Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 652858, 11 pages
doi:10.1155/2012/652858
Research Article
Univalence Criteria for Two Integral Operators
Laura Filofteia Stanciu1 and Daniel Breaz2
1
Department of Mathematics, University of Piteşti, Târgul din Vale Street No. 1,
110040 Piteşti, Argeş, Romania
2
Department of Mathematics, “1 Decembrie 1918” University of Alba Iulia, Street N. Iorga,
No. 11-13, 510000 Alba Iulia, Romania
Correspondence should be addressed to Daniel Breaz, breazdaniel@yahoo.com
Received 6 October 2011; Accepted 17 December 2011
Academic Editor: Ibrahim Sadek
Copyright q 2012 L. F. Stanciu and D. Breaz. 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 study the univalence conditions for two integral operators to be univalent in the open unit
disk. Many known univalence conditions are written to prove our main results.
1. Introduction and Preliminaries
Let A denote the class of functions of the form:
fz z ∞
an zn ,
1.1
n2
which are analytic in the open unit disk:
U {z ∈ C : |z| < 1},
1.2
and satisfy the following usual normalization condition:
f0 f 0 − 1 0.
1.3
We denote by S the subclass of A consisting of functions fz which are univalent in U.
In 1, for 0 < b ≤ 1, Silverman considered the class:
Gb f ∈ A
zf z zf z zf z ,
< b
−
: 1 f z
fz fz z∈U .
1.4
2
Abstract and Applied Analysis
Here, in our present investigation, we consider two general families of integral ope-
rators:
I f1 , . . . , fn ; g 1 , . . . , gn z 1/β
z
n fi t αi −1/Mi γi
β−1
,
β t
gi t dt
t
0
i1
1.5
αi , γi ∈ C; β ∈ C \ {0}; fi , gi ∈ A, Mi ≥ 1 for all i ∈ {1, 2, . . . , n};
n
1 αi
J f1 , . . . , fn ; g1 , . . . , gn z z
n
0 i1
i1
1/1ni1 αi α γ
fi t i gi t i dt
,
1.6
αi , γi ∈ C; fi , gi ∈ A for all i ∈ {1, 2, . . . , n}.
Many authors have studied the problem of integral operators which preserve the class
S see, e.g., 2–5.
In the present paper, we study the univalence conditions involving the general families
of integral operators defined by 1.5 and 1.6.
In the proof of our main results Theorem 2.1 and Theorem 3.1, we need the following
univalence criterion. The univalence criterion, asserted by Theorem 1.1, is a generalization of
Ahlfor’s and Becker’s univalence criterion; it was proven by Pescar 6.
Theorem 1.1 see Pescar 6. Let β ∈ C with Re β > 0, c ∈ C with |c| ≤ 1, c /
− 1. If f ∈ A
satisfies
c|z|2β 1 − |z|2β zf z ≤ 1,
1.7
βf z for all z ∈ U, then the integral operator,
z
1/β
β−1 Fβ z β t f t dt
,
1.8
0
is in the class S.
Finally, in the present investigation, one also needs the familiar Schwarz Lemma see,
for details, 7.
Lemma 1.2 General Schwarz Lemma see 7. 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 .
1.9
The equality can hold only if
fz eiθ
where θ is constant.
M m
z ,
Rm
1.10
Abstract and Applied Analysis
3
2. Univalence Conditions for If1 , . . . , fn ; g1 , . . . , gn z
Theorem 2.1. Let Mi ≥ 1 (i ∈ {1, 2, . . . , n}) and β, αi , γi be complex numbers with Re β ≥ 0 and
Re β ≥
n
3|αi − 1| γi 2bi 1 ,
2.1
i1
and let c ∈ C be such that
n
1 3|αi − 1| γi 2bi 1 .
Re β i1
|c| ≤ 1 −
2.2
If for all i ∈ {1, 2, . . . , n}, fi ∈ A satisfy the conditions:
fi z ≤ Mi
z ∈ U,
z2 f z
2M − 1
i
i
− 1 ≤
2
f z
M
i
i
z ∈ U,
2.3
and gi ∈ Gbi , 0 < bi ≤ 1 with
zg z
i
− 1 < 1
gi z
z ∈ U,
2.4
then the integral operator If1 , . . . , fn ; g1 , . . . , gn z defined by 1.5 is in the class S.
Proof. We begin by setting
hz z n 0 i1
fi t
t
αi −1/Mi
γ
gi t i dt,
2.5
and then we calculate for hz the derivates of the first and second orders.
From 2.5, we obtain
h z n fi z αi −1/Mi i1
z
gi z
γi
,
n
zgi z
zh z αi − 1 zfi z
− 1 γi h z
Mi
fi z
gi z
i1
n
αi − 1 zfi z
−1
Mi
fi z
i1
zgi z zgi z
zgi z
−
1 γi
−1 .
γi
gi z
gi z
gi z
2.6
4
Abstract and Applied Analysis
Thus, we have
n
zh z |αi − 1| zfi z ≤
1
h z fi z Mi
i1
zg z
zgi z zgi z
i
γi −
1 γi − 1
gi z
gi z
gi z
2 n
z f z
fi z |αi − 1|
i
1
≤
− 1 1 2
f z
Mi
z i1
i
2.7
zg z
zgi z zgi z
i
γi −
1 γi − 1 .
gi z
gi z
gi z
From the hypothesis 2.3 of Theorem 2.1, we have
fi z ≤ Mi
z ∈ U; Mi ≥ 1,
2M − 1
z2 f z
i
i
− 1 ≤
2
f z
M
i
i
z ∈ U; Mi ≥ 1
2.8
for all i ∈ {1, 2, . . . , n}.
By applying the General Schwarz Lemma, we thus obtain
fi z ≤ Mi |z|
z ∈ U; i ∈ {1, 2, . . . , n}.
2.9
Since gi ∈ Gbi , 0 < bi ≤ 1 for all i ∈ {1, 2, . . . , n}, from 1.4, 2.4, we obtain
n
zgi z zgi z
zh z ≤
γ
γ
b
3|α
−
1
−
1|
i
i
i
i
h z gi z
gi z
i1
≤
n
i1
≤
n
i1
≤
n
i1
≤
zg z
zgi z
i
3|αi − 1| γi bi − 1 1 γi − 1
gi z
gi z
zgi z
zgi z
3|αi − 1| γi bi − 1 γi bi γi − 1
gi z
gi z
zgi z
3|αi − 1| γi bi γi − 1 γi bi
gi z
n 3|αi − 1| γi 2bi 1 ,
i1
2.10
Abstract and Applied Analysis
5
which readily shows that
n
c|z|2β 1 − |z|2β zh z ≤ |c| 1
3|αi − 1| γi 2bi 1
β
βh z
i1
≤ |c| n
1 3|αi − 1| γi 2bi 1
Re β i1
2.11
≤ 1,
where we have also used the hypothesis 2.2 of Theorem 2.1.
Finally, by applying Theorem 1.1, we conclude that the integral operator If1 , . . . ,
fn ; g1 , . . . , gn z defined by 1.5 is in the class S. This evidently completes the proof of
Theorem 2.1.
Setting α1 α2 · · · αn 1 in Theorem 2.1, we have the following result.
Corollary 2.2. Let Mi ≥ 1 (i ∈ {1, 2, . . . , n}) and β, γi be complex numbers with Re β ≥ 0 and
Re β ≥
n γi 2bi 1
2.12
i1
and let c ∈ C be such that
|c| ≤ 1 −
n 1 γi 2bi 1 .
Re β i1
2.13
If for all i ∈ {1, 2, . . . , n}, gi ∈ Gbi , 0 < bi ≤ 1 with
zg z
i
− 1 < 1
gi z
z ∈ U,
2.14
then the integral operator,
1/β
z
n
γi
β−1
I f1 , . . . , fn ; g1 , . . . , gn z β t
,
gi t dt
0
2.15
i1
is in the class S.
Setting γi 1 for all i ∈ {1, 2, . . . , n} in Theorem 2.1, we have the following result.
Corollary 2.3. Let Mi ≥ 1 (i ∈ {1, 2, . . . , n}) and β, αi be complex numbers with Re β ≥ 0 and
Re β ≥
n
i1
3|αi − 1| 2bi 1,
2.16
6
Abstract and Applied Analysis
and let c ∈ C be such that
|c| ≤ 1 −
n
1 3|αi − 1| 2bi 1.
Re β i1
2.17
If for all i ∈ {1, 2, . . . , n}, f i ∈ A satisfy the conditions:
fi z ≤ Mi
z ∈ U,
z2 f z
2M − 1
i
i
− 1 ≤
2
f z
M
i
i
z ∈ U
2.18
and gi ∈ Gbi , 0 < bi ≤ 1 with
zg z
i
− 1 < 1
gi z
z ∈ U,
2.19
then the integral operator
1/β
z
n fi t αi −1/Mi β−1
I f1 , . . . , fn ; g1 , . . . , gn z β t
gi t dt
t
0
i1
2.20
is in the class S.
Setting n 1 in Theorem 2.1, we have the following result.
Corollary 2.4. Let M ≥ 1 and β, α, γ be complex numbers with Re β ≥ 0 and
Re β ≥ 3|α − 1| γ 2b 1 ,
2.21
and let c ∈ C be such that
|c| ≤ 1 −
1 3|α − 1| γ 2b 1 .
Re β
2.22
If the function f ∈ A satisfies the conditions:
fz ≤ M
z ∈ U,
z2 f z
2M − 1
− 1 ≤
2
f z
M
z ∈ U,
2.23
and g ∈ Gb , 0 < b ≤ 1 with
zg z
gz − 1 < 1
z ∈ U,
2.24
Abstract and Applied Analysis
7
then the integral operator,
1/β
z
α−1/M
γ
β−1 ft
,
I f; g z β t
g t dt
t
0
2.25
is in the class S.
3. Univalence Conditions for Jf1 , . . . , fn ; g1 , . . . , gn z
Theorem 3.1. Let Mi ≥ 1 (i ∈ {1, 2, . . . , n}) and β, αi , γi be complex numbers, β 1 Re β ≥ 0 and
Re β ≥
n
i1
n
|αi | γi bi 12Mi 1 γi bi ,
αi ,
3.1
i1
and let c ∈ C be such that
|c| ≤ 1 −
n
1 |αi | γi bi 12Mi 1 γi bi .
Re β i1
3.2
If for all i ∈ {1, 2, . . . , n}, fi ∈ A satisfy the condition:
zf z
i
− 1 < 1
fi z
z ∈ U,
3.3
and gi ∈ Gbi , 0 < bi ≤ 1 with
z2 g z
i
− 1 < 1 z ∈ U,
2
g z
i
gi z ≤ Mi z ∈ U; i ∈ {1, 2, . . . , n},
3.4
3.5
then the integral operator Jf1 , . . . , fn ; g1 , . . . , gn z defined by 1.6 is in the class S.
Proof. We begin by observing that the integral operator Jf1 , . . . , fn ; g1 , . . . , gn z defined by
1.6 can be rewritten as follows:
J f1 , . . . , fn ; g1 , . . . , gn z 1
n
i1
where fi ∈ A for all i ∈ {1, 2, . . . , n}.
z
αi
0
1/1
n fi t αi γi
αi
i1
t
gi t dt
t
i1
n
n
i1
αi ,
3.6
8
Abstract and Applied Analysis
Defining the function hz by
hz z n 0 i1
fi t
t
αi
gi t
γi
dt,
3.7
we take the same steps as in the proof of Theorem 2.1, and we have
n
zf z
zgi z
zh z i
≤
− 1 γi bi γi − 1 γi bi .
|αi |
h z gi z
fi z
i1
3.8
Thus, we have
c|z|2β 1 − |z|2β zh z βh z n
z2 gi z gi z 1 1 γi bi .
≤ |c| |αi | γi bi γi 2
β
g z z i1
i
3.9
Furthermore, from the hypothesis 3.4 of Theorem 3.1, we have
z2 g z
i
− 1 < 1 z ∈ U ,
2
g z
i
gi z ≤ Mi z ∈ U; i ∈ {1, 2, . . . , n}.
3.10
By applying the General Schwarz Lemma, we obtain
gi z ≤ Mi |z|
z ∈ U; i ∈ {1, 2, . . . , n}.
3.11
So, we obtain
c|z|2β 1 − |z|2β zh z βh z 2 n
z g z
1 i
≤ |c| − 1 1 Mi 1 γi bi
|αi | γi bi γi 2
β
g z
i1
i
n
1 ≤ |c| |αi | γi bi γi 2Mi 1 γi bi
β
i1
≤ |c| n
1 |αi | γi bi γi 2Mi 1 γi bi
Re β i1
≤ 1.
3.12
Abstract and Applied Analysis
9
Finally, by applying Theorem 1.1, we conclude that the integral operator Jf1 , . . . ,
fn ; g1 , . . . , gn z defined by 1.6 is in the class S. This evidently completes the proof of
Theorem 3.1.
Setting α1 α2 · · · αn 1 in Theorem 3.1, we have
Corollary 3.2. Let Mi ≥ 1 (i ∈ {1, 2, . . . , n}) and β, γi be complex numbers, β 1 n, Re β ≥ 0
and
Re β ≥
n
1 γi bi 12Mi 1 γi bi ,
3.13
i1
and let c ∈ C be such that
|c| ≤ 1 −
n
1 1 γi bi 12Mi 1 γi bi .
Re β i1
3.14
If for all i ∈ {1, 2, . . . , n}, fi ∈ A satisfy the condition:
zf z
i
− 1 < 1
fi z
z ∈ U,
3.15
and gi ∈ Gbi , 0 < bi ≤ 1 with
z2 g z
i
− 1 < 1
2
g z
i
gi z ≤ Mi
z ∈ U ,
z ∈ U,
3.16
then the integral operator,
J f1 , . . . , fn ; g1 , . . . , gn z 1 n
z n
0 i1
1/1n
γ
fi t gi t i dt
,
3.17
is in the class S.
Setting γi 1 for all i ∈ {1, 2, . . . , n} in Theorem 3.1, we have the following result.
Corollary 3.3. Let Mi ≥ 1 (i ∈ {1, 2, . . . , n}) and β, αi be complex numbers, β 1 Re β ≥ 0 and
Re β ≥
n
|αi | bi 12Mi 1 bi ,
i1
n
i1
αi ,
3.18
10
Abstract and Applied Analysis
and let c ∈ C be such that
|c| ≤ 1 −
n
1 |αi | bi 12Mi 1 bi .
Re β i1
3.19
If for all i ∈ {1, 2, . . . , n}, fi ∈ A satisfy the condition:
zf z
i
− 1 < 1
fi z
z ∈ U,
3.20
and gi ∈ Gbi , 0 < bi ≤ 1 with
z2 g z
− 1 < 1
2i
g z
i
gi z ≤ Mi
z ∈ U ,
z ∈ U,
3.21
then the integral operator,
J f1 , . . . , fn ; g1 , . . . , gn z 1
n
z
αi
i1
n
fi t
αi 0 i1
gi t dt
1/1ni1 αi ,
3.22
is in the class S.
Setting n 1 in Theorem 3.1, we have the following result.
Corollary 3.4. Let M ≥ 1 and β, α, γ be complex numbers, β 1 α, Re β 0 and
Re β ≥ |α| γ b 12M 1 γ b ,
3.23
and let c ∈ C be such that
|c| ≤ 1 −
1 |α| γ b 12M 1 γ b .
Re β
3.24
If one has that the function f ∈ A satisfies the condition:
zf z
≤1
−
1
fz
3.25
z ∈ U,
and g ∈ Gb , 0 < b ≤ 1 with
z2 g z
− 1 < 1
2
g z
z ∈ U ,
gz ≤ M
z ∈ U,
3.26
Abstract and Applied Analysis
11
then the integral operator,
J f; g z 1 α
z
ft
1/1α
γ
g t dt
α ,
3.27
0
is in the class S.
Acknowledgment
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
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2 A. Baricz and B. A. Frasin, “Univalence of integral operators involving Bessel functions,” Applied
Mathematics Letters, vol. 23, no. 4, pp. 371–376, 2010.
3 V. Pescar, “On the univalence of some integral operators,” General Mathematics, vol. 14, no. 2, pp. 77–84,
2006.
4 D. Blezu and R. N. Pascu, “Univalence criteria for integral operators,” Glasnik Matematički. Serija III,
vol. 36, no. 2, pp. 241–245, 2001.
5 D. Breaz and N. Breaz, “Univalence conditions for certain integral operators,” Universitatis Babeş-Bolyai,
Studia, Mathematica, vol. 47, no. 2, pp. 9–15, 2002.
6 V. Pescar, “A new generalization of Ahlfors’s and Becker’s criterion of univalence,” Malaysian
Mathematical Society, Bulletin (Second Series), vol. 19, no. 2, pp. 53–54, 1996.
7 Z. Nehari, Conformal Mapping, McGraw-Hill, New York, NY, USA, 1952.
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