Some sufficient conditions for
univalence of two integral operators
Laura Stanciu and Daniel Breaz
Abstract. New conditions for the univalence of two complex integral operators in the open unit disk are obtained. These original conditions derived
by means of specializing certain parameters involved in the definitions of
the integral operators involved.
M.S.C. 2010: 30C45, 30C75.
Key words: Analytic functions; open unit disk; integral operators; univalence conditions; general Schwarz Lemma.
1
Two families of integral operators
Let C be the set of complex numbers. Let A be the class of functions
f (z) = z +
∞
X
ak z k ,
k=2
which are analytic in the open unit disk
U = {z : z ∈ C, |z| < 1}
and satisfy the following usual normalization condition
f (0) = f 0 (0) − 1 = 0.
We denote by P the class of the functions p(z) which are analytic in U and satisfy
the conditions
p(0) = 1, Re{p(z)} > 0, z ∈ U.
Let S denote the subclass of A consisting of functions f (z) which are univalent in U.
In this paper, we define two families of integral operators. The first family of integral
operators is defined by
(1.1)
Kn (z) =
à n
X αi Z
i=1
γi
Applied Sciences, Vol.15, 2013, pp.
0
z
t
−1
n
Y
(fi (t))
αi
γi
! Pn 1
gi (t)dt
i=1
104-111.
c Balkan Society of Geometers, Geometry Balkan Press 2013.
°
αi
i=1 γi
Some sufficient conditions for univalence of two integral operators
105
αi , γi ∈ C, γi 6= 0, fi ∈ A and gi ∈ P for all i ∈ {1, 2, ..., n}.
Remark 1.1. For α1 = α2 = ... = αn = 1 and g1 = g2 = ... = gn = 1, we obtain the
integral operator
Ã
Jγ1 ,γ2 , ...,γn (z) =
! Pn 1 1
Z
n
n
X
i=1 γi
1
1 z −1 Y
t
(fi (t)) γi dt
γ
i=1 i 0
i=1
studied in [6].
Remark 1.2. For n = 1, α1 = 1 and g1 = 1, from (1.1) we obtain the integral
operator
µ Z z
¶γ
1
1
Jγ (z) =
t−1 (f (t)) γ dt
γ 0
studied in [6].
The second family of integral operators is defined by
Z
(1.2)
Ln (z) =
n µ
zY
0 i=1
fi (t)
t
¶ αγ i
i
gi (t)dt
αi , γi ∈ C, γi 6= 0, fi ∈ A and gi ∈ P for all i ∈ {1, 2, ..., n}.
Remark 1.3. From (1.2), for n = 1, g1 = 1, α1 = 1 and
integral operator Kim-Merkes
¶α
Z zµ
f (t)
Hα (z) =
dt
t
0
1
γ1
= α, we obtain the
studied in [3].
In the present paper, we obtain some sufficient conditions for the integral operators
Kn (z) and Ln (z) to be in the class S.
2
Preliminary results
In the proof of our main results (Theorem 3.1 and Theorem 3.3 below), we need the
Definition 2.1. (see [2]) A function f (z) ∈ A is said to be a member of the class
B(µ) if and only if
¯
¯ 2 0
¯
¯ z f (z)
¯ ≤ 1 − µ, z ∈ U; 0 ≤ µ < 1.
¯
−
1
(2.1)
¯
¯ [f (z)]2
Theorem 2.1. (see [1]) If the function f (z) = z + a2 z 2 + ... is regular in unit disk U
and
³
´ ¯ 00 ¯
2 ¯ zf (z) ¯¯
1 − |z| ¯¯ 0
≤ 1,
z∈U
f (z) ¯
then the function f is univalent in U.
106
Laura Stanciu and Daniel Breaz
Theorem 2.2. (see [5]) Let the parameters β, c ∈ C be constrained by
Reβ > 0
and
|c| ≤ 1,
c 6= −1.
If f ∈ A satisfies the inequality
¯
¯
³
´ 00
¯
¯
¯c |z|2β + 1 − |z|2β zf (z) ¯ ≤ 1,
¯
0
βf (z) ¯
z∈U
then the function Fβ (z) given by
Fβ (z) =
µ Z
β
z
¶ β1
tβ−1 f 0 (t)dt
= z + ...
0
is analytic and univalent in U.
Finally, in our present investigation, we shall also need the familiar Schwarz
Lemma (see, for details, [4]).
Lemma 2.3. (General Schwarz Lemma) [4] Let the function f (z) be regular in
the disk
UR = {z : z ∈ C and
|z| < R
(R > 0)},
with |f (z)| < M , for a fixed number M > 0. If the function f (z) has one zero with
multiplicity order bigger than a positive integer m for z = 0, then
(2.2)
|f (z)| ≤
M
m
|z| ,
Rm
z ∈ UR .
The equality (2.2) can hold true only if
f (z) = eiθ
M m
z ,
Rm
where θ is a real constant.
3
Univalence conditions
Our main univalence conditions for the integral operators Kn (z), Ln (z) defined by
(1.1), (1.2) are asserted by Theorem 3.1 and Theorem 3.3 below.
Theorem 3.1. Let the functions fi (z) ∈ A be in the class B(µi ), 0 ≤P
µi < 1 satisfy
n
the hypothesis (2.1) of Definition 2.1 and αi , γi ∈ C, γi 6= 0 with β = i=1 Re αγii for
all i ∈ {1, 2, ..., n}. Suppose that for all i ∈ {1, 2, ..., n}
Mi ≥ 1,
and
Reβ ≥
Ni ≥ 1,
n µ
X
|αi |
i=1
|γi |
gi (z) ∈ P,
z∈U
¶
((2 − µi ) Mi + 1) + Ni .
Some sufficient conditions for univalence of two integral operators
107
If
(3.1)
|fi (z)| ≤ Mi
¯ 0 ¯
¯ zgi (z) ¯
¯
¯
¯ gi (z) ¯ ≤ Ni
z ∈ U,
z∈U
and
(3.2)
¶
n µ
1 X |αi |
|c| ≤ 1 −
((2 − µi ) Mi + 1) + Ni
Reβ i=1 |γi |
(c ∈ C; c 6= −1) ,
for all i ∈ {1, 2, ..., n}, then the integral operator Kn (z) defined by (1.1) is in the class
S.
Proof. We begin by observing that the integral operator Kn (z) in (1.1) can be rewritten in the form
Kn (z) =
à n
X αi Z
i=1
γi
αi
i=1 γi
t
¶ i
n µ
Y
fi (t) γi
α
z Pn
−1
0
t
i=1
! Pn 1
gi (t)dt
αi
i=1 γi
.
Let us define the function h(z) by
Z
h(z) =
n µ
zY
0 i=1
fi (t)
t
¶ αγ i
i
g(t)dt
(fi ∈ A; gi ∈ P) .
The functions fi (z) are indeed regular in U and satisfy the following usual normalization condition
f (0) = f 0 (0) − 1 = 0.
Now, calculating the derivatives of h(z) of the first and second orders, we readily
obtain
¶ i
n µ
Y
fi (z) γi
α
0
(3.3)
h (z) =
z
i=1
gi (z)
and
h00 (z) =
n
X
i=1
(3.4)
Ã
αi
γi
µ
fi (z)
z
!
¶
µ
¶ αi
zfi0 (z) − fi (z)
fi (z) γi 0
gi (z) ×
gi (z) +
z2
z
¶ αk
n µ
Y
fk (z) γk
×
gk (z).
z
k=1
¶ αγ i −1 µ
i
k6=i
We easily find, from (3.3) and (3.4), that
n
(3.5)
zh00 (z) X
=
h0 (z)
i=1
µ
αi
γi
µ
¶
¶
zfi0 (z)
zgi0 (z)
−1 +
,
fi (z)
gi (z)
108
Laura Stanciu and Daniel Breaz
which readily shows that
¯
³
´ 00 ¯¯
¯
¯c |z|2β + 1 − |z|2β zh (z) ¯
¯
βh0 (z) ¯
¯
µ 0
¶
¶¯¯
n µ
¯
³
´1X
0
α
zf
(z)
zg
(z)
¯
¯
i
2β
2β
i
= ¯c |z| + 1 − |z|
−1 + i
¯
¯
β i=1 γi
fi (z)
gi (z) ¯
¯
¯¯
¶ ¯ 0 ¯¶
µ¯
n µ
¯ zg (z) ¯
1 X |αi | ¯¯ z 2 fi0 (z) ¯¯ ¯¯ fi (z) ¯¯
(3.6)
+
1
+ ¯¯ i ¯¯ .
≤ |c| +
¯
¯
¯
¯
2
|β| i=1 |γi |
[fi (z)]
z
gi (z)
Furthermore, from the hypothesis (3.1) of Theorem 3.1, we have
¯ 0 ¯
¯ zgi (z) ¯
¯
¯
|fi (z)| ≤ Mi ,
z ∈ U and
z ∈ U.
¯ gi (z) ¯ ≤ Ni ,
By applying the General Schwarz Lemma, we thus obtain
|fi (z)| ≤ Mi |z| ,
z ∈ U.
Next, by making use of (3.6), we find
¯
³
´ 00 ¯¯
¯
¯c |z|2β + 1 − |z|2β zh (z) ¯
¯
βh0 (z) ¯
¯
µ
µµ¯
¶
¶
¶
n
¯ z 2 fi0 (z)
¯
1 X |αi |
¯
¯ + 1 Mi + 1 + Ni
≤ |c| +
−
1
¯ [fi (z)]2
¯
|β|
|γi |
i=1
¶
n µ
1 X |αi |
((2 − µi ) Mi + 1) + Ni
≤ |c| +
|β| i=1 |γi |
¶
n µ
1 X |αi |
((2 − µi ) Mi + 1) + Ni
≤ |c| +
Reβ i=1 |γi |
≤1
where we have also used the hypothesis (3.2) of Theorem 3.1.
Finally, by applying Theorem 2.2, we conclude that the integral operator Kn (z) defined by (1.1) is in the class S. This evidently completes the proof of Theorem 3.1.
¤
Setting n = 1 in Theorem 3.1, we immediately arrive at the following application of
Theorem 3.1.
Corollary 3.2. Let the function f (z) ∈ A be in the class B(µ), 0 ≤ µ < 1 and satisfy
the hypothesis (2.1) of Definition 2.1 and α, γ ∈ C, γ 6= 0 with β = Re αγ . Suppose
that
M ≥ 1,
N ≥ 1,
g(z) ∈ P, z ∈ U
and
Reβ ≥
|α|
[(2 − µ) M + 1] + N.
|γ|
Some sufficient conditions for univalence of two integral operators
If
|f (z)| ≤ M
and
|c| ≤ 1 −
¯ 0 ¯
¯ zg (z) ¯
¯
¯
¯ g(z) ¯ ≤ N
z ∈ U,
1 |α|
N
[(2 − µ) M + 1] −
Reβ |γ|
Reβ
109
z∈U
(c ∈ C; c 6= −1) ,
then the integral operator
µ
K(z) =
α
γ
Z
z
t
−1
¶ αγ
(f (t)) g(t)dt
α
γ
0
is in the class S.
Theorem 3.3. Let αi , γi be complex numbers, γi 6= 0 and fi (z) ∈ A for all i ∈
{1, 2, ..., n}. Suppose that for all i ∈ {1, 2, ..., n}
Mi ≥ 1,
Ni ≥ 1,
If for all i ∈ {1, 2, ..., n}
¯ 0
¯
¯ zfi (z)
¯
¯
¯ ≤ Mi
(3.7)
−
1
¯ fi (z)
¯
gi (z) ∈ P,
z ∈ U.
¯ 0 ¯
¯ gi (z) ¯
¯
¯
¯ gi (z) ¯ ≤ Ni
z ∈ U,
z∈U
and
(3.8)
n
X
Ã
|αi | ≤
i=1
n
X
9
√ −
Ni
2 3 i=1
!
n
X
|γi |
i=1
Mi
then the integral operator Ln (z) defined by (1.2) is in the class S.
Proof. Defining the function
Z
h(z) =
n µ
zY
0 i=1
fi (t)
t
¶ αγ i
i
gi (t)dt
we take the same steps as in the proof of Theorem 3.1. Then, from (3.5), we obtain
¯ 00 ¯ X
¯
¯ ¯ 0 ¯¶
n µ
¯ zh (z) ¯
¯ ¯ zg (z) ¯
|αi | ¯¯ zfi0 (z)
¯
¯
(3.9)
− 1¯¯ + ¯¯ i ¯¯ .
¯ h0 (z) ¯ ≤
¯
|γi | fi (z)
gi (z)
i=1
From (3.7) and (3.9), we have
¯ 00 ¯ X
¶
n µ
¯ zh (z) ¯
|αi |
¯
¯≤
Mi + Ni |z| .
¯ h0 (z) ¯
|γi |
i=1
Applying General Schwarz Lemma, we find
¯ 00 ¯ X
¶
n µ
¯ zh (z) ¯
|αi |
¯
¯≤
Mi |z| + Ni |z|
¯ h0 (z) ¯
|γi |
i=1
110
Laura Stanciu and Daniel Breaz
and hence, we get
(3.10)
¶
n µ
³
´ ¯ 00 ¯ ³
´ X
|αi |
2 ¯¯ zh (z) ¯¯
2
1 − |z| ¯ 0
≤ 1 − |z| |z|
Mi + Ni .
h (z) ¯
|γi |
i=1
Since
2
max{(1 − |z| ) |z|} =
|z|≤1
√
2 3
,
9
from (3.8) and (3.10), we deduce
³
(3.11)
1 − |z|
2
´ ¯¯ zh00 (z) ¯¯
¯
¯
¯ h0 (z) ¯ ≤ 1.
From (3.11) and by Theorem 2.1, we obtain that the integral operator Ln (z) defined
by (1.2) is in the class S.
¤
Setting n = 1 in Theorem 3.3, we obtain the following consequence of Theorem 3.3.
Corollary 3.4. Let α, γ be complex numbers, γ 6= 0 and f (z) ∈ A. Suppose that
M ≥ 1,
If
N ≥ 1,
¯ 0
¯
¯ zf (z)
¯
¯
¯≤M
−
1
¯ f (z)
¯
and
g(z) ∈ P,
z ∈ U,
|γ|
|α| ≤
M
µ
z ∈ U.
¯ 0 ¯
¯ g (z) ¯
¯
¯
¯ g(z) ¯ ≤ N
9
√ −N
2 3
z∈U
¶
then the integral operator
Z
Ln (z) =
0
z
µ
f (t)
t
¶ αγ
g(t)dt
is in the class S.
Acknowledgements. This work was partially supported by the strategic project
POSDRU 107/1.5/S/77265, inside POSDRU Romania 2007-2013 co-financed by the
European Social Fund-Investing in People.
References
[1] J. Becker, Lëownersche Differentialgleichung und quasikonform fortsetzbare
schlichte Funktionen, J. Reine Angew. Math. 255 (1972), 23-43.
[2] B. A. Frasin, M. Darus, On certain analytic univalent functions, IJMMS 25:5
(2001), 305-310.
[3] Y. J. Kim, E. P. Merkes, On an Integral of Powers of a Spirallike Function,
Kyungpook Math. J., 12 (1972), 249-253.
[4] Z. Nehari, Conformal Mapping, McGraw-Hill, 1952.
Some sufficient conditions for univalence of two integral operators
111
[5] V. Pescar, A new generalization of Ahlfors’s and Becker’s criterion of univalence,
Bull. Malaysian Math. Soc. (Second Ser.) 19 (1996), 53-54.
[6] V. Pescar, D. Breaz, N. Breaz, Certain sufficient conditions for univalence, General Mathematics vol. 17, 4 (2009), 97-109.
Author’s address:
Laura Stanciu
University of Piteşti, Department of Mathematics
Târgul din Vale Str., No.1, 110040, Piteşti, Argeş, România.
E-mail: laura stanciu 30@yahoo.com
Daniel Breaz
”1 Decembrie 1918” University of Alba Iulia, Department of Mathematics,
Alba Iulia, 11-13 N. Iorga Str., RO-510000, Romania.
E-mail: dbreaz@uab.ro