Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 924645, 9 pages
doi:10.1155/2012/924645
Research Article
The Univalence Conditions of
Some Integral Operators
Laura Stanciu
Department of Mathematics, University of Piteşti, Argeş, Romania
Correspondence should be addressed to Laura Stanciu, laura stanciu 30@yahoo.com
Received 26 January 2012; Revised 4 March 2012; Accepted 8 March 2012
Academic Editor: Ondřej Došlý
Copyright q 2012 Laura Stanciu. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
We introduce new integral operators of analytic functions f and g defined in the open unit disk U.
For these operators, we discuss some univalence conditions.
1. Introduction and Preliminaries
Let A denote the class of all functions of the form
fz z ∞
an zn ,
1.1
U {z ∈ C : |z| < 1}
1.2
n2
which are analytic in the open unit disk
and satisfy the following usual normalization condition:
f0 f 0 − 1 0.
1.3
Also, let S denote the subclass of A consisting of functions f, which are univalent in U see,
for details 1; see also 2, 3.
In 4, 5, Pescar gave the following univalence conditions for the functions f ∈ A.
2
Abstract and Applied Analysis
Theorem 1.1 see 4. Let α be a complex number, Re α > 0, and c a complex number, |c| ≤
1, c /
− 1, and fz z · · · a regular function in U. If
c|z|2α 1 − |z|2α zfz ≤ 1,
αf z 1.4
for all z ∈ U, then the function
Fα z z
1/α
z ···
α tα−1 f tdt
1.5
0
is regular and univalent in U.
Theorem 1.2 see 5. Let α be a complex number, Re α > 0, and c a complex number, |c| ≤
1, c /
− 1, and f ∈ A. If
1 − |z|2Re α zfz ≤ 1 − |c|
Re α f z 1.6
for all z ∈ U, then for any complex number β, Re β ≥ Re α, the function
1/β
z
Fβ z β tβ−1 f tdt
1.7
0
is in the class S.
On the other hand, for the functions f ∈ A, Ozaki and Nunokawa 6 proved another
univalence condition asserted by Theorem 1.3.
Theorem 1.3 see 6. Let f ∈ A satisfy the condition
z2 f z
−
1
<1
fz 2
z ∈ U.
1.8
Then f is univalent in U.
In the paper 7, Pescar determined some univalence conditions for the following
integral operators.
Theorem 1.4 see 7. Let the function g satisfy 1.8, M a positive real number fixed, and c a
complex number. If α ∈ 2M 1/2M 2, 2M 1/2M,
α − 1
2M 1,
|c| ≤ 1 − α gz ≤ M
c/
− 1,
1.9
Abstract and Applied Analysis
3
for all z ∈ U, then the function
Gα z 1/α
z
α gtα−1 dt
1.10
0
is in the class S.
Theorem 1.5 see 7. Let g ∈ A, α a real number, α ≥ 1, and c a complex number, |c| ≤
1/α, c / − 1. If
g z g z ≤ 1
z ∈ U,
1.11
then the function
Hα z 1/α
z
α−1
dt
α
tg t
1.12
0
is in the class S.
Theorem 1.6 see 7. Let g ∈ A satisfies 1.8, α a complex number, M > 1 fixed, Re α > 0, and
c a complex number, |c| < 1. If |gz| ≤ M for all z ∈ U, then for any complex number β
Re β ≥ Re α ≥
2M 1
|α|1 − |c|
1.13
the function
z
1/α 1/β
β−1 gt
Hβ z β t
dt
t
0
1.14
is in the class S.
In this paper, we introduce the following integral operators as follows:
F1 f, g z z α−1 1/α
ftegt
dt
α
f, g ∈ A; α ∈ C ,
1.15
0
z α−1 1/α
G1 f, g z α
tf tegt
dt
f, g ∈ A; α ∈ C ,
1.16
f, g ∈ A; α, β ∈ C − {0} .
1.17
0
z
1/α 1/β
β−1 ft gt
e
dt
H1 f, g z β t
t
0
Remark 1.7. For egz 1 and fz gz, the integral operators 1.15, 1.16, and 1.17
would reduce to the integral operators 1.10, 1.12, and 1.14.
4
Abstract and Applied Analysis
In this paper, we generalize the integral operators given by Pescar 7, and we study
the univalence conditions for the integral operators defined by 1.15, 1.16, and 1.17.
For this purpose, we need the following result.
Lemma 1.8 General Schwarz Lemma 8. Let the function f be regular in the disk UR {z ∈
C : |z| < R}, with |fz| < M for fixed M. If f has one zero with multiplicity order bigger than m for
z 0, then
fz ≤ M |z|m
Rm
z ∈ UR .
1.18
The equality can hold only if
fz eiθ
M m
z ,
Rm
1.19
where θ is constant.
2. Main Results
Theorem 2.1. Let f, g ∈ A, where g satisfies the condition 1.8, M1 and M2 are real positive
numbers, and α a complex number, Re α > 0. If
f z gz ≤ M2 z ∈ U,
fz ≤ M1 z ∈ U,
α − 1 M1 2M2 1 , c ∈ C, c /
− 1,
|c| ≤ 1 − 2
α 2.1
2.2
then the integral operator F1 f, gz defined by 1.15 is in the class S.
Proof. From 1.15, we have
z
α−1 1/α
α−1 ft gt
e
F1 f, g z α t
dt
.
t
0
2.3
Let us consider the function
hz z 0
ft gt
e
t
α−1
dt.
2.4
Abstract and Applied Analysis
5
The function h is regular in U. From 2.4, we get
h z fz gz
e
h z α − 1
z
α−2 fz gz
e
z
α−1
,
zf z − fz gz fz gz
g ze
.
e
z
z2
2.5
Also, a simple computation yields
zh z
α − 1
h z
zf z
− 1 zg z ,
fz
2.6
which readily shows that
zf z
c|z|2α 1 − |z|2α zh z c|z|2α 1 − |z|2α α − 1
−
1
zg
z
αh z
α
fz
2 2 α − 1
zf z z g z gz ≤ |c| z ∈ U.
2 fz 1 α gz z 2.7
From the hypothesis of Theorem 2.1, we have
f z fz ≤ M1
z ∈ U,
gz ≤ M2
z ∈ U,
2.8
then by General Schwarz Lemma for the function g, we obtain
gz ≤ M2 |z|
z ∈ U.
2.9
Using the inequality 2.7, we have
2 2
c|z|2α 1 − |z|2α zh z ≤ |c| α − 1 M1 1 z g z
− 1 1 M2 .
α gz
2
αh z 2.10
From 2.10 and since g satisfies the condition 1.8, we have
c|z|2α 1 − |z|2α zh z ≤ |c| α − 1 M1 2M2 1 ,
2
αh z
α
2.11
from which, by 2.2, we get
c|z|2α 1 − |z|2α zh z ≤ 1
αh z z ∈ U.
2.12
6
Abstract and Applied Analysis
Applying Theorem 1.1, we conclude that the integral operator F1 f, gz defined by 1.15 is
in the class S.
Setting M1 1 and M2 1 in Theorem 2.1, we immediately arrive at the following
application of Theorem 2.1.
Corollary 2.2. Let f, g ∈ A, where g satisfies the condition 1.8 and α a complex number, Re α > 0.
If
f z fz ≤ 1
gz ≤ 1
z ∈ U,
4α − 1 ,
|c| ≤ 1 − α
z ∈ U,
2.13
c ∈ C, c /
− 1,
then the integral operator F1 f, gz defined by 1.15 is in the class S.
Theorem 2.3. Let f, g ∈ A, where g satisfies the inequality |gz| ≤ M, M ≥ 1. Also, let α be a real
number, α ≥ 1, and c a complex number with
α − 1
M 1,
|c| ≤ 1 − α c
/ − 1.
2.14
If
f z f z ≤ 1
z ∈ U,
g z gz ≤ 1
z ∈ U,
2.15
then the integral operator G1 f, gz defined by 1.16 is in the class S.
Proof. We observe that
z
α−1 1/α
α−1
gt
G1 f, g z α t
f te
dt
.
2.16
0
Let us consider the function
hz z f tegt
α−1
dt.
2.17
0
The function h is regular in U. From 2.17, we have
α−1
,
h z f zegz
zf z
zh z
α − 1
zg z ,
h z
f z
2.18
Abstract and Applied Analysis
7
which readily shows that
c|z|2α 1 − |z|2α zh z ≤ |c| α − 1 zf z zg z gz
α f z gz αh z z ∈ U.
2.19
From 2.19 and the conditions of Theorem 2.3, we get
c|z|2α 1 − |z|2α zh z ≤ |c| α − 1 1 M ≤ 1
αh z
α z ∈ U.
2.20
Applying Theorem 1.1, we conclude that the integral operator G1 f, gz defined by 1.16
is in the class S.
Setting M 1 in Theorem 2.3, we obtain the following consequence of Theorem 2.3.
Corollary 2.4. Let f, g ∈ A, where g satisfies the condition |g z/gz| ≤ 1, α a real number,
α ≥ 1, and c a complex number with |c| ≤ 1 − |2α − 1/α|, c /
− 1. If
f z f z ≤ 1
gz ≤ 1
z ∈ U,
z ∈ U,
2.21
then the integral operator G1 f, gz defined by 1.16 is in the class S.
Theorem 2.5. Let f, g ∈ A, where g satisfies the condition 1.8, α a complex number, Re α > 0, M1
and M2 are real positive numbers, and c a complex number, |c| < 1. If
f z fz ≤ M1
gz ≤ M2
z ∈ U,
z ∈ U,
2.22
then for any complex number β,
Re β ≥ Re α ≥
M1 2M22 1
,
|α|1 − |c|
2.23
the integral operator H1 f, gz defined by 1.17 is in the class S.
Proof. Let us consider the function
hz z 0
ft gt
e
t
1/α
dt.
2.24
8
Abstract and Applied Analysis
The function h is regular in U. From 2.24, we have
fz gz 1/α
e
h z ,
z
zf z
zh z 1
−
1
zg
,
z
h z
α
fz
2.25
which readily shows that
z2 g z gz
2 1 − |z|2 Re α zh z 1 − |z|2 Re α zf z 1 ≤
2
Re α
h z
fz
z |α| Re α
gz
1 − |z|2 Re α
≤
|α| Re α
gz
2 2 zf z 1 z g z
− 1 1 .
fz z gz
2
2.26
By the General Schwarz Lemma for the function g, we obtain
gz ≤ M2 |z|
z ∈ U,
2.27
and using the inequality 2.26, we have
z2 g z
1 − |z|2 Re α zh z 1 − |z|2 Re α
2
≤
M1 1 − 1 1 M2 .
gz
2
Re α h z |α| Re α
2.28
From 2.28 and since g satisfies the condition 1.8, we get
1 − |z|2 Re α zh z 1 − |z|2 Re α M1 2M22 1
≤
Re α h z Re α
|α|
M1 2M22 1
≤
.
|α| Re α
2.29
From 2.23, we have
M1 2M22 1
≤ 1 − |c|,
|α| Re α
2.30
and using 2.29, we obtain
1 − |z|2 Re α zh z ≤ 1 − |c|
Re α h z z ∈ U.
2.31
Applying Theorem 1.2, we conclude that the integral operator H1 f, gz defined by 1.17
is in the class S.
Abstract and Applied Analysis
9
Setting M1 1 and M2 1 in Theorem 2.5, we obtain the following corollary.
Corollary 2.6. Let f, g ∈ A, where g satisfies the condition 1.8, α a complex number, Re α > 0,
and c a complex number, |c| < 1. If
f z fz ≤ 1
z ∈ U,
gz ≤ 1
z ∈ U,
2.32
then for any complex number β,
Re β ≥ Re α ≥
4
,
|α|1 − |c|
2.33
the integral operator H1 f, gz defined by 1.17 is in the class S.
Acknowledgments
This paper was partially supported by the strategic project POSDRU 107/1.5/S/77265, inside
POSDRU Romania 2007–2013 cofinanced by the European Social Fund-Investing in People.
References
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