Surveys in Mathematics and its Applications
ISSN 1842-6298 (electronic), 1843-7265 (print)
Volume 7 (2012), 117 – 124
SOME CRITERIA FOR TWO NEW INTEGRAL
OPERATORS
Laura Stanciu and Daniel Breaz
Abstract. Using the Hadamard product, we define two new integral operators. The main
object of the present paper is to discuss some univalence conditions for these operators. Several
corollaries of the main results are also considered.
1
Introduction
Let A denote the class of functions of the form
∞
X
f (z) = z +
an z n ,
n=2
which are analytic in the open unit disk
U = {z : z ∈ C
and
|z| < 1}
and satisfy the following usual normalization condition
f (0) = f 0 (0) − 1 = 0,
C being the set of complex numbers.
We denote by S the subclass of A consisting of functions f (z) which are univalent
in U.
For two functions, f (z) ∈ A and g(z) given by
g(z) = z +
∞
X
bn z n ,
(1.1)
n=2
their Hadamard product (or convolution) is defined by
(f ∗ g)(z) = z +
∞
X
an bn z n .
(1.2)
n=2
2010 Mathematics Subject Classification: Primary 30C45; Secondary 30C75.
Keywords: Analytic functions; Univalence conditions; Integral Operators; General Schwarz
Lemma.
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118
L. Stanciu and D. Breaz
For a function g ∈ A defined by (1.1), where bn ≥ 0, (n ≥ 2), we define the family
S(g, p) so that it consists of functions f ∈ A satisfying the condition
2
z (f ∗ g)0 (z)
≤ p |z|2
−
1
(z ∈ U; 0 < p ≤ 2) ,
(1.3)
[(f ∗ g)(z)]2
provided that (f ∗ g)(z) 6= 0.
z
Note that S( 1−z
, p) = S(p), where the class S(p), 0 < p ≤ 2 of analytic and univalent
functions was defined by D. Yang and J. Liu [4].
Also, if f ∈ S(p) then the following property is true
2 0
z f (z)
≤ p |z|2
−
1
(z ∈ U)
(1.4)
[f (z)]2
relation proved in [3].
Using the Hadamard product defined by (1.2), we define two families of integral
operators:
n
zY
Z
Fn ,α (f, g)(z) =
(1 + n(α − 1))
!
α−1
((fi ∗ egi )(t))
1
1+n(α−1)
dt
(1.5)
0 i=1
αi ∈ C; fi , gi ∈ A for all i ∈ {1, 2, ..., n}.
Z
Gn ,β (f, g)(z) =
β
z
t
β−nα−1
0
n
Y
!1
β
α
((fi ∗ gi )(t)) dt
(1.6)
i=1
αi ∈ C; fi , gi ∈ A; M ≥ 1 for all i ∈ {1, 2, ..., n}.
In the present paper, we study the univalence conditions involving the families of
integral operators defined by (1.5) and (1.6).
In the proof of our main results (Theorem 3 and Theorem 6) we need the following
univalence criterion. The univalence criterion, asserted by Theorem 1 below, was
proven by Pescar [2].
Theorem 1. [2] Let α ∈ C with Reα > 0, c ∈ C with |c| ≤ 1, c 6= −1. If f ∈ A
satisfies
00 c |z|2α + 1 − |z|2α zf (z) ≤ 1,
αf 0 (z) for all z ∈ U then the integral operator
Z
Fα (z) = α
z
1
α
f (t)dt
α−1 0
t
0
is in the class S.
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Surveys in Mathematics and its Applications 7 (2012), 117 – 124
http://www.utgjiu.ro/math/sma
119
Some criteria for two new integral operators
Finally, in our present investigation, we shall also need the familiar Schwarz
Lemma (see, for details, [1]).
Lemma 2. (General Schwarz Lemma) [1] Let the function f be regular in the
disk UR = {z ∈ C : |z| < R}, with |f (z)| < M for fixed M . If f has one zero with
multiplicity order bigger than m for z = 0, then
|f (z)| ≤
M
|z|m
m
R
(z ∈ UR ) .
The equality can hold only if
f (z) = eiθ
M m
z ,
Rm
where θ is constant.
2
Main Results
Theorem 3. Let fi ∈ A for all i ∈ {1, 2, ..., n}, α ∈ C and Mi ≥ 1 with
n
α − 1 X
|c| ≤ 1 − [(pi + 1)Mi + 1],
α c ∈ C, c 6= −1.
(2.1)
i=1
If for all i ∈ {1, 2, ..., n}, fi ∈ S(gi , pi ), 0 < pi ≤ 2 and
|(fi ∗ egi )(z)| ≤ Mi ,
(z ∈ U)
(2.2)
then the integral operator Fn ,α (f, g)(z) defined by (1.5) is in the class S.
Proof. We begin by observing that the integral operator Fn ,α (f, g)(z) in (1.5) can
be rewritten as follows:
Z
Fn ,α (f, g)(z) =
z
(1 + n(α − 1))
t
n(α−1)
0
n Y
(fi ∗ egi )(t) α−1
i=1
t
!
dt
1
1+n(α−1)
.
Let us define the function h(z) by
Z
h(z) =
n zY
0 i=1
(fi ∗ egi )(t)
t
α−1
dt
fi , gi ∈ A for all i ∈ {1, 2, ..., n}.
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Surveys in Mathematics and its Applications 7 (2012), 117 – 124
http://www.utgjiu.ro/math/sma
120
L. Stanciu and D. Breaz
The function h(z) is indeed regular in U and satisfies the following usual normalization
condition:
h(0) = h0 (0) − 1 = 0.
Now, calculating the derivatives of h(z) of the first and second orders, we readily
obtain
n Y
(fi ∗ egi )(z) α−1
0
h (z) =
z
i=1
and
n
X
zh00 (z)
=
(α
−
1)
h0 (z)
i=1
z(fi ∗ egi )0 (z)
−1 ,
(fi ∗ egi )(z)
(z ∈ U)
which readily shows that
00 c |z|2α + 1 − |z|2α zh (z) αh0 (z) n X
z(fi ∗ egi )0 (z)
2α
2α α − 1
−1 = c |z| + 1 − |z|
α
(fi ∗ egi )(z)
i=1
n α − 1 X
z(fi ∗ egi )0 (z) +1
≤ |c| + (fi ∗ egi )(z) α i=1
n 2
z (fi ∗ egi )0 (z) (fi ∗ egi )(z) α − 1 X
+1 .
≤ |c| + [(fi ∗ egi )(z)]2 α z
(2.3)
i=1
Furthermore, from the hypothesis (2.2) of Theorem 3, we have
|(fi ∗ egi )(z)| ≤ Mi
(z ∈ U)
then by General Schwarz Lemma, we thus obtain
|(fi ∗ egi )(z)| ≤ Mi |z|
(z ∈ U)
for all i ∈ {1, 2, ..., n}.
Next, by making use of (2.3), we have
00 c |z|2α + 1 − |z|2α zh (z) αh0 (z) n 2
z (fi ∗ egi )0 (z)
α − 1 X
≤ |c| + [(fi ∗ egi )(z)]2 − 1 + 1 Mi + 1
α i=1
n
α − 1 X
≤ |c| + [(pi + 1) Mi + 1]
α i=1
≤1
(z ∈ U, c ∈ C, Mi ≥ 1)
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Surveys in Mathematics and its Applications 7 (2012), 117 – 124
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121
Some criteria for two new integral operators
where we have also used the hypothesis (2.1) of Theorem 3.
Finally, by applying Theorem 1, we conclude that the integral operator Fn ,α (f, g)(z)
defined by (1.5) is in the class S.
Setting g1 = g2 = ... = gn = 0 in Theorem 3 we have
Corollary 4. Let fi ∈ A for all i ∈ {1, 2, ..., n}, α ∈ C and Mi ≥ 1 with
n
α − 1 X
|c| ≤ 1 − [(pi + 1)Mi + 1],
c ∈ C, c 6= −1.
α i=1
If for all i ∈ {1, 2, ..., n}, fi ∈ S(pi ), 0 < pi ≤ 2 satisfy the condition (1.4) and
|fi (z)| ≤ Mi
(z ∈ U)
then the integral operator
Z
Fn ,α (f )(z) =
(1 + n(α − 1))
n zY
0 i=1
fi (t)
t
!
α−1
1
1+n(α−1)
dt
is in the class S.
Setting n = 1 and M = 1 in Corollary 4 we have
Corollary 5. Let f ∈ A, α ∈ C and
α − 1
(p + 2),
|c| ≤ 1 − α c ∈ C, c 6= −1.
If f ∈ S(p), 0 < p ≤ 2 satisfies the condition (1.4) and
|f (z)| ≤ 1
(z ∈ U)
then the integral operator
z
Z
Fα (f )(z) =
α
0
f (t)
t
!1
α−1
α
dt
is in the class S.
Theorem 6. Let fi ∈ A, α ∈ C and Mi ≥ 1 for all i ∈ {1, 2, ..., n}. If for all
i ∈ {1, 2, ..., n}, fi ∈ S(gi , pi ), 0 < pi ≤ 2 and
|(fi ∗ gi )(z)| ≤ Mi
(z ∈ U)
(2.4)
[(pi + 1)Mi + 1]
(2.5)
then for any complex number β,
Reβ ≥ |α|
n
X
i=1
the integral operator Gn ,β (f, g)(z) defined by (1.6) is in the class S.
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Surveys in Mathematics and its Applications 7 (2012), 117 – 124
http://www.utgjiu.ro/math/sma
122
L. Stanciu and D. Breaz
Proof. We define a function
Z
h(z) =
n zY
0 i=1
(fi ∗ gi )(t)
t
α
dt
(2.6)
and we obtain that h(0) = h0 (0) − 1 = 0. We calculate the derivatives of the first
and second orders. From (2.6) we have
n Y
(fi ∗ gi )(z) α
0
h (z) =
z
i=1
and
n
X
zh00 (z)
=α
0
h (z)
i=1
z(fi ∗ gi )0 (z)
−1
(fi ∗ gi )(z)
which readily shows that
00 (z) zh
2β
2β
c |z| + (1 − |z| )
βh0 (z) n 0 (z)
X
α
z(f
∗
g
)
i
i
= c |z|2β + (1 − |z|2β )
−1 β
(fi ∗ gi )(z)
i=1
n
|α| X z(fi ∗ gi )0 (z) ≤ |c| +
(fi ∗ gi )(z) + 1
|β|
i=1
n |α| X z 2 (fi ∗ gi )0 (z) (fi ∗ gi )(z) ≤ |c| +
+1 .
[(fi ∗ gi )(z)]2 |β|
z
i=1
Since |(fi ∗ gi )(z)| ≤ Mi , z ∈ U and fi ∈ S(gi , pi ), 0 < pi ≤ 2 for all i ∈ {1, 2, ..., n},
then from General Schwarz Lemma and (1.3), we obtain
00 (z) zh
2β
2β
c |z| + (1 − |z| )
βh0 (z) n
|α| X
≤ |c| +
|β|
i=1
≤ |c| +
n
|α| X
|β|
2
z (fi ∗ gi )0 (z)
[(fi ∗ gi )(z)]2 − 1 + 1 Mi + 1
[(pi + 1) Mi + 1]
(z ∈ U)
i=1
which, in the light of the hypothesis (2.5), we have
00
c |z|2β + (1 − |z|2β ) zh (z) ≤ 1
βh0 (z) (z ∈ U) .
Applying Theorem 1 for the function h(z), we obtain that the integral operator
Gn ,β (f, g)(z) defined by (1.6) is in the class S.
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Surveys in Mathematics and its Applications 7 (2012), 117 – 124
http://www.utgjiu.ro/math/sma
123
Some criteria for two new integral operators
Setting g1 = g2 = ... = gn = 0 in Theorem 6, we have
Corollary 7. Let fi ∈ A, α ∈ C and Mi ≥ 1 for all i ∈ {1, 2, ..., n}. If for all
i ∈ {1, 2, ..., n}, fi ∈ S(pi ), 0 < pi ≤ 2 satisfy the inequality (1.4) and
|fi (z)| ≤ Mi
(z ∈ U)
then for any complex number β,
Reβ ≥ |α|
n
X
[(pi + 1)Mi + 1]
i=1
the integral operator
Z
Gn ,β (f, g)(z) =
z
tβ−nα−1
β
0
n
Y
!1
β
(fi (t))α dt
i=1
is in the class S.
Setting n = 1 and M = 1 in Corollary 7, we have
Corollary 8. Let f ∈ A, α ∈ C and M ≥ 1. If f ∈ S(p), 0 < p ≤ 2 satisfies the
inequality (1.4) and
|f (z)| ≤ M
(z ∈ U)
then for any complex number β,
Reβ ≥ |α| [(p + 1)M + 1]
the integral operator
Z
G1 ,β (f, g)(z) = β
z
β−α−1
t
1
β
(f (t)) dt
α
0
is in the class S.
Acknowledgement. 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.
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Surveys in Mathematics and its Applications 7 (2012), 117 – 124
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124
L. Stanciu and D. Breaz
References
[1] Z. Nehari, Conformal mapping, McGraw-Hill Book Company, New York, 1952.
MR45823(13,640h)30.0X. Zbl 0048.31503.
[2] V. Pescar, A new generalization of Ahlfors’s and Becker’s criterion of univalence,
Bull. Malaysian Math. Soc. (Second Series) 19 (1996), 53-54. Zbl 0880.30020.
[3] V. Singh, On a class of univalent functions, Internat. J. Math. Math. Sci.
23(2000), 855-857. MR1769871(2001b:30025). Zbl 0962.30004.
[4] D. Yang and J. Liu, On a class of univalent functions, Int. J. Math. Math. Sci.
22(3)(1999), 605-610. MR1717183(2000h:30025). Zbl 0963.30008.
Laura Stanciu
Daniel Breaz
Department of Mathematics,
Department of Mathematics,
University of Piteşti,
”1 Decembrie 1918” University of Alba Iulia,
Târgul din Vale Str., No.1, 110040, Piteşti,
Alba Iulia, Str. N. Iorga, 510000, No. 11-13,
Argeş, România.
România.
e-mail: laura stanciu 30@yahoo.com
e-mail: dbreaz@uab.ro
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Surveys in Mathematics and its Applications 7 (2012), 117 – 124
http://www.utgjiu.ro/math/sma