paper 07-18-2009. 104KB Jun 04 2011 12:03:06 AM
ACTA UNIVERSITATIS APULENSIS
No 18/2009
UNIVALENCE CONDITION FOR A NEW GENERALIZATION
OF THE FAMILY OF INTEGRAL OPERATORS
Serap Bulut
Abstract. In [3], Breaz et al. gave an univalence condition of the integral
operator Gn,α introduced in [2]. The purpose of this paper is to generalize
the definition of Gn,α by means of the Al-Oboudi differential operator and
investigate univalence condition of this generalized integral operator. Our
results generalize the results of [3].
2000 Mathematics Subject Classification: 30C45.
Keywords and phrases: Analytic functions, Univalent functions, Integral
operator, Differential operator, Schwarz lemma.
1.Introduction
Let A denote the class of all functions of the form
∞
X
f (z) = z +
ak z k
(1)
k=2
which are analytic in the open unit disk U = {z ∈ C : |z| < 1}, and
S = {f ∈ A : f is univalent in U} .
For f ∈ A, Al-Oboudi [1] introduced the following operator:
D0 f (z) = f (z),
(2)
D1 f (z) = (1 − δ)f (z) + δzf ′ (z) = Dδ f (z),
n
D f (z) = Dδ (D
n−1
f (z)),
δ≥0
(3)
(n ∈ N := {1, 2, 3, . . .}).
(4)
If f is given by (1), then from (3) and (4) we see that
n
D f (z) = z +
∞
X
[1 + (k − 1)δ]n ak z k ,
k=2
71
(n ∈ N0 := N ∪ {0}),
(5)
S. Bulut - Univalence condition for a new generalization of the family ...
with Dn f (0) = 0.
Remark 1. When δ = 1, we get S˘al˘agean’s differential operator [8].
The following results will be required in our investigation.
Schwarz Lemma [4]. Let the analytic function f be regular in the open
unit disk U and let f (0) = 0. If
|f (z)| ≤ 1
(z ∈ U) ,
|f (z)| ≤ |z|
(z ∈ U),
then
where the equality holds true only if
f (z) = Kz
(z ∈ U)
and
|K| = 1.
Theorem A [6]. Let
α ∈ C ( Re α > 0)
and
c ∈ C ( |c| ≤ 1; c 6= −1).
Suppose also that the function f (z) given by (1) is analytic in U. If
′′
c |z|2α + 1 − |z|2α zf (z) ≤ 1 (z ∈ U),
αf ′ (z)
then the function Fα (z) defined by
Z
Fα (z) := α
α1
f (t)dt
= z + ···
z
α−1 ′
t
0
(6)
is analytic and univalent in U.
Theoem B [5]. Let f ∈ A satisfy the following inequality:
2 ′
z f (z)
[f (z)]2 − 1 ≤ 1 (z ∈ U).
72
(7)
S. Bulut - Univalence condition for a new generalization of the family ...
Then f is univalent in U.
Theorem C [7]. Let the function g ∈ A satisfies the inequality (7). Also
let
3
α∈R
α ∈ 1,
and c ∈ C.
2
If
|c| ≤
3 − 2α
α
(c 6= −1)
and
|g(z)| ≤ 1 (z ∈ U),
then the function Gα (z) defined by
Z
Gα (z) := α
z
α−1
(g(t))
0
α1
dt
(8)
is in the univalent function class S.
In [2], Breaz and Breaz considered the integral operator
Gn,α (z) :=
[n (α − 1) + 1]
Z
z
α−1
(g1 (t))
α−1
· · · (gn (t))
0
1
n(α−1)+1
dt
(g1 , . . . , gn ∈ A)
(9)
and proved that the function Gn,α (z) is univalent in U.
Remark 2. We note that for n = 1, we obtain the integral operator in
(8).
In [3], Breaz et al. proved the following theorem.
Theorem D [3]. Let M ≥ 1 and suppose that each of the functions gj ∈
A (j ∈ {1, . . . , n}) satisfies the inequality (7). Also let
(2M + 1)n
and c ∈ C.
α∈R
α ∈ 1,
(2M + 1)n − 1
If
|c| ≤ 1 +
1−α
α
73
(2M + 1)n
S. Bulut - Univalence condition for a new generalization of the family ...
and
|gj (z)| ≤ M
(z ∈ U; j ∈ {1, . . . , n}),
then the function Gn,α (z) defined by (9) is in the univalent function class S.
Now we introduce a new general integral operator by means of the AlOboudi differential operator.
Definition 1. Let n ∈ N, m ∈ N0 and α ∈ C. We define the integral
operator Gn,m,α : An → A by
Gn,m,α (z) :=
(
[n (α − 1) + 1]
Z
0
where g1 , . . . , gn ∈ A and D
m
n
zY
α−1
(Dm gj (t))
j=1
1
) n(α−1)+1
dt
(z ∈ U),
(10)
is the Al-Oboudi differential operator.
Remark 3. In the special case n = 1, we obtain the integral operator
Z
Gm,α (z) := α
z
m
α−1
(D g(t))
0
α1
dt
(z ∈ U).
(11)
Remark 4. If we set m = 0 in (10) and (11), then we obtain the integral
operators defined in (9) and (8), respectively.
2.Main results
Theorem 1. Let M ≥ 1 and suppose that each of the functions gj ∈
A (j ∈ {1, . . . , n}) satisfies the inequality
z 2 (Dm g (z))′
j
(12)
− 1 ≤ 1 (z ∈ U; m ∈ N0 ).
2
m
(D gj (z))
Also let
α∈R
If
α ∈ 1,
|c| ≤ 1 +
(2M + 1)n
(2M + 1)n − 1
1−α
n(α − 1) + 1
74
and
(2M + 1)n
c ∈ C.
S. Bulut - Univalence condition for a new generalization of the family ...
and
|Dm gj (z)| ≤ M
(z ∈ U; j ∈ {1, . . . , n}),
then the integral operator Gn,m,α (z) defined by (10) is in the univalent function
class S.
Proof. Since gj ∈ A (j ∈ {1, . . . , n}), by (5), we have
∞
X
Dm gj (z)
=1+
[1 + (k − 1)δ]m ak,j z k−1
z
k=2
and
(m ∈ N0 )
Dm gj (z)
6= 0
z
for all z ∈ U.
Also we note that
(
Gn,m,α (z) =
[n(α − 1) + 1]
Z
z
tn(α−1)
0
Define a function
f (z) =
Z
0
Then we obtain
′
f (z) =
α−1
n m
Y
D gj (t)
t
j=1
n
zY
j=1
Dm gj (t)
t
α−1
α−1
n m
Y
D gj (z)
z
j=1
ln f (z) = (α − 1)
n
X
j=1
ln
.
Dm gj (z)
z
or equivalently
ln f ′ (z) = (α − 1)
n
X
j=1
75
dt
.
dt.
It is clear that f (0) = f ′ (0) − 1 = 0.
The equality (13) implies that
′
1
) n(α−1)+1
(ln Dm gj (z) − ln z) .
(13)
S. Bulut - Univalence condition for a new generalization of the family ...
By differentiating above equality, we get
n
X
(Dm gj (z))′ 1
f ′′ (z)
.
= (α − 1)
−
f ′ (z)
Dm gj (z)
z
j=1
Hence we obtain
n
X
z (Dm gj (z))′
zf ′′ (z)
= (α − 1)
−1 ,
f ′ (z)
Dm gj (z)
j=1
which readily shows that
zf ′′ (z)
c |z|2[n(α−1)+1] + 1 − |z|2[n(α−1)+1]
′
[n(α − 1) + 1] f (z)
X
n
′
m
z
(D
g
(z))
α
−
1
j
−1
= c |z|2[n(α−1)+1] + 1 − |z|2[n(α−1)+1]
n(α − 1) + 1 j=1
Dm gj (z)
X
n 2
z (Dm gj (z))′ Dm gj (z)
α−1
+1 .
≤ |c| +
n(α − 1) + 1 j=1 (Dm gj (z))2
z
From the hypothesis, we have |gj (z)| ≤ M (j ∈ {1, . . . , n} ; z ∈ U), then by
the Schwarz lemma, we obtain that
|gj (z)| ≤ M |z|
(j ∈ {1, . . . , n} ; z ∈ U).
Then we find
′′
zf
(z)
2[n(α−1)+1]
2[n(α−1)+1]
c |z|
+
1
−
|z|
[n(α − 1) + 1] f ′ (z)
X
n 2
z (Dm gj (z))′
α−1
M + 1
≤ |c| +
n(α − 1) + 1 j=1 (Dm gj (z))2
X
n 2
z (Dm gj (z))′
α−1
− 1 M + M + 1
≤ |c| +
n(α − 1) + 1 i=1 (Dm gj (z))2
α−1
≤ |c| +
(2M + 1)n ≤ 1
n(α − 1) + 1
1−α
since |c| ≤ 1 + n(α−1)+1
(2M + 1)n. Applying Theorem A, we obtain that
Gn,m,α is in the univalent function class S.
76
S. Bulut - Univalence condition for a new generalization of the family ...
Remark 5. If we set m = 0 in Theorem ??, then we have Theorem D.
Corollary 2. Let each of the functions gj ∈ A (j ∈ {1, . . . , n}) satisfies
the inequality (12). Suppose also that
3n
α∈R
α ∈ 1,
and c ∈ C.
3n − 1
If
|c| ≤ 1 + 3
and
1−α
n(α − 1) + 1
n
|Dm gj (z)| ≤ 1 (z ∈ U; j ∈ {1, . . . , n}),
then the integral operator Gn,m,α (z) defined by (10) is in the univalent function
class S.
Proof. In Theorem 1, we consider M = 1.
Remark 6. If we set m = 0 in Corollary 2, then we have Corollary 1 in
[3].
Corollary 3. Let M ≥ 1 and suppose that the functions g ∈ A satisfies
the inequality (12). Also let
2M + 1
and c ∈ C.
α∈R
α ∈ 1,
2M
If
|c| ≤ 1 +
and
1−α
α
|Dm g(z)| ≤ M
(2M + 1)
(z ∈ U),
then the integral operator Gm,α (z) defined by (11) is in the univalent function
class S.
Proof. In Theorem 1, we consider n = 1.
Remark 7. If we set m = 0 in Corollary 3, then we have Corollary 2 in
[3].
77
S. Bulut - Univalence condition for a new generalization of the family ...
Remark 8. If we set M = n = 1 and m = 0 in Theorem 1, then we obtain
Theorem C.
References
[1] F. M. Al-Oboudi, On univalent functions defined by a generalized S˘al˘agean
operator, Int. J. Math. Math. Sci. 2004, no. 25-28, 1429-1436.
[2] D. Breaz and N. Breaz, Univalence of an integral operator, Mathematica
(Cluj) 47 (2005), no. 70, 35-38.
[3] D. Breaz, N. Breaz and H. M. Srivastava, An extension of the univalent
condition for a family of integral operators, Appl. Math. Lett. 22 (2009), no.
1, 41-44.
[4] Z. Nehari, Conformal Mapping, Dover, New York, NY, USA, 1975.
[5] S. Ozaki and M. Nunokawa, The Schwarzian derivative and univalent
functions, Proc. Amer. Math. Soc. 33 (1972), 392-394.
[6] V. Pescar, A new generalization of Ahlfors’s and Becker’s criterion of
univalence, Bull. Malaysian Math. Soc. (Ser. 2) 19 (1996), 53-54.
[7] V. Pescar, On the univalence of some integral operators, J. Indian Acad.
Math. 27 (2005), 239-243.
[8] G. S¸. S˘al˘agean, Subclasses of univalent functions, Complex AnalysisFifth Romanian-Finnish seminar, Part 1 (Bucharest, 1981), Lecture Notes in
Math., vol. 1013, Springer, Berlin, 1983, pp. 362-372.
Author:
Serap Bulut
Civil Aviation College
Kocaeli University
Arslanbey Campus
˙
41285 Izmit-Kocaeli
Turkey
e-mail:[email protected]
78
No 18/2009
UNIVALENCE CONDITION FOR A NEW GENERALIZATION
OF THE FAMILY OF INTEGRAL OPERATORS
Serap Bulut
Abstract. In [3], Breaz et al. gave an univalence condition of the integral
operator Gn,α introduced in [2]. The purpose of this paper is to generalize
the definition of Gn,α by means of the Al-Oboudi differential operator and
investigate univalence condition of this generalized integral operator. Our
results generalize the results of [3].
2000 Mathematics Subject Classification: 30C45.
Keywords and phrases: Analytic functions, Univalent functions, Integral
operator, Differential operator, Schwarz lemma.
1.Introduction
Let A denote the class of all functions of the form
∞
X
f (z) = z +
ak z k
(1)
k=2
which are analytic in the open unit disk U = {z ∈ C : |z| < 1}, and
S = {f ∈ A : f is univalent in U} .
For f ∈ A, Al-Oboudi [1] introduced the following operator:
D0 f (z) = f (z),
(2)
D1 f (z) = (1 − δ)f (z) + δzf ′ (z) = Dδ f (z),
n
D f (z) = Dδ (D
n−1
f (z)),
δ≥0
(3)
(n ∈ N := {1, 2, 3, . . .}).
(4)
If f is given by (1), then from (3) and (4) we see that
n
D f (z) = z +
∞
X
[1 + (k − 1)δ]n ak z k ,
k=2
71
(n ∈ N0 := N ∪ {0}),
(5)
S. Bulut - Univalence condition for a new generalization of the family ...
with Dn f (0) = 0.
Remark 1. When δ = 1, we get S˘al˘agean’s differential operator [8].
The following results will be required in our investigation.
Schwarz Lemma [4]. Let the analytic function f be regular in the open
unit disk U and let f (0) = 0. If
|f (z)| ≤ 1
(z ∈ U) ,
|f (z)| ≤ |z|
(z ∈ U),
then
where the equality holds true only if
f (z) = Kz
(z ∈ U)
and
|K| = 1.
Theorem A [6]. Let
α ∈ C ( Re α > 0)
and
c ∈ C ( |c| ≤ 1; c 6= −1).
Suppose also that the function f (z) given by (1) is analytic in U. If
′′
c |z|2α + 1 − |z|2α zf (z) ≤ 1 (z ∈ U),
αf ′ (z)
then the function Fα (z) defined by
Z
Fα (z) := α
α1
f (t)dt
= z + ···
z
α−1 ′
t
0
(6)
is analytic and univalent in U.
Theoem B [5]. Let f ∈ A satisfy the following inequality:
2 ′
z f (z)
[f (z)]2 − 1 ≤ 1 (z ∈ U).
72
(7)
S. Bulut - Univalence condition for a new generalization of the family ...
Then f is univalent in U.
Theorem C [7]. Let the function g ∈ A satisfies the inequality (7). Also
let
3
α∈R
α ∈ 1,
and c ∈ C.
2
If
|c| ≤
3 − 2α
α
(c 6= −1)
and
|g(z)| ≤ 1 (z ∈ U),
then the function Gα (z) defined by
Z
Gα (z) := α
z
α−1
(g(t))
0
α1
dt
(8)
is in the univalent function class S.
In [2], Breaz and Breaz considered the integral operator
Gn,α (z) :=
[n (α − 1) + 1]
Z
z
α−1
(g1 (t))
α−1
· · · (gn (t))
0
1
n(α−1)+1
dt
(g1 , . . . , gn ∈ A)
(9)
and proved that the function Gn,α (z) is univalent in U.
Remark 2. We note that for n = 1, we obtain the integral operator in
(8).
In [3], Breaz et al. proved the following theorem.
Theorem D [3]. Let M ≥ 1 and suppose that each of the functions gj ∈
A (j ∈ {1, . . . , n}) satisfies the inequality (7). Also let
(2M + 1)n
and c ∈ C.
α∈R
α ∈ 1,
(2M + 1)n − 1
If
|c| ≤ 1 +
1−α
α
73
(2M + 1)n
S. Bulut - Univalence condition for a new generalization of the family ...
and
|gj (z)| ≤ M
(z ∈ U; j ∈ {1, . . . , n}),
then the function Gn,α (z) defined by (9) is in the univalent function class S.
Now we introduce a new general integral operator by means of the AlOboudi differential operator.
Definition 1. Let n ∈ N, m ∈ N0 and α ∈ C. We define the integral
operator Gn,m,α : An → A by
Gn,m,α (z) :=
(
[n (α − 1) + 1]
Z
0
where g1 , . . . , gn ∈ A and D
m
n
zY
α−1
(Dm gj (t))
j=1
1
) n(α−1)+1
dt
(z ∈ U),
(10)
is the Al-Oboudi differential operator.
Remark 3. In the special case n = 1, we obtain the integral operator
Z
Gm,α (z) := α
z
m
α−1
(D g(t))
0
α1
dt
(z ∈ U).
(11)
Remark 4. If we set m = 0 in (10) and (11), then we obtain the integral
operators defined in (9) and (8), respectively.
2.Main results
Theorem 1. Let M ≥ 1 and suppose that each of the functions gj ∈
A (j ∈ {1, . . . , n}) satisfies the inequality
z 2 (Dm g (z))′
j
(12)
− 1 ≤ 1 (z ∈ U; m ∈ N0 ).
2
m
(D gj (z))
Also let
α∈R
If
α ∈ 1,
|c| ≤ 1 +
(2M + 1)n
(2M + 1)n − 1
1−α
n(α − 1) + 1
74
and
(2M + 1)n
c ∈ C.
S. Bulut - Univalence condition for a new generalization of the family ...
and
|Dm gj (z)| ≤ M
(z ∈ U; j ∈ {1, . . . , n}),
then the integral operator Gn,m,α (z) defined by (10) is in the univalent function
class S.
Proof. Since gj ∈ A (j ∈ {1, . . . , n}), by (5), we have
∞
X
Dm gj (z)
=1+
[1 + (k − 1)δ]m ak,j z k−1
z
k=2
and
(m ∈ N0 )
Dm gj (z)
6= 0
z
for all z ∈ U.
Also we note that
(
Gn,m,α (z) =
[n(α − 1) + 1]
Z
z
tn(α−1)
0
Define a function
f (z) =
Z
0
Then we obtain
′
f (z) =
α−1
n m
Y
D gj (t)
t
j=1
n
zY
j=1
Dm gj (t)
t
α−1
α−1
n m
Y
D gj (z)
z
j=1
ln f (z) = (α − 1)
n
X
j=1
ln
.
Dm gj (z)
z
or equivalently
ln f ′ (z) = (α − 1)
n
X
j=1
75
dt
.
dt.
It is clear that f (0) = f ′ (0) − 1 = 0.
The equality (13) implies that
′
1
) n(α−1)+1
(ln Dm gj (z) − ln z) .
(13)
S. Bulut - Univalence condition for a new generalization of the family ...
By differentiating above equality, we get
n
X
(Dm gj (z))′ 1
f ′′ (z)
.
= (α − 1)
−
f ′ (z)
Dm gj (z)
z
j=1
Hence we obtain
n
X
z (Dm gj (z))′
zf ′′ (z)
= (α − 1)
−1 ,
f ′ (z)
Dm gj (z)
j=1
which readily shows that
zf ′′ (z)
c |z|2[n(α−1)+1] + 1 − |z|2[n(α−1)+1]
′
[n(α − 1) + 1] f (z)
X
n
′
m
z
(D
g
(z))
α
−
1
j
−1
= c |z|2[n(α−1)+1] + 1 − |z|2[n(α−1)+1]
n(α − 1) + 1 j=1
Dm gj (z)
X
n 2
z (Dm gj (z))′ Dm gj (z)
α−1
+1 .
≤ |c| +
n(α − 1) + 1 j=1 (Dm gj (z))2
z
From the hypothesis, we have |gj (z)| ≤ M (j ∈ {1, . . . , n} ; z ∈ U), then by
the Schwarz lemma, we obtain that
|gj (z)| ≤ M |z|
(j ∈ {1, . . . , n} ; z ∈ U).
Then we find
′′
zf
(z)
2[n(α−1)+1]
2[n(α−1)+1]
c |z|
+
1
−
|z|
[n(α − 1) + 1] f ′ (z)
X
n 2
z (Dm gj (z))′
α−1
M + 1
≤ |c| +
n(α − 1) + 1 j=1 (Dm gj (z))2
X
n 2
z (Dm gj (z))′
α−1
− 1 M + M + 1
≤ |c| +
n(α − 1) + 1 i=1 (Dm gj (z))2
α−1
≤ |c| +
(2M + 1)n ≤ 1
n(α − 1) + 1
1−α
since |c| ≤ 1 + n(α−1)+1
(2M + 1)n. Applying Theorem A, we obtain that
Gn,m,α is in the univalent function class S.
76
S. Bulut - Univalence condition for a new generalization of the family ...
Remark 5. If we set m = 0 in Theorem ??, then we have Theorem D.
Corollary 2. Let each of the functions gj ∈ A (j ∈ {1, . . . , n}) satisfies
the inequality (12). Suppose also that
3n
α∈R
α ∈ 1,
and c ∈ C.
3n − 1
If
|c| ≤ 1 + 3
and
1−α
n(α − 1) + 1
n
|Dm gj (z)| ≤ 1 (z ∈ U; j ∈ {1, . . . , n}),
then the integral operator Gn,m,α (z) defined by (10) is in the univalent function
class S.
Proof. In Theorem 1, we consider M = 1.
Remark 6. If we set m = 0 in Corollary 2, then we have Corollary 1 in
[3].
Corollary 3. Let M ≥ 1 and suppose that the functions g ∈ A satisfies
the inequality (12). Also let
2M + 1
and c ∈ C.
α∈R
α ∈ 1,
2M
If
|c| ≤ 1 +
and
1−α
α
|Dm g(z)| ≤ M
(2M + 1)
(z ∈ U),
then the integral operator Gm,α (z) defined by (11) is in the univalent function
class S.
Proof. In Theorem 1, we consider n = 1.
Remark 7. If we set m = 0 in Corollary 3, then we have Corollary 2 in
[3].
77
S. Bulut - Univalence condition for a new generalization of the family ...
Remark 8. If we set M = n = 1 and m = 0 in Theorem 1, then we obtain
Theorem C.
References
[1] F. M. Al-Oboudi, On univalent functions defined by a generalized S˘al˘agean
operator, Int. J. Math. Math. Sci. 2004, no. 25-28, 1429-1436.
[2] D. Breaz and N. Breaz, Univalence of an integral operator, Mathematica
(Cluj) 47 (2005), no. 70, 35-38.
[3] D. Breaz, N. Breaz and H. M. Srivastava, An extension of the univalent
condition for a family of integral operators, Appl. Math. Lett. 22 (2009), no.
1, 41-44.
[4] Z. Nehari, Conformal Mapping, Dover, New York, NY, USA, 1975.
[5] S. Ozaki and M. Nunokawa, The Schwarzian derivative and univalent
functions, Proc. Amer. Math. Soc. 33 (1972), 392-394.
[6] V. Pescar, A new generalization of Ahlfors’s and Becker’s criterion of
univalence, Bull. Malaysian Math. Soc. (Ser. 2) 19 (1996), 53-54.
[7] V. Pescar, On the univalence of some integral operators, J. Indian Acad.
Math. 27 (2005), 239-243.
[8] G. S¸. S˘al˘agean, Subclasses of univalent functions, Complex AnalysisFifth Romanian-Finnish seminar, Part 1 (Bucharest, 1981), Lecture Notes in
Math., vol. 1013, Springer, Berlin, 1983, pp. 362-372.
Author:
Serap Bulut
Civil Aviation College
Kocaeli University
Arslanbey Campus
˙
41285 Izmit-Kocaeli
Turkey
e-mail:[email protected]
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