Applied Mathematics Letters 22 (2009) 4144 [612107]

Applied Mathematics Letters 22 (2009) 41–44
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Applied Mathematics Letters
journal homepage: www.elsevier.com/locate/aml
An extension of the univalent condition for a family of integral operators
Daniel Breaza, Nicoleta Breaza, H.M. Srivastavab,
aDepartment of Mathematics and Informatics, “1 Decembrie 1918” University of Alba Iulia, RO-510009 Alba Iulia, Romania
bDepartment of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3R4, Canada
a r t i c l e i n f o
Article history:
Received 25 November 2007
Accepted 25 November 2007
Keywords:
Analytic functions
Open unit disk
Integral operators
Univalent functions
Univalent conditions
Schwarz Lemmaa b s t r a c t
The main object of this work is to extend the univalent condition for a family of integral
operators. Several other closely-related results are also considered. A number of known
univalent conditions would follow upon specializing the parameters involved in our main
result.
'2008 Elsevier Ltd. All rights reserved.
1. Introduction and preliminaries
LetAdenote the class of functions fof the form:
f.z/DzC1X
nD2anzn; (1.1)
which are analytic in the open unit disk
UD
zVz2Candjzj<1
and satisfy the following usual normalization condition:
f.0/Df0.0/1D0:
We denote by Sthe subclass of Aconsisting of functions which are also univalent in U. For some recent investigations of
various interesting subclasses of the univalent function class S, see the works by (for example) Altıntaș et al. [ 1], Gao et al. [ 4],
and Owa et al. [ 6].
The following univalent condition was proven by Pescar [ 8].
Theorem 1 (Pescar [ 8]).Let
2C.R./>0/
and
c2C.jcj51Ic6D 1/:
Corresponding author.
E-mail addresses: [anonimizat] (D. Breaz), [anonimizat] (N. Breaz), [anonimizat] (H.M. Srivastava).
0893-9659/$ – see front matter '2008 Elsevier Ltd. All rights reserved.
doi:10.1016/j.aml.2007.11.008

42 D. Breaz et al. / Applied Mathematics Letters 22 (2009) 41–44
Suppose also that the function f.z/given by (1.1) is analytic in U. If
cjzj2C
1jzj2zf00.z/
f0.z/51.z2U/;
then the function F.z/defined by
F.z/VD
Zz
0t1f0.t/dt1

DzC



is analytic and univalent in U.
Ozaki and Nunokawa [ 7], on the other hand, proved another univalent condition asserted by Theorem 2 .
Theorem 2 (Ozaki and Nunokawa [ 7]).Letf2Asatisfy the following inequality:
z2f0.z/
Tf.z/U2151.z2U/: (1.2)
Then fis univalent in U.
Yet another univalent condition was given by Pescar [ 9] as follows.
Theorem 3 (Pescar [ 9]).Let the function g2Asatisfy the inequality (1.2) . Also let
2R
2
1;3
2
and c2C:
If
jcj532
.c6D 1/
and
jg.z/j51.z2U/;
then the function G.z/defined by
G.z/VD
Zz
0[g.t/]1dt1

(1.3)
is in the univalent function class S.
Finally, Breaz and Breaz [ 3] (see also [ 2]) considered the following family of integral operators and proved that the
function Gn;.z/defined by
Gn;.z/VD
[n.1/C1]Zz
0[g1.t/]1


[gn.t/]1dt 1
n.1/C1.g1;:::; gn2A/ (1.4)
is univalent in U.
Remark 1. In its special case when nD1, the integral operator in (1.4) would obviously reduce to the integral operator in
(1.3) .
In view of Remark 1 , we propose to investigate further univalent conditions involving the general family of integral
operators defined by (1.4) . The following familiar result is of fundamental importance in our investigation.
Schwarz Lemma (See, for Example, [ 5]).Let the analytic function fbe regular in the open unit disk Uand let f.0/D0. If
jf.z/j51.z2U/;
then
jf.z/j5jzj.z2U/; (1.5)
where the equality holds true only if
f.z/DKz.z2U/andjKjD1: (1.6)

D. Breaz et al. / Applied Mathematics Letters 22 (2009) 41–44 43
2. The main univalent condition
In this section, we first state the main univalent condition involving the general integral operator given by (1.4) .
Theorem 4. LetM=1and suppose that each of the functions gj2A.j2f1;:::; ng/satisfies the inequality (1.2) . Also let
2R
2
1;.2MC1/n
.2MC1/n1
and c2C:
If
jcj51C1

.2MC1/n (2.1)
and
gj.z/5M.z2UIj2f1;:::; ng/;
then the function Gn;.z/defined by (1.4) is in the univalent function class S.
Proof. We begin by setting
f.z/DZz
0nY
jD1gj.t/
t1
dt;
so that, obviously,
f0.z/DnY
jD1gj.z/
z1
(2.2)
and
f00.z/D.1/
nX
jD1gj.z/
z2
zg0
j.z/gj.z/
z2!

nY
kD1.k6Dj/gk.z/
z1
: (2.3)
We thus find from (2.2) and (2.3) that
zf00.z/
f0.z/D.1/nX
jD1
zg0
j.z/
gj.z/1!
;
which readily shows that
cjzj2C
1jzj2zf00.z/
f0.z/Dcjzj2C
1jzj21
nX
jD1
zg0
j.z/
gj.z/1!
5jcjC1


nX
jD1 z2g0
j.z/
Tgj.z/U2
gj.z/
jzjC1!
:
Since
gj.z/5M.z2UIj2f1;:::; ng/;
by using the inequality (1.2) , we obtain
cjzj2C
1jzj2zf00.z/
f0.z/5jcjC1

.2MC1/n.z2U/;
which, in the light of the hypothesis (2.1) , yields
cjzj2C
1jzj2zf00.z/
f0.z/51.z2U/:
Finally, by applying Theorem 1 , we conclude that the function Gn;.z/defined by (1.4) is in the univalent function class
S. This evidently completes the proof of Theorem 4 .

44 D. Breaz et al. / Applied Mathematics Letters 22 (2009) 41–44
3. Applications of Theorem 4
First of all, upon setting MD1 in Theorem 4 , we immediately arrive at the following application of Theorem 4 .
Corollary 1. Let each of the functions gj2A.j2f1;:::; ng/satisfy the condition (1.2) . Suppose also that
2R
2
1;3n
3n1
and c2C:
If
jcj51C31

n
and
gj.z/51.z2UIj2f1;:::; ng/;
then the function Gn;.z/defined by (1.4) is in the univalent function class S.
Next we set nD1 in Theorem 4 . We thus obtain the following interesting consequence of Theorem 4 .
Corollary 2. LetM=1and suppose that the function g2Asatisfies the condition (1.2) . Also let
2R
2
1;2MC1
2M
and c2C:
If
jcj51C1

.2MC1/
and
jg.z/j5M.z2U/;
then the function G.z/defined by (1.3) is in the univalent function class S.
Remark 2. Corollary 2 provides an extension of Theorem 3 due of Pescar [ 9].
Remark 3. If, in Theorem 4 , we set MDnD1, we obtain Theorem 3 due of Pescar [ 9].
Many other interesting corollaries and consequences of Theorem 4 can be deduced from Theorem 4 in a similar manner.
Acknowledgements
The present investigation was supported, in part, by the Natural Sciences and Engineering Research Council of Canada under
Grant OGP0007353.
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