DOI: 10.1515/auom-2015-0014
An. Şt. Univ. Ovidius Constanţa
Vol. 23(1),2015, 213–224
Univalence conditions for a general integral
operator
Adriana Oprea and Daniel Breaz
Abstract
For analytic functions in the open unit disk U , we define a new
general integral operator. The main object of the this paper is to study
this new integral operator and to determine univalence conditions of it.
Several corollaries of the main results are also considered.
1
Introduction
Let A denote the class of function of the form:
f (z) = z +
∞
X
an z n ,
(1)
i=2
which are analytic in the open unit disc U = {z : |z| < 1}
Further, by S we shall denote the class of functions in A which are univalent
in U.
In [9], Silverman define the class Gb .
For 0 < b ≤ 1, he considered the class:
0 zf (z) zf 00 (z) zf 0 (z) Gb = f ∈ A : 1 + 0
−
< b
,z ∈ U
(2)
f (z)
f (z) f (z) Key Words: Analytic functions; Integral Operators; General Schwarz Lemma
2010 Mathematics Subject Classification: Primary 30C45; Secondary 30C75
Received: 2 May, 2014.
Revised: 30 May, 2014.
Accepted: 27 June, 2014.
213
214
UNIVALENCE CONDITIONS FOR A GENERAL INTEGRAL OPERATOR
Let αi ∈ C∗ , for all i = 1, 2, ...n, n ∈ N∗ ,γ ∈ C, with Reγ > 0.
We consider Iγ,αi : An × An → A be the integral operator defined by:
Iγ,αi (f1 , ...fn , g1 , ...gn )(z) =
( Z
γ
z
t
γ−1
0
f1 (t)
t
1− α1
1
1
(g10 (t)) α1
....
fn (t)
t
(3)
1− α1
n
1
(gn0 (t)) αn
) γ1
dt
In the present paper we study the univalence conditions for the integral
operator Iγ,αi (f1 , ...fn , g1 , ...gn ), when gi ∈ Gbi and fi ∈ A for all i = 1, ..., n.
Also, the univalence of some integral operators has been studied by other authors like V. Pescar and D. Breaz in work [8].
In order to derive our main results, we have to recall here the following univalence criteria.
Theorem 1. [5] Let f ∈ A and γ ∈ C. If Reγ > 0 and
2Reγ
00 zf (z) f 0 (z) ≤ 1,
1 − |z|
Reγ
for all z ∈ U, then the integral operator
Z
Fγ (z) = γ
z
t
γ1
f (t)dt
γ−1 0
0
is in the class S.
Theorem 2. [6] Let δ ∈ C with Reδ > 0. If f ∈ A satisfies
1 − |z|2Reδ zf 00 (z) f 0 (z) ≤ 1,
Reδ
for all z ∈ U, then, for any complex γ with Reγ ≥ Reδ, the integral operator
Z
Fγ (z) = γ
z
t
γ1
f (t)dt
γ−1 0
0
is in the class S.
Theorem 3. [7] Let γ be a complex number, Reγ > 0, and c a complex
number, |c| ≤ 1, c 6= −1. If f ∈ A satisfies
00 c |z|2γ + 1 − |z|2γ zf (z) ≤ 1,
γf 0 (z) UNIVALENCE CONDITIONS FOR A GENERAL INTEGRAL OPERATOR
215
for all z ∈ U, then the integral operator
Z
Fγ (z) = γ
z
t
γ1
f (t)dt
γ−1 0
0
is in the class S.
Also, we need the following general Schwarz Lemma
Lemma 1. (General Schwarz-Lemma) [4] Let the function f be regular in the
disk UR = {z ∈ C : |z| < R}, with |f (z)| < M , M fixed. If f has one zero
with multiplicity greater then m for z = 0, then
M
m
|z| ,
Rm
|f (z)| ≤
z ∈ UR .
(4)
the equality can hold only if
f (z) = eiθ
M m
z ,
Rm
where θ is constant.
2
Univalence conditions for Iγ,αi (f1 , ...fn , g1 , ...gn )(z)
First, we prove:
Theorem 4. Let γ ∈ C and αi ∈ C∗ for all i = 1, ...n, with
Reγ ≥
n
X
1
(|αi − 1| + 2bi + 1)
|αi |
(5)
i=1
If for all i = 1, ..., n, gi ∈ Gbi ; 0 < bi ≤ 1, fi ∈ A, and
0
0
zgi (z)
zfi (z)
fi (z) − 1 < 1, z ∈ U gi (z) − 1 < 1, z ∈ U
(6)
then the integral operator Iγ,αi (f1 , ...fn , g1 , ...gn ), defined by (3) is in the class
S.
Proof. We define:
Z
h(z) =
n
zY
0 i=1
"
fi (t)
t
1− α1
i
(gi0 (t))
1
αi
#
dt,
216
UNIVALENCE CONDITIONS FOR A GENERAL INTEGRAL OPERATOR
so, that, obviously, we get:
#
"
1− α1
n
Y
1
i
fi (z)
0
αi
h (z) =
(gi (z))
z
i=1
0
and
=
n X
i=1
n
zh00 (z) X
=
h0 (z)
i=1
1−
1
αi
(7)
0
zfi (z)
1 zgi00 (z)
1
−1 +
1−
αi
fi (z)
αi gi0 (z)
(8)
zfi0 (z)
zgi0 (z)
1 zgi00 (z)
1 zgi0 (z)
−1 +
−
+
1
+
−
1
fi (z)
αi
gi0 (z)
gi (z)
αi
gi (z)
Since gi ∈ Gbi ;0 < bi ≤ 1 for all i = 1, ...n, from (2) and (6) we obtain:
00
zh (z) h0 (z) 00
0
n 0
X
|αi − 1| zfi0 (z)
+ 1 zgi (z) − zgi (z) + 1 + 1 zgi (z) − 1
−
1
≤
0
|αi |
fi (z)
|αi | gi (z)
gi (z)
|αi | gi (z)
i=1
n
X
zg 0 (z) 0
|αi − 1| zfi0 (z)
1
+ 1 zgi (z) − 1
≤
− 1 +
bi i
gi (z)
|α
f
|α
g
|α
i|
i (z)
i|
i (z)
i|
i=1
0
0
n X
|αi − 1| zfi0 (z)
+ 1 bi zgi (z) − 1 + 1 + 1 zgi (z) − 1
−
1
≤
|α
|
f
(z)
|α
|
g
(z)
|α
|
g
(z)
i
i
i
i
i
i
i=1
n
X
0
0
|αi − 1| zfi0 (z)
+ 1 bi zgi (z) − 1 + 1 bi + 1 zgi (z) − 1
=
−
1
|αi |
fi (z)
|αi |
gi (z)
|αi |
|αi | gi (z)
i=1
0
n X
|αi − 1| zfi0 (z)
+ 1 (bi + 1) zgi (z) − 1 + 1 bi
=
−
1
|αi |
gi (z)
|αi |
|αi | fi (z)
i=1
n
X |αi − 1|
1
1
≤
+
(bi + 1) +
bi
|α
|α
|α
i|
i|
i|
i=1
≤
n
X
1
(|αi − 1| + 2bi + 1) ,
|α
i|
i=1
which shows that:
2Reγ 2Reγ
00
1 − |z|
zh (z) ≤ 1 − |z|
h0 (z) Reγ
Reγ
1
≤
Reγ
(9)
n
X
1
(|αi − 1| + 2bi + 1)
|α
i|
i=1
!
n
X
1
(|αi − 1| + 2bi + 1)
|α
i|
i=1
!
≤1
Applying Theorem 1 for the function h, we prove that the integral operator
Iγ,αi (f1 , ...fn , g1 , ...gn ) is in the class S.
UNIVALENCE CONDITIONS FOR A GENERAL INTEGRAL OPERATOR
217
Let αi = 1 for all i = 1, 2, ...n in the Theorem 4, we have:
Corollary 1. Let γ ∈ C with
n
X
Reγ ≥
(2bi + 1)
(10)
i=1
If for all i = 1, ..., n, gi ∈ Gbi ; 0 < bi ≤ 1 and
0
zgi (z)
gi (z) − 1 < 1, z ∈ U,
(11)
then the integral operator
( Z
Iγ (g1 , ..., gn )(z) = γ
z
t
γ−1
0
n
Y
) γ1
(gi0 (t)) dt
(12)
i=1
is in the class S.
Let n = 1, α1 = α, b1 = b and f1 = f , g1 = g in Theorem 4, we have:
Corollary 2. Let γ ∈ C and α ∈ C∗ with
Reγ ≥
1
(|α − 1| + 2b + 1)
|α|
If f ∈ A, g ∈ Gb ; 0 < b ≤ 1 and
0
zf (z)
f (z) − 1 < 1,
(13)
0
zg (z)
g(z) − 1 < 1, z ∈ U,
(14)
then the integral operator defined by
( Z
Iγ,α (f, g)(z) = γ
0
z
t
1
γ−1+ α
g 0 (t)
f (t)
α1
) γ1
f (t)
dt
t
(15)
is in the class S.
Using Theorem 2 and Schwarz Lemma, we prove:
Theorem 5. Let αi ∈ C∗ , Mi ≥ 1, Ni ≥ 1, for all i = 1, ..., n and δ ∈ C with
Reδ ≥
n
X
1
[|αi − 1|(2Mi + 1) + (bi + 1)(2Ni + 1) + bi ]
|αi |
i=1
(16)
UNIVALENCE CONDITIONS FOR A GENERAL INTEGRAL OPERATOR
If for all i = 1, ..., n, fi ∈ A, gi ∈ Gbi ; 0 < bi ≤ 1 satisfy
z 2 f 0 (z)
z 2 g 0 (z)
i
i
<
1,
−
1
−
1
< 1, z ∈ U
[fi (z)]2
[gi (z)]2
218
(17)
and
|fi (z)| ≤ Mi , |gi (z)| ≤ Ni , z ∈ U, i = 1, ...n,
then for any complex number γ with Reγ ≥ Reδ, the integral operator
Iγ,αi (f1 , ...fn , g1 , ...gn ) defined by (3) is in the class S.
Proof. From the proof of Theorem 4, we have:
00 X
0
n zh (z) |αi − 1| zfi0 (z)
≤
+ 1 (bi + 1) zgi (z) − 1 + 1 bi
−
1
|αi |
gi (z)
|αi |
h0 (z) |αi | fi (z)
i=1
(18)
Thus, we obtain:
1 − |z|2Reδ zh00 (z) h0 (z) Reδ
≤
0
n 1 − |z|2Reδ X |αi − 1| zfi0 (z)
+ 1 (bi + 1) zgi (z) − 1 + 1 bi
−
1
|αi |
gi (z)
|αi |
Reδ
|αi | fi (z)
i=1
n 1 − |z|2Reδ X |αi − 1| z 2 fi0 (z) fi (z) ≤
+
1
+
[fi (z)]2 z Reδ
|αi |
i=1
2 0 z gi (z) gi (z) 1
1
(bi + 1) +1 +
bi
+
|αi |
[gi (z)]2 z |αi |
Since |fi (z)| ≤ Mi , |gi (z)| ≤ Ni , z ∈ U, i = 1, ...n and each fi , each gi satisfy
conditions (17) for all i = 1, ..., n, then applying general Schwarz Lemma, we
have:
1 − |z|2Reδ zh00 (z) h0 (z) ≤
Reδ
n 1 − |z|2Reδ X |αi − 1| z 2 fi0 (z)
Mi + Mi + 1 +
−
1
[fi (z)]2
Reδ
|αi |
i=1
2 0
z gi (z)
1
1
(bi + 1) − 1 Ni + Ni + 1 +
bi
+
|αi |
[gi (z)]2
|αi |
n 1 X |αi − 1|
1
1
≤
(2Mi + 1) +
(bi + 1)(2Ni + 1) +
bi
Reδ i=1
|αi |
|αi |
|αi |
UNIVALENCE CONDITIONS FOR A GENERAL INTEGRAL OPERATOR
≤
219
n
1 X 1
[|αi − 1|(2Mi + 1) + (bi + 1)(2Ni + 1) + bi ] , z ∈ U,
Reδ i=1 |αi |
and from the hypothesis of the Theorem 5, we get:
1 − |z|2Reδ zh00 (z) h0 (z) ≤ 1, z ∈ U
Reδ
Applying Theorem 2 for the function h, we prove that
Iγ,αi (f1 , ..., fn , g1 , ...gn ) ∈ S.
Let αi = 1 for all i = 1, ..., n in Theorem 5, we have:
Corollary 3. Let Ni ≥ 1, for all i = 1, ..., n and δ ∈ C with
Reδ ≥
n
X
[(bi + 1)(2Ni + 1) + bi ]
(19)
i=1
If for all i = 1, ..., n, gi ∈ Gbi ; 0 < bi ≤ 1 satisfy
2 0
z gi (z)
< 1, z ∈ U
−
1
[gi (z)]2
(20)
and |gi (z)| ≤ Ni , z ∈ U, i = 1, 2, ..., n, then for any complex number γ with
Reγ ≥ Reδ, the integral operator Iγ (g1 , ..., gn ) defined by (12) is in the class
S.
Let n = 1, α1 = α, b1 = b, M1 = M , N1 = N and f1 = f , g1 = g in
Theorem 5, we have:
Corollary 4. Let α ∈ C∗ , M ≥ 1, N ≥ 1 and δ ∈ C with
Reδ ≥
1
[|α − 1|(2M + 1) + (b + 1)(2N + 1) + b]
|α|
If f ∈ A, g ∈ Gb ; 0 < b ≤ 1 satisfy
2 0
2 0
z f (z)
< 1, z g (z) − 1 < 1, z ∈ U
−
1
[f (z)]2
[g(z)]2
(21)
(22)
and
|f (z)| ≤ M, |g(z)| ≤ N, z ∈ U,
then for any complex γ with Reγ ≥ Reδ, the integral operator Iγ,α defined by
(15) is in the class S.
UNIVALENCE CONDITIONS FOR A GENERAL INTEGRAL OPERATOR
220
Theorem 6. Let αi ∈ C∗ for all i = 1, 2, ...n and γ ∈ C with
Reγ ≥
n
X
1
(|αi − 1| + 2bi + 1)
|αi |
(23)
i=1
and let c ∈ C be such that:
|c| ≤ 1 −
n
1 X 1
(|αi − 1| + 2bi + 1)
Reγ i=1 |αi |
If for all i = 1, ...n, fi ∈ A, gi ∈ Gbi ; 0 < bi ≤ 1, and
0
0
zfi (z)
zgi (z)
fi (z) − 1 < 1, gi (z) − 1 < 1 z ∈ U
(24)
(25)
then the integral operator Iγ,αi (f1 , ..., fn , g1 , ...gn ) defined by (3) is in the class
S.
Proof. From (9) we deduce that
00 00 2γ c |z|2γ + 1 − |z|2γ zh (z) ≤ |c| + 1 − |z| zh (z) 0
0
γh (z)
γ
h (z) n
1 − |z|2γ X 1
≤ |c| + (|αi − 1| + 2bi + 1)
γ
|αi |
i=1
n
≤ |c| +
≤ |c| +
1 X 1
(|αi − 1| + 2bi + 1)
|γ| i=1 |αi |
n
1 X 1
(|αi − 1| + 2bi + 1) ≤ 1
Reγ i=1 |αi |
Finally, by applying Theorem 3, we get that Iγ,αi (f1 , ..., fn , g1 , ..., gn ) ∈ S
Let αi = 1 for all i = 1, ..., n in Theorem 6, we have
Corollary 5. Let γ ∈ C∗ with
Reγ ≥
n
X
(2bi + 1)
(26)
n
1 X
(2bi + 1)
Reγ i=1
(27)
i=1
and let c ∈ C be such that
|c| ≤ 1 −
UNIVALENCE CONDITIONS FOR A GENERAL INTEGRAL OPERATOR
If for all i = 1, ..., n, gi ∈ Gbi ; 0 < bi ≤ 1 and
0
zgi (z)
< 1, z ∈ U
−
1
gi (z)
221
(28)
then the integral operator Iγ (g1 , g2 , ..., gn ) defined by (12) is in the class S.
Let n = 1, α1 = α, b1 = b and f1 = f , g1 = g in Theorem 6, then we have:
Corollary 6. Let α ∈ C∗ and γ ∈ C with
Reγ ≥
1
(|α − 1| + 2b + 1)
|α|
(29)
and let c ∈ C be such that
|c| ≤ 1 −
1 1
(|α − 1| + 2b + 1)
Reγ |α|
If f ∈ A, g ∈ Gb ; 0 < b ≤ 1 satisfy
0
zf (z)
< 1,
−
1
f (z)
0
zg (z)
<1z∈U
−
1
g(z)
(30)
(31)
then the integral operator Iγ,α (f1 , ..., fn , g1 , ...gn ) defined by (15) is in the class
S.
Theorem 7. Let αi ∈ C∗ , Mi ≥ 1, Ni ≥ 1 for all i = 1, ..., n and γ ∈ C with
Reγ ≥
n
X
1
(|αi − 1|(2Mi + 1) + (bi + 1)(2Ni + 1) + bi )
|αi |
(32)
i=1
and let c ∈ C be such that
|c| ≤ 1 −
n
1 X 1
(|αi − 1|(2Mi + 1) + (bi + 1)(2Ni + 1) + bi )
Reγ i=1 |αi |
If for all i = 1, ..., n, fi ∈ A, gi ∈ Gbi ; 0 < bi ≤ 1 satisfy
2 0
2 0
z fi (z)
z gi (z)
[fi (z)]2 − 1 < 1, [gi (z)]2 − 1 < 1, z ∈ U
(33)
(34)
then the integral operator Iγ,αi (f1 , ..., fn , g1 , ..., gn ) defined by (3) is in the class
S.
UNIVALENCE CONDITIONS FOR A GENERAL INTEGRAL OPERATOR
222
Proof. From the proof of the Theorem 5 we have
00 c |z|2γ + 1 − |z|2γ zh (z) ≤
γh0 (z) n
1 X 1
|c| +
(|αi − 1|(2Mi + 1) + (bi + 1)(2Ni + 1) + bi ) , z ∈ U
Reγ i=1 |αi |
which, from the hypothesis of the Theorem 7, we get:
00 c |z|2γ + 1 − |z|2γ zh (z) ≤ 1
γh0 (z) Applying Theorem 6, for the function h, we prove that
Iγ,αi (f1 , ..., fn , g1 , ..., gn ) ∈ S.
Let αi = 1 for all i = 1, ..., n in Theorem 4, we have:
Corollary 7. Let Ni ≥ 1 for all i = 1, ..., n and γ ∈ C with
Reγ ≥
n
X
[(bi + 1)(2Ni + 1) + bi ]
(35)
i=1
and let c ∈ C be such that
|c| ≤ 1 −
n
1 X
[(bi + 1)(2Ni + 1) + bi ]
|Reγ i=1
If for all i = 1, ..., n, gi ∈ Gbi ; 0 < bi ≤ 1, satisfy
z 2 g 0 (z)
i
− 1 < 1, z ∈ U,
2
[gi (z)]
(36)
(37)
then the integral operator Iγ (g1 , ..., gn ) defined by(12) is in the class S.
Let n = 1, α1 = α, b1 = b, M1 = M and f1 = f , g1 = g in Theorem 7,
then we have
Corollary 8. Let α ∈ C∗ , M ≥ 1, N ≥ 1 and γ ∈ C with
Reγ ≥
1
[|α − 1|(2M + 1) + (b + 1)(2N + 1) + b]
|α|
(38)
223
UNIVALENCE CONDITIONS FOR A GENERAL INTEGRAL OPERATOR
and let c ∈ C be such that
|c| ≤ 1 −
1
[|α − 1|(2M + 1) + (b + 1)(2N + 1) + b]
Reγ
If f ∈ A, g ∈ Gb ;0 < b ≤ 1 satisfy:
2 0
z f (z)
[f (z)]2 − 1 < 1,
(39)
2 0
z g (z)
[g(z)]2 − 1 < 1 z ∈ U,
(40)
then the integral operator Iγ,α (f, g) defined by (15) is in the class S.
Remark 1. If βi =
the integral operator
1
αi
and gi = fi , for all i = 1, ..., n, in relation (3), we get
Iγβi (f1 , ...fn , g1 , ...gn )(z) =
( Z
γ
0
z
tγ−1
f1 (t)
t
1−β1
(f10 (t))
β1
....
fn (t)
t
1−βn
) γ1
βn
(fn0 (t))
dt
,
studied by B. A. Frasin in [3].
and gi0 (t) = 1 for all i = 1, ..., n, in relation (3),
βi γ1
Rz
Qn we get the integral operator G(z) = γ 0 tγ−1 i=1 fit(t)
dt , z ∈ U
Remark 2. For βi = 1 −
1
αi
studied by D. Breaz and N. Breaz in [1].
Remark 3. If instead of 1 − α1i we consider αi , gi0 (t) = 1, and if instead of
fi (t) we have Dn,γ
λ fi (t) for all i = 1, ..., n in relation (3), we get the integral
operator
Iβn,γ (f1 , ..., fm ) : Am → A,
Iβn,γ (f1 , f2 , ..., fm )(z)
( Z
= β
0
z
t
β−1
α
m n,γ
Y
D fi (t) i
λ
i=1
t
) β1
dt
, z ∈ U,
where Dn,γ
is the generalized Al-Oboudi differenatial operator, defined and
λ
studied by S. Bulut in [2].
Acknowledgment
This work was partially supported by the strategic project PERFORM,
POSDRU 159/1.5/S/138963, inside POSDRU Romania 2014, co-financed by
the European Social Fund-Investing in People.
UNIVALENCE CONDITIONS FOR A GENERAL INTEGRAL OPERATOR
224
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Bolyai, Mathematica, Cluj Napoca, 3 (2002),13-21 .
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[3] Frasin, B.A.,Univalence criteria for general integral operator, Mathematical Communications, 16 (2011),115-124.
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[5] Pascu, N.N., On a univalence criterion II, Itinerant Seminar on
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[7] Pescar, V., A new generalization of Ahlfors’s and Becker’s criterion of
univalence, Bull. Malaysian Math. Soc. (2) 19 (1996), no.2, 53-54.
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75-79.
Adriana OPREA,
Department of Mathematics,
University of Piteşti,
Târgul din Vale Str., No. 1, 110040, Piteşti, Argeş, Romania.
Email: adriana oprea@yahoo.com
Daniel BREAZ ,
Department of Mathematics,
”1 Decembrie 1918” University of Alba Iulia,
N. Iorga Str., No 11-13, 510009 Alba Iulia, Alba, Romania.
Email: dbreaz@uab.ro