNIVALENCE CONDITIONS FOR A FAMILY OF INTEGRAL OPERATORS 47
4
Re()nX
i=1ji1j: (2.14)
Using the condition (a). and (2.14) we have
1jzj2Re()
Re()zh00(z)
h0(z)1 (2.15)
for allz2U:
i2):Re()1:
We have
1jzj2Re()
Re()1jzj2(2.16)
for allz2U:
From (2.12) and (2.16) we have
1jzj2Re()
Re()zh00(z)
h0(z)
1jzj2
2
(1jzj)nX
i=1ji1j
4nX
i=1ji1j: (2.17)
Using the condition (b). and (2.17) we obtain
1jzj2Re()
Re()zh00(z)
h0(z)1 (2.18)
for allz2U:
From (2:15) and (2:18) we obtain that the function G1;2;


;n(z) dened by
(1:2) is in the classS:

Settingn= 1 in Theorem 2.3 we have
Corollary 2.4. Letf2A andbe complex number with Re()0:If
(a):4j1jRe();Re()2(0;1)
or
(b):j1j1
4;Re()2[1;2]
then the function
G(z) =
Zz
0(f(t))1dt1

is in the classS:
Theorem 2.5. Let the functions fi2A;(i2f1;


;ng)andi,be complex
numbers with Re()0for alli2f1;2;


;ng. If
(a):4nX
i=1jijRe();Re()2(0;1)

48 L. STANCIU, D. BREAZ
or
(b):nX
i=1jij1
4;Re()2[1;1)
then the function H1;2;


;n;(z)dened by (1.3) is in the class S:
Proof. Dening the function h(z) by
h(z) =Zz
0nY
i=1fi(t)
ti
dt:
we take the same steps as in the proof. of Theorem 2.1.
Then we obtain that
1jzj2Re()
Re()zh00(z)
h0(z)=1jzj2Re()
Re()nX
i=1izf0
i(z)
fi(z)1
1jzj2Re()
Re()nX
i=1jijzf0
i(z)
fi(z)1
1jzj2Re()
Re()nX
i=1jijzf0
i(z)
fi(z)+ 1
: (2.19)
From (2.2) and (2.19) we obtain
1jzj2Re()
Re()zh00(z)
h0(z)1jzj2Re()
Re()nX
i=1jij1 +jzj
1jzj+ 1
1jzj2Re()
Re()nX
i=1jij2
1jzj
1jzj2Re()
Re()2
1jzjnX
i=1jij: (2.20)
Now, we consider the cases
i1):0<Re()<1:
We have
1jzj2Re()1jzj2(2.21)
for allz2U:
From (2.20) and (2.21), we have
1jzj2Re()
Re()zh00(z)
h0(z)1jzj2
Re()2
1jzjnX
i=1jij
4
Re()nX
i=1jij: (2.22)
Using the condition (a). and (2.22) we have
1jzj2Re()
Re()zh00(z)
h0(z)1 (2.23)

UNIVALENCE CONDITIONS FOR A FAMILY OF INTEGRAL OPERATORS 49
for allz2U:
i2):Re()1:
We have
1jzj2Re()
Re()1jzj2(2.24)
for allz2U:
From (2.24) and (2.20) we have
1jzj2Re()
Re()zh00(z)
h0(z)
1jzj2
2
(1jzj)nX
i=1jij
4nX
i=1jij: (2.25)
Using the condition (b). and (2.25) we obtain
1jzj2Re()
Re()zh00(z)
h0(z)1 (2.26)
for allz2U:
From (2.23) and (2.26) we obtain that the function H1;2;


;n;(z) dened
by (1.3) is in the class S: 
Settingn= 1 in Theorem 2.5 we have
Corollary 2.6. Letf2A and; be complex numbers with Re()0:If
(a):jjRe()
4;Re()2(0;1)
or
(b):jj1
4;Re()2[1;2]
then the function
H(z) =
Zz
0t1f(t)
t1

dt!1

is in the classS:
Acknowledgement. This work was partially supported by the strategic
project POSDRU 107/1.5/S/77265, inside POSDRU Romania 2007-2013 co-nanc
ed by the European Social Fund-Investing in People.
References
1. N.N. Pascu, On a univalence criterion II , in Itinerant Seminar on Functional Euations,
Approximation and Convexity Cluj-Napoca, 1985, 153{154, Preprint 86-6, Univ. Babes-
Bolyai, Cluj-Napoca, 1985.
2. D. Breaz and N. Breaz, The univalent conditions for an integral operator on the classes
S(p) and T2, J. Approx. Theory Appl. 1(2005), no. 2, 93{98.
3. D. Breaz and N. Breaz, Univalence of an integral operator , Mathematica 47(70) (2005), no.
1, 35{38.

50 L. STANCIU, D. BREAZ
4. D. Yang and J. Liu, On a class of univalent functions, Int. J. Math. Math. Sci. 22(1999),
no. 3, 605{610.
1Department of Mathematics,University of Pites ti, T ^argul din Vale Str.,
No.1, 110040, Pites ti, Arges , Rom ^ania.
E-mail address :laura stanciu 30@yahoo.com
2Department of Mathematics, "1 Decembrie 1918"' University of Alba Iu-
lia,Alba Iulia, Str. N. Iorga, 510000, No. 11-13, Rom ^ania.
E-mail address :dbreaz@uab.ro