Ann. Funct. Anal. 2 (2011), no. 2, 42–50
A nnals of F unctional A nalysis
ISSN: 2008-8752 (electronic)
URL: www.emis.de/journals/AFA/
THE UNIVALENCE CONDITIONS FOR A FAMILY OF
INTEGRAL OPERATORS
LAURA STANCIU1∗ AND DANIEL BREAZ2
Communicated by M. S. Moslehian
Abstract. The main object of the present paper is to discuss some univalence conditions for a family of integral operators. Several other closely-related
results are also considered.
1. Introduction and preliminaries
Let U={z ∈ C : |z| < 1} be the open unit disk and A denote the class of functions
of the form
∞
X
f (z) = z +
an z n ,
n=2
which are analytic in U and satisfy the following usual normalization condition
f (0) = f 0 (0) − 1 = 0.
Also, let S denote the subclass of A consisting of functions f (z) which are univalent in U.
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 f ∈ A.
Theorem 1.1. (Pascu [1]). Let f ∈ A and β ∈ C. If Re(β) > 0 and
1 − |z|2Re(β) zf 00 (z) ≤ 1 (z ∈ U)
Re(β) f 0 (z) Date: Received: 12 June 2011; Accepted: 25 July 2011.
∗
Corresponding author.
2010 Mathematics Subject Classification. 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 Fβ (z) given by
Z
Fβ (z) = β
β1
f (t)dt
z
β−1 0
t
0
is in the univalent function class S in U.
In this paper, we consider three general families of integral operators.
The first family of integral operators, studied by Breaz and Breaz [2], is defined
as follows:
α1 ! β1
Z z
n Y
fi (t) i
(1.1)
dt
Fα1 ,α2 , · · · ,αn ,β (z) = β
tβ−1
t
0
i=1
The second family of integral operators was introduced by Breaz and Breaz [3]
and it has the following form:
!Z n
! P 1
n
zY
X
( ni=1 (αi −1)+1)
αi −1
Gα1 ,α2 , · · · ,αn (z) =
(αi − 1) + 1
(fi (t))
dt
0
i=1
i=1
(1.2)
Finally, Breaz and Breaz [4] considered the following family of integral operators
Z
Hα1 ,α2 , · · · ,αn ,β (z) =
z
β
tβ−1
0
α
n Y
fi (t) i
t
i=1
! β1
dt
(1.3)
In the present paper, we propose to investigate further univalence conditions
involving the general families of integral operators defined by (1.1), (1.2) and
(1.3).
2. Main results
Theorem 2.1. Let the functions fi ∈ A, (i ∈ {1, · · · , n}) and αi , β be complex
numbers with Re(β) ≥ 0 for all i ∈ {1, 2, · · · , n}. If
(a).
n
X
1
4
≤ Re(β),
|αi |
i=1
Re(β) ∈ (0, 1)
or
(b).
n
X
1
1
≤ ,
|αi |
4
i=1
Re(β) ∈ [1, ∞)
then the function Fα1 ,α2 , · · · ,αn ,β (z) defined by (1.1) is in the class S.
Proof. We define the function
Z
h(z) =
0
n zY
i=1
fi (t)
t
α1
i
dt.
(2.1)
44
L. STANCIU, D. BREAZ
Because fi ∈ S, for i ∈ {1, 2, · · · , n}, we have
0 zfi (z) 1 + |z|
fi (z) ≤ 1 − |z|
(2.2)
for all z ∈ U.
Now, we calculate for h(z) the derivates of the first and second order.
From (2.1) we obtain
α1
n Y
i
f
(z)
i
h0 (z) =
z
i=1
and
n 1 X fi (z)
h (z) =
αi i=1
z
00
1−αi
αi
zfi0 (z) − fi (z)
z2
Y
n k=1
k6=i
fk (z)
z
α1
i
.
After the calculus we obtain that
n
zh00 (z) X 1
=
h0 (z)
αi
i=1
zfi0 (z)
−1
fi (z)
which readily shows that
1 − |z|2Re(β) zh00 (z) 1 − |z|2Re(β)
=
Re(β) h0 (z) Re(β)
n
X
1 zfi0 (z)
−1 αi fi (z)
i=1
n
1 − |z|2Re(β) X 1 zfi0 (z)
≤
− 1
Re(β)
|αi | fi (z)
i=1
n
1 − |z|2Re(β) X 1 zfi0 (z) ≤
fi (z) + 1 .
Re(β)
|α
|
i
i=1
(2.3)
From (2.2) and (2.3) we obtain
n
1 − |z|2Re(β) zh00 (z) 1 − |z|2Re(β) X 1
1 + |z|
≤
+1
Re(β) h0 (z) Re(β)
|αi | 1 − |z|
i=1
n
1 − |z|2Re(β) X 1
2
≤
Re(β)
|αi | 1 − |z|
i=1
1 − |z|2Re(β)
≤
Re(β)
2
1 − |z|
X
n
i=1
1
.
|αi |
(2.4)
Now, we consider the cases
i1).
0 < Re(β) < 1.
We have
1 − |z|2Re(β) ≤ 1 − |z|2
(2.5)
UNIVALENCE CONDITIONS FOR A FAMILY OF INTEGRAL OPERATORS
45
for all z ∈ U.
From (2.4) and (2.5), we have
X
n
2
1
1 − |z|2Re(β) zh00 (z) 1 − |z|2
≤
0
Re(β)
h (z)
Re(β) 1 − |z| i=1 |αi |
n
4 X 1
≤
.
Re(β) i=1 |αi |
(2.6)
Using the condition (a). and (2.6) we have
1 − |z|2Re(β) zh00 (z) ≤1
Re(β) h0 (z) for all z ∈ U.
i2). Re(β) ≥ 1.
We have
1 − |z|2Re(β)
≤ 1 − |z|2
Re(β)
for all z ∈ U.
From (2.8) and (2.4) we have
X
n
1 − |z|2Re(β) zh00 (z) 2
1
2
≤ 1 − |z|
0
Re(β)
h (z)
(1 − |z|) i=1 |αi |
≤4
(2.7)
(2.8)
n
X
1
.
|α
|
i
i=1
Using the condition (b). and (2.9) we obtain
1 − |z|2Re(β) zh00 (z) ≤1
Re(β) h0 (z) (2.9)
(2.10)
for all z ∈ U.
From (2.7) and (2.10) we obtain that the function Fα1 ,α2 , · · · ,αn ,β (z) defined by
(1.1) is in the class S.
Setting n = 1 in Theorem 2.1 we have
Corollary 2.2. Let f ∈ A and α, β be complex numbers with Re(β) ≥ 0. If
4
(a).
≤ Re(β), Re(β) ∈ (0, 1)
|α|
or
1
1
(b).
≤ , Re(β) ∈ [1, 2]
|α|
4
then the function
α1 ! β1
Z z
f
(t)
F (z) = β
tβ−1
dt
t
0
46
L. STANCIU, D. BREAZ
is in the class S.
Theorem 2.3. Let the functions fi ∈PA (i ∈ {1, 2, · · · , n}) , β, αi be complex
numbers for all i ∈ {1, 2, · · · , n}, β = ( ni=1 (αi − 1) + 1) and Re(β) ≥ 0. If
n
X
(a). 4
|αi − 1| ≤ Re(β), Re(β) ∈ (0, 1)
i=1
or
(b).
n
X
i=1
1
|αi − 1| ≤ ,
4
Re(β) ∈ [1, ∞)
then the function Gα1 ,α2 , · · · ,αn (z) defined by 1.2 is in the class S.
Proof. Defining the function h(z) by
α −1
Z zY
n fi (t) i
h(z) =
dt
t
0 i=1
we take the same steps as in the proof. of Theorem 2.1.
Then, we obtain that
0
n
1 − |z|2Re(β) zh00 (z) 1 − |z|2Re(β) X
zfi (z)
=
(αi − 1)
−1 Re(β) h0 (z) Re(β) i=1
fi (z)
0
n
zfi (z)
1 − |z|2Re(β) X
|αi − 1| − 1
≤
Re(β)
fi (z)
i=1
n
zfi0 (z) 1 − |z|2Re(β) X
+1 .
≤
|αi − 1| (2.11)
Re(β)
f
(z)
i
i=1
From (2.2) and (2.11) we obtain
n
1 + |z|
1 − |z|2Re(β) zh00 (z) 1 − |z|2Re(β) X
≤
|αi − 1|
+1
Re(β) h0 (z) Re(β)
1 − |z|
i=1
n
1 − |z|2Re(β) X
2
≤
|αi − 1|
Re(β)
1 − |z|
i=1
X
n
1 − |z|2Re(β)
2
≤
|αi − 1| .
Re(β)
1 − |z| i=1
(2.12)
Now, we consider the cases
i1).
0 < Re(β) < 1.
We have
1 − |z|2Re(β) ≤ 1 − |z|2
for all z ∈ U.
From (2.12) and (2.13), we have
X
n
1 − |z|2Re(β) zh00 (z) 1 − |z|2
2
≤
|αi − 1|
Re(β) h0 (z) Re(β) 1 − |z| i=1
(2.13)
UNIVALENCE CONDITIONS FOR A FAMILY OF INTEGRAL OPERATORS
47
n
4 X
≤
|αi − 1| .
Re(β) i=1
Using the condition (a). and (2.14) we have
1 − |z|2Re(β) zh00 (z) ≤1
Re(β) h0 (z) (2.14)
(2.15)
for all z ∈ U.
i2).
Re(β) ≥ 1.
We have
1 − |z|2Re(β)
≤ 1 − |z|2
Re(β)
(2.16)
for all z ∈ U.
From (2.12) and (2.16) we have
X
n
2
1 − |z|2Re(β) zh00 (z) 2
≤ 1 − |z|
|αi − 1|
Re(β) h0 (z) (1 − |z|) i=1
≤4
n
X
|αi − 1| .
(2.17)
i=1
Using the condition (b). and (2.17) we obtain
1 − |z|2Re(β) zh00 (z) ≤1
Re(β) h0 (z) (2.18)
for all z ∈ U.
From (2.15) and (2.18) we obtain that the function Gα1 ,α2 , · · · ,αn (z) defined by
(1.2) is in the class S.
Setting n = 1 in Theorem 2.3 we have
Corollary 2.4. Let f ∈ A and α be complex number with Re(α) ≥ 0. If
(a).
4 |α − 1| ≤ Re(α),
Re(α) ∈ (0, 1)
or
(b).
1
|α − 1| ≤ ,
4
Re(α) ∈ [1, 2]
then the function
Z
G(z) = α
0
is in the class S.
z
α−1
(f (t))
α1
dt
48
L. STANCIU, D. BREAZ
Theorem 2.5. Let the functions fi ∈ A, (i ∈ {1, · · · , n}) and αi , β be complex
numbers with Re(β) ≥ 0 for all i ∈ {1, 2, · · · , n}. If
n
X
(a). 4
|αi | ≤ Re(β),
Re(β) ∈ (0, 1)
i=1
or
(b).
n
X
i=1
1
|αi | ≤ ,
4
Re(β) ∈ [1, ∞)
then the function Hα1 ,α2 , · · · ,αn ,β (z) defined by (1.3) is in the class S.
Proof. Defining the function h(z) by
α
Z zY
n fi (t) i
h(z) =
dt.
t
0 i=1
we take the same steps as in the proof. of Theorem 2.1.
Then we obtain that
0
n
2Re(β) 2Re(β) X
00
1 − |z|
zfi (z)
zh (z) = 1 − |z|
αi
−1 0
Re(β)
h (z)
Re(β) fi (z)
i=1
≤
1 − |z|2Re(β)
Re(β)
1 − |z|2Re(β)
≤
Re(β)
0
n
X
zfi (z)
|αi | − 1
fi (z)
i=1
0 n
X
zfi (z) +1 .
|αi | f
(z)
i
i=1
From (2.2) and (2.19) we obtain
n
1 + |z|
1 − |z|2Re(β) zh00 (z) 1 − |z|2Re(β) X
≤
|αi |
+1
Re(β) h0 (z) Re(β)
1
−
|z|
i=1
n
1 − |z|2Re(β) X
2
≤
|αi |
Re(β)
1 − |z|
i=1
X
n
1 − |z|2Re(β)
2
≤
|αi |.
Re(β)
1 − |z| i=1
(2.19)
(2.20)
Now, we consider the cases
i1).
0 < Re(β) < 1.
We have
1 − |z|2Re(β) ≤ 1 − |z|2
for all z ∈ U.
From (2.20) and (2.21), we have
X
n
1 − |z|2Re(β) zh00 (z) 1 − |z|2
2
≤
|αi |
Re(β) h0 (z) Re(β) 1 − |z| i=1
(2.21)
UNIVALENCE CONDITIONS FOR A FAMILY OF INTEGRAL OPERATORS
49
n
≤
4 X
|αi |.
Re(β) i=1
(2.22)
Using the condition (a). and (2.22) we have
1 − |z|2Re(β) zh00 (z) ≤1
Re(β) h0 (z) (2.23)
for all z ∈ U.
Re(β) ≥ 1.
i2).
We have
1 − |z|2Re(β)
≤ 1 − |z|2
Re(β)
(2.24)
for all z ∈ U.
From (2.24) and (2.20) we have
X
n
2
1 − |z|2Re(β) zh00 (z) 2
≤ 1 − |z|
|αi |
Re(β) h0 (z) (1 − |z|) i=1
≤4
n
X
|αi |.
(2.25)
i=1
Using the condition (b). and (2.25) we obtain
1 − |z|2Re(β) zh00 (z) ≤1
Re(β) h0 (z) (2.26)
for all z ∈ U.
From (2.23) and (2.26) we obtain that the function Hα1 ,α2 , · · · ,αn ,β (z) defined
by (1.3) is in the class S.
Setting n = 1 in Theorem 2.5 we have
Corollary 2.6. Let f ∈ A and α, β be complex numbers with Re(β) ≥ 0. If
(a).
or
(b).
|α| ≤
Re(β)
,
4
1
|α| ≤ ,
4
Re(β) ∈ (0, 1)
Re(β) ∈ [1, 2]
then the function
Z
H(z) =
β
0
z
tβ−1
f (t)
t
α1
! β1
dt
is in the class S.
Acknowledgement. This work was partially supported by the strategic
project POSDRU 107/1.5/S/77265, inside POSDRU Romania 2007-2013 co-financ
ed by the European Social Fund-Investing in People.
50
L. STANCIU, D. BREAZ
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. BabesBolyai, 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.
4. D. Yang and J. Liu, On a class of univalent functions, Int. J. Math. Math. Sci. 22 (1999),
no. 3, 605–610.
1
Department of Mathematics,University of Piteşti, Târgul din Vale Str.,
No.1, 110040, Piteşti, Argeş, România.
E-mail address: laura stanciu 30@yahoo.com
2
Department of Mathematics, ”1 Decembrie 1918”’ University of Alba Iulia,Alba Iulia, Str. N. Iorga, 510000, No. 11-13, România.
E-mail address: dbreaz@uab.ro