Paper-18-22-2010. 112KB Jun 04 2011 12:03:11 AM
Acta Universitatis Apulensis
ISSN: 1582-5329
No. 22/2010
pp. 207-213
INTEGRAL OPERATORS ON A CERTAIN CLASS OF UNIVALENT
FUNCTIONS
Virgil Pescar
Abstract. In
work is considered the class T2 of univalent functions defined
2this
z f ′ (z)
by the condition f 2 (z) − 1 < 1 for |z| < 1, where f (z) = z +a3 z 3 +... is analytic in
the open unit disk U = {z ∈ C : |z| < 1}. The integral operators Gγ , Jγ , Jγ1 ,γ2 ,...,γn ,
Dα,β , Lα,β , Kγ1 ,γ2 ,...,γn and Hγ1 ,γ2 ,...,γn ,β,δ , for the functions f ∈ T2 are considered.
In the present paper we obtain univalence conditions of these integral operators.
2000 Mathematics Subject Classification: 30C45.
Key words and phrases: Integral operator, univalence.
1. Introduction
Let A be the class of the functions f which are analitic in the open unit disk
U = {z ∈ C : |z| < 1} and f (0) = f ′ (0) − 1 = 0.
We denote by S the class of the functions f ∈ A which are univalent in U.
We consider the integral operators
Gγ (z) =
Zz
f (u)
u
1
γ
(1.1)
du,
0
Jγ (z) =
1
γ
Zz
0
1
γ
γ
u−1 (f (u)) du ,
n1
P 1
Zz
n
n
1
X
Y
γ
1
j=1 j
−1
γj
Jγ1 ,γ2 ,...,γn (z) =
,
u
(fj (u)) du
γj
j=1
j=1
0
207
(1.2)
(1.3)
V. Pescar - Integral operators on a certain class of univalent functions
Dα,β (z) = β
Lα,β (z) = β
Zz
u
β−1
f (u)
u
1
β
1
α
0
Zz
u
n
Y
fj (u) α
u
j=1
0
Kγ1 ,γ2 ,...,γn (z) =
(1.5)
du ,
1
Zz Y
n
fj (u) γj
u
0 j=1
1
β
1
β−1
(1.4)
du ,
(1.6)
du,
for f ∈ A, α, β, γ complex numbers, α 6= 0, β 6= 0, γ 6= 0 and fj ∈ A, γj complex
numbers, γj 6= 0, j = 1, n.
In [1], [2], [4], [7], [8], [9], [10], [11] we have certain the univalence conditions of
these integral operators.
We define a general integral operator
Hγ1 ,γ2 ,...,γn ,β,δ (z) = βδ
Zz
1
uβδ−1
n
Y
fj (u) γj
j=1
0
u
du
1
βδ
,
(1.7)
for fj ∈ A, β, δ, γj complex numbers, βδ 6= 0, γj 6= 0, j = 1, n, n ∈ N − {0}.
For β, δ, γj , n ∈ N − {0}, j = 1, n, in the particular cases, from (1.7) we obtain
the integral operators Gγ , Jγ , Jγ1 ,γ2 ,...,γn , Dα,β , Lα,β , Kγ1 ,γ2 ,...,γn .
2. Preliminary results
We need the following theorems.
Theorem 2.1. [6].Let α be a complex number, Reα > 0 and f ∈ A. If
1 − |z|2Reα zf ′′ (z)
f ′ (z) ≤ 1,
Reα
(2.1)
for all z ∈ U, then for any complex number β, Reβ ≥ Reα, the function
Fβ (z) = β
Zz
1
β
u
β−1 ′
0
208
f (u) du
(2.2)
V. Pescar - Integral operators on a certain class of univalent functions
is in the class S.
Theorem 2.2. (Schwarz [3]). Let f be the function regular in the disk UR =
{z ∈ C : |z| < R} with |f (z)| < M , M fixed. If f (z) has in z = 0 one zero with
multiply ≥ m, then
M
|z|m , (z ∈ UR ) ,
Rm
the equality (in the inequality (2.3) for z 6= 0) can hold only if
|f (z)| ≤
f (z) = eiθ
(2.3)
M m
z ,
Rm
where θ is constant.
Theorem 2.3. [5]. Assume that the function f ∈ A satisfies the condition
2 ′
z f (z)
< 1, (z ∈ U),
(2.4)
−
1
f 2 (z)
then the function f is univalent in U.
3. Main results
Theorem 3.1. Let γj , α complex numbers, γj 6= 0, j = 1, n, Reα > 0, Mj
∞
P
akj z k , j = 1, n, n ∈ N − {0}.
positive real numbers and fj ∈ T2 , fj (z) = z +
k=3
If
|fj (z)| ≤ Mj , (j = 1, n; z ∈ U)
(3.1)
n
X
2Mj + 1
≤ Reα,
|γj |
(3.2)
and
j=1
then for any complex numbers β and δ, Reβδ ≥ Reα, the function
1
1
1
Zz
βδ
γ
γ
n
1
fn (u)
βδ−1 f1 (u)
Hγ1 ,γ2 ,...,γn ,β,δ (z) = βδ u
...
du
u
u
0
209
(3.3)
V. Pescar - Integral operators on a certain class of univalent functions
is in the class S.
Proof. We consider the function
h(z) =
Zz
f1 (u)
u
1
γ1
...
fn (u)
u
1
γn
du.
(3.4)
0
The function h is regular in U.
We have
′′
n
zh (z) X
1 zfj′ (z)
=
− 1 , (z ∈ U).
h′ (z)
|γj | fj (z)
(3.5)
j=1
We obtain
′
2 ′
z fj (z) fj (z)
zfj (z)
fj (z) − 1 ≤ f 2 (z) z + 1 ≤
j
z 2 f ′ (z)
|f (z)| |f (z)|
j
j
j
+
+ 1, (j = 1, n; z ∈ U).
− 1
≤ 2
fj (z)
|z|
|z|
Since fj ∈ T2 and by Theorem 2.2, from (3.6) we get
′
zfj (z)
fj (z) − 1 ≤ 2Mj + 1 j = 1, n; z ∈ U .
(3.6)
(3.7)
From (3.5) and (3.7) we obtain
n
1 − |z|2Reα zh′′ (z) 1 − |z|2Reα X 2Mj + 1
, (z ∈ U)
≤
h′ (z)
Reα
Reα
|γj |
(3.8)
j=1
and hence, by (3.2) we have
1 − |z|Reα zh′′ (z)
≤ 1,
Reα h′ (z)
for all z ∈ U.
So, by Theorem 2.1, the integral operator Hγ1 ,γ2 ,...,γn ,β,δ is in the class S.
210
(3.9)
V. Pescar - Integral operators on a certain class of univalent functions
Corollary 3.2. Let γj , α complex numbers, Reγj 6= 0, j = 1, n,
n
P
j=1
Re γ1j ≥
Reα > 0, Mj positive real numbers and fj ∈ T2 , fj (z) = z + a3j z 3 + ..., j = 1, n,
n ∈ N − {0}.
If
|fj (z)| ≤ Mj , (j = 1, n; z ∈ U)
(3.10)
n
X
2Mj + 1
≤ Reα,
|γj |
(3.11)
and
j=1
then the integral operator Jγ1 ,γ2 ,...,γn given by (1.3) is in the class S.
Proof. For βδ =
n
P
j=1
1
γj
from Theorem 3.1 we obtain Corollary 3.2.
Remark 3.3. From Corollary 3.2, for n = 1, γ1 = γ, f1 = f , we obtain the
integral operator Jγ defined by (1.2) is in the class S.
Corollary 3.4. Let γj , α complex numbers, γj 6= 0, j = 1, n, 0 < Reα ≤ 1, Mj
positive real numbers and fj ∈ T2 , fj (z) = z + a3j z 3 + ..., j = 1, n, n ∈ N − {0}.
If
|fj (z)| ≤ Mj , j = 1, n, (z ∈ U)
(3.12)
and
n
X
2Mj + 1
≤ Reα,
|γj |
(3.13)
j=1
then the function
Kγ1 ,γ2 ,...,γn (z) =
Zz
f1 (u)
u
1
γ1
...
fn (u)
u
1
γn
du
(3.14)
0
is in the class S.
Proof. For βδ = 1, from Theorem 3.1, we obtain Corollary 3.4.
Remark 3.5. If we take n = 1, γ1 = γ, f1 = f , from Corollary 3.4 we have the
integral operator Gγ given by (1.1) is in the class S.
211
V. Pescar - Integral operators on a certain class of univalent functions
Corollary 3.6. Let α, γ complex numbers, α 6= 0, Reγ > 0, Mj positive real
numbers and fj ∈ T2 , fj (z) = z + a3j z 3 + ..., j = 1, n, n ∈ N − {0}.
If
|fj (z)| ≤ Mj , (j = 1, n, z ∈ U),
(3.15)
n
X
2Mj + 1
≤ Reγ,
|α|
(3.16)
and
j=1
then for any complex number β, Reβ ≥ Reγ, the function
1
z
1
1 β
Z
f1 (u) α
fn (u) α
...
du
Lα,β (z) = β uβ−1
u
u
(3.17)
0
is in the class S.
Proof. For δ = 1, from Theorem 3.1, we have Corollary 3.6.
Remark 3.7. If take n = 1, f1 = f in Corollary 3.6, we obtain that the integral
operator Dα,β defined by (1.4) is in the class S.
References
[1] D. Breaz, N. Breaz, Two integral operators, Studia Universitatis ”Babe¸sBolyai”, Mathematica, Cluj-Napoca, No.3, 2002, 13-21.
[2] Y.J. Kim, E.P. Merkes, On an integral of powers of a spirallike function,
Kyungpook Math., J., 12 (1972), 2, 249-253.
[3] O. Mayer, The Functions Theory of One Variable Complex, Bucure¸sti, 1981.
[4] S.S. Miller, P.T. Mocanu, Differential Subordonations, Theory and Aplications, Monographs and Text Books in Pure and Applied Mathematics, 225, Marcel
Dekker, New York, 2000.
[5] S. Ozaki, M. Nunokawa, The Schwartzian derivative and univalent functions,
Proc., Amer. Math. Soc., 33 (2), (1972), 392-394.
[6] N.N. Pascu, An improvement of Becker’s Univalence Criterion, Proceedings
of the Commemorative Session Simion Stoilow, University of Bra¸sov, 1987, 43-48.
[7] N.N. Pascu, V.Pescar, On the integral operators of Kim-Mekes and Pfaltzgraff,
Mathematica, Univ. ”Babe¸s-Bolyai”, Cluj-Napoca, 32 (55), 2(1990), 185-192.
[8] V. Pescar, Univalence conditions for certain integral operators, Journal of
Inequalities in Pure and Applied Mathematics, Volume 7, Issue 4, Article 147, 2006.
212
V. Pescar - Integral operators on a certain class of univalent functions
[9] V. Pescar, Integral operators on a certain class of analytic functions, Libertas
Mathematica, Vol.XXXVII (2007), 95-98.
[10] V. Pescar, On the univalence of some integral operators, Acta Universitatis
Apulensis, Mathematics - Informatics, No. 16/2008, 157-164.
[11] V. Pescar, S. Owa, Univalence problems for integral operators by Kim-Merkes
and Pfaltzgraff, Journal of Approximation Theory and Applications, Vol.3, No.1-2,
2007, 17-21.
Virgil Pescar
Department of Mathematics
”Transilvania” University of Bra¸sov
Faculty of Mathematics and Computer Science
500091 Bra¸sov, Romania
email: [email protected]
213
ISSN: 1582-5329
No. 22/2010
pp. 207-213
INTEGRAL OPERATORS ON A CERTAIN CLASS OF UNIVALENT
FUNCTIONS
Virgil Pescar
Abstract. In
work is considered the class T2 of univalent functions defined
2this
z f ′ (z)
by the condition f 2 (z) − 1 < 1 for |z| < 1, where f (z) = z +a3 z 3 +... is analytic in
the open unit disk U = {z ∈ C : |z| < 1}. The integral operators Gγ , Jγ , Jγ1 ,γ2 ,...,γn ,
Dα,β , Lα,β , Kγ1 ,γ2 ,...,γn and Hγ1 ,γ2 ,...,γn ,β,δ , for the functions f ∈ T2 are considered.
In the present paper we obtain univalence conditions of these integral operators.
2000 Mathematics Subject Classification: 30C45.
Key words and phrases: Integral operator, univalence.
1. Introduction
Let A be the class of the functions f which are analitic in the open unit disk
U = {z ∈ C : |z| < 1} and f (0) = f ′ (0) − 1 = 0.
We denote by S the class of the functions f ∈ A which are univalent in U.
We consider the integral operators
Gγ (z) =
Zz
f (u)
u
1
γ
(1.1)
du,
0
Jγ (z) =
1
γ
Zz
0
1
γ
γ
u−1 (f (u)) du ,
n1
P 1
Zz
n
n
1
X
Y
γ
1
j=1 j
−1
γj
Jγ1 ,γ2 ,...,γn (z) =
,
u
(fj (u)) du
γj
j=1
j=1
0
207
(1.2)
(1.3)
V. Pescar - Integral operators on a certain class of univalent functions
Dα,β (z) = β
Lα,β (z) = β
Zz
u
β−1
f (u)
u
1
β
1
α
0
Zz
u
n
Y
fj (u) α
u
j=1
0
Kγ1 ,γ2 ,...,γn (z) =
(1.5)
du ,
1
Zz Y
n
fj (u) γj
u
0 j=1
1
β
1
β−1
(1.4)
du ,
(1.6)
du,
for f ∈ A, α, β, γ complex numbers, α 6= 0, β 6= 0, γ 6= 0 and fj ∈ A, γj complex
numbers, γj 6= 0, j = 1, n.
In [1], [2], [4], [7], [8], [9], [10], [11] we have certain the univalence conditions of
these integral operators.
We define a general integral operator
Hγ1 ,γ2 ,...,γn ,β,δ (z) = βδ
Zz
1
uβδ−1
n
Y
fj (u) γj
j=1
0
u
du
1
βδ
,
(1.7)
for fj ∈ A, β, δ, γj complex numbers, βδ 6= 0, γj 6= 0, j = 1, n, n ∈ N − {0}.
For β, δ, γj , n ∈ N − {0}, j = 1, n, in the particular cases, from (1.7) we obtain
the integral operators Gγ , Jγ , Jγ1 ,γ2 ,...,γn , Dα,β , Lα,β , Kγ1 ,γ2 ,...,γn .
2. Preliminary results
We need the following theorems.
Theorem 2.1. [6].Let α be a complex number, Reα > 0 and f ∈ A. If
1 − |z|2Reα zf ′′ (z)
f ′ (z) ≤ 1,
Reα
(2.1)
for all z ∈ U, then for any complex number β, Reβ ≥ Reα, the function
Fβ (z) = β
Zz
1
β
u
β−1 ′
0
208
f (u) du
(2.2)
V. Pescar - Integral operators on a certain class of univalent functions
is in the class S.
Theorem 2.2. (Schwarz [3]). Let f be the function regular in the disk UR =
{z ∈ C : |z| < R} with |f (z)| < M , M fixed. If f (z) has in z = 0 one zero with
multiply ≥ m, then
M
|z|m , (z ∈ UR ) ,
Rm
the equality (in the inequality (2.3) for z 6= 0) can hold only if
|f (z)| ≤
f (z) = eiθ
(2.3)
M m
z ,
Rm
where θ is constant.
Theorem 2.3. [5]. Assume that the function f ∈ A satisfies the condition
2 ′
z f (z)
< 1, (z ∈ U),
(2.4)
−
1
f 2 (z)
then the function f is univalent in U.
3. Main results
Theorem 3.1. Let γj , α complex numbers, γj 6= 0, j = 1, n, Reα > 0, Mj
∞
P
akj z k , j = 1, n, n ∈ N − {0}.
positive real numbers and fj ∈ T2 , fj (z) = z +
k=3
If
|fj (z)| ≤ Mj , (j = 1, n; z ∈ U)
(3.1)
n
X
2Mj + 1
≤ Reα,
|γj |
(3.2)
and
j=1
then for any complex numbers β and δ, Reβδ ≥ Reα, the function
1
1
1
Zz
βδ
γ
γ
n
1
fn (u)
βδ−1 f1 (u)
Hγ1 ,γ2 ,...,γn ,β,δ (z) = βδ u
...
du
u
u
0
209
(3.3)
V. Pescar - Integral operators on a certain class of univalent functions
is in the class S.
Proof. We consider the function
h(z) =
Zz
f1 (u)
u
1
γ1
...
fn (u)
u
1
γn
du.
(3.4)
0
The function h is regular in U.
We have
′′
n
zh (z) X
1 zfj′ (z)
=
− 1 , (z ∈ U).
h′ (z)
|γj | fj (z)
(3.5)
j=1
We obtain
′
2 ′
z fj (z) fj (z)
zfj (z)
fj (z) − 1 ≤ f 2 (z) z + 1 ≤
j
z 2 f ′ (z)
|f (z)| |f (z)|
j
j
j
+
+ 1, (j = 1, n; z ∈ U).
− 1
≤ 2
fj (z)
|z|
|z|
Since fj ∈ T2 and by Theorem 2.2, from (3.6) we get
′
zfj (z)
fj (z) − 1 ≤ 2Mj + 1 j = 1, n; z ∈ U .
(3.6)
(3.7)
From (3.5) and (3.7) we obtain
n
1 − |z|2Reα zh′′ (z) 1 − |z|2Reα X 2Mj + 1
, (z ∈ U)
≤
h′ (z)
Reα
Reα
|γj |
(3.8)
j=1
and hence, by (3.2) we have
1 − |z|Reα zh′′ (z)
≤ 1,
Reα h′ (z)
for all z ∈ U.
So, by Theorem 2.1, the integral operator Hγ1 ,γ2 ,...,γn ,β,δ is in the class S.
210
(3.9)
V. Pescar - Integral operators on a certain class of univalent functions
Corollary 3.2. Let γj , α complex numbers, Reγj 6= 0, j = 1, n,
n
P
j=1
Re γ1j ≥
Reα > 0, Mj positive real numbers and fj ∈ T2 , fj (z) = z + a3j z 3 + ..., j = 1, n,
n ∈ N − {0}.
If
|fj (z)| ≤ Mj , (j = 1, n; z ∈ U)
(3.10)
n
X
2Mj + 1
≤ Reα,
|γj |
(3.11)
and
j=1
then the integral operator Jγ1 ,γ2 ,...,γn given by (1.3) is in the class S.
Proof. For βδ =
n
P
j=1
1
γj
from Theorem 3.1 we obtain Corollary 3.2.
Remark 3.3. From Corollary 3.2, for n = 1, γ1 = γ, f1 = f , we obtain the
integral operator Jγ defined by (1.2) is in the class S.
Corollary 3.4. Let γj , α complex numbers, γj 6= 0, j = 1, n, 0 < Reα ≤ 1, Mj
positive real numbers and fj ∈ T2 , fj (z) = z + a3j z 3 + ..., j = 1, n, n ∈ N − {0}.
If
|fj (z)| ≤ Mj , j = 1, n, (z ∈ U)
(3.12)
and
n
X
2Mj + 1
≤ Reα,
|γj |
(3.13)
j=1
then the function
Kγ1 ,γ2 ,...,γn (z) =
Zz
f1 (u)
u
1
γ1
...
fn (u)
u
1
γn
du
(3.14)
0
is in the class S.
Proof. For βδ = 1, from Theorem 3.1, we obtain Corollary 3.4.
Remark 3.5. If we take n = 1, γ1 = γ, f1 = f , from Corollary 3.4 we have the
integral operator Gγ given by (1.1) is in the class S.
211
V. Pescar - Integral operators on a certain class of univalent functions
Corollary 3.6. Let α, γ complex numbers, α 6= 0, Reγ > 0, Mj positive real
numbers and fj ∈ T2 , fj (z) = z + a3j z 3 + ..., j = 1, n, n ∈ N − {0}.
If
|fj (z)| ≤ Mj , (j = 1, n, z ∈ U),
(3.15)
n
X
2Mj + 1
≤ Reγ,
|α|
(3.16)
and
j=1
then for any complex number β, Reβ ≥ Reγ, the function
1
z
1
1 β
Z
f1 (u) α
fn (u) α
...
du
Lα,β (z) = β uβ−1
u
u
(3.17)
0
is in the class S.
Proof. For δ = 1, from Theorem 3.1, we have Corollary 3.6.
Remark 3.7. If take n = 1, f1 = f in Corollary 3.6, we obtain that the integral
operator Dα,β defined by (1.4) is in the class S.
References
[1] D. Breaz, N. Breaz, Two integral operators, Studia Universitatis ”Babe¸sBolyai”, Mathematica, Cluj-Napoca, No.3, 2002, 13-21.
[2] Y.J. Kim, E.P. Merkes, On an integral of powers of a spirallike function,
Kyungpook Math., J., 12 (1972), 2, 249-253.
[3] O. Mayer, The Functions Theory of One Variable Complex, Bucure¸sti, 1981.
[4] S.S. Miller, P.T. Mocanu, Differential Subordonations, Theory and Aplications, Monographs and Text Books in Pure and Applied Mathematics, 225, Marcel
Dekker, New York, 2000.
[5] S. Ozaki, M. Nunokawa, The Schwartzian derivative and univalent functions,
Proc., Amer. Math. Soc., 33 (2), (1972), 392-394.
[6] N.N. Pascu, An improvement of Becker’s Univalence Criterion, Proceedings
of the Commemorative Session Simion Stoilow, University of Bra¸sov, 1987, 43-48.
[7] N.N. Pascu, V.Pescar, On the integral operators of Kim-Mekes and Pfaltzgraff,
Mathematica, Univ. ”Babe¸s-Bolyai”, Cluj-Napoca, 32 (55), 2(1990), 185-192.
[8] V. Pescar, Univalence conditions for certain integral operators, Journal of
Inequalities in Pure and Applied Mathematics, Volume 7, Issue 4, Article 147, 2006.
212
V. Pescar - Integral operators on a certain class of univalent functions
[9] V. Pescar, Integral operators on a certain class of analytic functions, Libertas
Mathematica, Vol.XXXVII (2007), 95-98.
[10] V. Pescar, On the univalence of some integral operators, Acta Universitatis
Apulensis, Mathematics - Informatics, No. 16/2008, 157-164.
[11] V. Pescar, S. Owa, Univalence problems for integral operators by Kim-Merkes
and Pfaltzgraff, Journal of Approximation Theory and Applications, Vol.3, No.1-2,
2007, 17-21.
Virgil Pescar
Department of Mathematics
”Transilvania” University of Bra¸sov
Faculty of Mathematics and Computer Science
500091 Bra¸sov, Romania
email: [email protected]
213