Volume 9 (2008), Issue 4, Article 115, 5 pp.
SOME PROPERTIES FOR AN INTEGRAL OPERATOR DEFINED BY
AL-OBOUDI DIFFERENTIAL OPERATOR
SERAP BULUT
KOCAELI U NIVERSITY
C IVIL AVIATION C OLLEGE
A RSLANBEY C AMPUS
41285 İ ZMIT-KOCAELI
TURKEY
serap.bulut@kocaeli.edu.tr
Received 08 July, 2008; accepted 12 November, 2008
Communicated by S.S. Dragomir
A BSTRACT. In this paper, we investigate some properties for an integral operator defined by
Al-Oboudi differential operator.
Key words and phrases: Analytic functions, Differential operator.
2000 Mathematics Subject Classification. Primary 30C45.
1. I NTRODUCTION
Let A denote the class of all functions of the form
∞
X
(1.1)
f (z) = z +
aj z j
j=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 [2] introduced the following operator:
D0 f (z) = f (z),
(1.2)
(1.3)
D1 f (z) = (1 − δ)f (z) + δzf 0 (z) = Dδ f (z),
(1.4)
Dn f (z) = Dδ (Dn−1 f (z)),
δ ≥ 0,
(n ∈ N := {1, 2, 3, . . .}).
If f is given by (1.1), then from (1.3) and (1.4) we see that
∞
X
n
(1.5)
D f (z) = z +
[1 + (j − 1)δ]n aj z j , (n ∈ N0 := N ∪ {0}),
j=2
n
with D f (0) = 0.
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S ERAP B ULUT
When δ = 1, we get Sălăgean’s differential operator [10].
A function f ∈ A is said to be starlike of order α if it satisfies the inequality:
0 zf (z)
Re
> α (z ∈ U)
f (z)
for some 0 ≤ α < 1. We say that f is in the class S ∗ (α) for such functions.
A function f ∈ A is said to be convex of order α if it satisfies the inequality:
00
zf (z)
+ 1 > α (z ∈ U)
Re
f 0 (z)
for some 0 ≤ α < 1. We say that f is in the class K(α) if it is convex of order α in U.
We note that f ∈ K(α) if and only if zf 0 ∈ S ∗ (α).
In particular, the classes
S ∗ (0) := S ∗
and
K(0) := K
are familiar classes of starlike and convex functions in U, respectively.
Now, we introduce two new classes S n (δ, α) and Mn (δ, β) as follows:
Let S n (δ, α) denote the class of functions f ∈ A which satisfy the condition
z (Dn f (z))0
Re
> α (z ∈ U)
Dn f (z)
for some 0 ≤ α < 1, δ ≥ 0, and n ∈ N0 .
It is clear that
S 0 (δ, α) ≡ S ∗ (α) ≡ S n (0, α),
S n (0, 0) ≡ S ∗ .
Let Mn (δ, β) be the subclass of A, consisting of the functions f , which satisfy the inequality
z (Dn f (z))0
Re
< β (z ∈ U)
Dn f (z)
for some β > 1, δ ≥ 0, and n ∈ N0 .
Also, let N (β) be the subclass of A, consisting of the functions f , which satisfy the inequality
00
zf (z)
Re
+ 1 < β (z ∈ U).
f 0 (z)
It is obvious that
M0 (δ, β) ≡ M(β) ≡ Mn (0, β).
The classes M(β) and N (β) were studied by Uralegaddi et al. in [11], Owa and Srivastava
in [9], and Breaz in [4].
Definition 1.1. Let n, m ∈ N0 and ki > 0, 1 ≤ i ≤ m. We define the integral operator
In (f1 , . . . , fm ) : Am → A
k
n
k
Z z n
D f1 (t) 1
D fm (t) m
In (f1 , . . . , fm )(z) :=
···
dt, (z ∈ U),
t
t
0
where fi ∈ A and Dn is the Al-Oboudi differential operator.
Remark 1.
(i) For n = 0, we have the integral operator
k
k
Z z
f1 (t) 1
fm (t) m
I0 (f1 , . . . , fm )(z) =
···
dt
t
t
0
introduced in [5]. More details about I0 (f1 , . . . , fm ) can be found in [3] and [4].
J. Inequal. Pure and Appl. Math., 9(4) (2008), Art. 115, 5 pp.
http://jipam.vu.edu.au/
S OME P ROPERTIES FOR AN I NTEGRAL O PERATOR
3
(ii) For n = 0, m = 1, k1 = 1, k2 = · · · = km = 0 and D0 f1 (z) := D0 f (z) = f (z) ∈ A,
we have the integral operator of Alexander
Z z
f (t)
I0 (f )(z) :=
dt
t
0
introduced in [1].
(iii) For n = 0, m = 1, k1 = k ∈ [0, 1], k2 = · · · = km = 0 and D0 f1 (z) := D0 f (z) =
f (z) ∈ S, we have the integral operator
k
Z z
f (t)
I(f )(z) :=
dt
t
0
studied in [8].
(iv) If ki ∈ C for 1 ≤ i ≤ m, then we have the integral operator In (f1 , . . . , fm ) studied in
[7].
In this paper, we investigate some properties for the operators In on the classes S n (δ, α) and
Mn (δ, β).
2. S OME P ROPERTIES FOR In
ON THE
C LASS S n (δ, α)
Theorem 2.1. Let fi ∈ S n (δ, αi ) for 1 ≤ i ≤ m with 0 ≤ αi < 1, δ ≥ 0 and n ∈ N0 . Also let
ki > 0, 1 ≤ i ≤ m. If
m
X
ki (1 − αi ) ≤ 1,
i=1
then In (f1 , . . . , fm ) ∈ K(λ) with λ = 1 +
Pm
i=1
ki (αi − 1).
Proof. By (1.5), for 1 ≤ i ≤ m, we have
∞
X
Dn fi (z)
=1+
[1 + (j − 1)δ]n aj,i z j−1 ,
z
j=2
and
(n ∈ N0 )
Dn fi (z)
6= 0
z
for all z ∈ U.
On the other hand, we obtain
0
In (f1 , . . . , fm ) (z) =
Dn f1 (z)
z
k 1
···
Dn fm (z)
z
k m
for z ∈ U. This equality implies that
ln In (f1 , . . . , fm )0 (z) = k1 ln
Dn f1 (z)
Dn fm (z)
+ · · · + km ln
z
z
or equivalently
ln In (f1 , . . . , fm )0 (z) = k1 [ln Dn f1 (z) − ln z] + · · · + km [ln Dn fm (z) − ln z] .
By differentiating the above equality, we get
n
m
In (f1 , . . . , fm )00 (z) X
(D fi (z))0 1
=
ki
−
.
In (f1 , . . . , fm )0 (z)
Dn fi (z)
z
i=1
J. Inequal. Pure and Appl. Math., 9(4) (2008), Art. 115, 5 pp.
http://jipam.vu.edu.au/
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S ERAP B ULUT
Thus, we obtain
m
m
X z (Dn fi (z))0 X
zIn (f1 , . . . , fm )00 (z)
+1=
ki
−
ki + 1.
In (f1 , . . . , fm )0
Dn fi (z)
i=1
i=1
This relation is equivalent to
X
X
m
m
z (Dn fi (z))0
zIn (f1 , . . . , fm )00 (z)
+1 =
ki Re
−
ki + 1.
Re
n f (z)
In (f1 , . . . , fm )0
D
i
i=1
i=1
Since fi ∈ S n (δ, αi ), we get
X
m
m
m
X
X
zIn (f1 , . . . , fm )00 (z)
Re
+1 >
ki αi −
ki + 1 = 1 +
ki (αi − 1).
In (f1 , . . . , fm )0
i=1
i=1
i=1
P
So, the integral operator In (f1 , . . . , fm ) is convex of order λ with λ = 1 + m
i=1 ki (αi − 1).
Corollary 2.2. Let fi ∈ S n (δ, α) for 1 ≤ i ≤ m with 0 ≤ α < 1, δ ≥ 0 and n ∈ N0 . Also let
ki > 0, 1 ≤ i ≤ m. If
m
X
1
ki ≤
,
1−α
i=1
P
then In (f1 , . . . , fm ) ∈ K(ρ) with ρ = 1 + (α − 1) m
i=1 ki .
Proof. In Theorem 2.1, we consider α1 = · · · = αm = α.
Corollary 2.3. Let f ∈ S n (δ, α) with 0 ≤ α < 1, δ ≥ 0 and n ∈ N0 . Also let 0 < k ≤
1/(1 − α). Then the function
k
Z z n
D f (t)
In (f )(z) =
dt
t
0
is in K(1 + k(α − 1)).
Proof. In Corollary 2.2, we consider m = 1 and k1 = k.
Corollary 2.4. Let f ∈ S n (δ, α). Then the integral operator
Z z
In (f )(z) =
(Dn f (t)/t) dt ∈ K(α).
0
Proof. In Corollary 2.3, we consider k = 1.
3. S OME P ROPERTIES FOR In
ON THE CLASS
Mn (δ, β)
Theorem 3.1.P
Let fi ∈ Mn (δ, βi ) for 1 ≤ i ≤ m with βi > 1. Then In (f1 , . . . , fm ) ∈ N (λ)
with λ = 1 + m
i=1 ki (βi − 1) and ki > 0, (1 ≤ i ≤ m).
Proof. Proof is similar to the proof of Theorem 2.1.
Remark 2. For n = 0, we have Theorem 2.1 in [4].
Corollary 3.2. Let fi P
∈ Mn (δ, β) for 1 ≤ i ≤ m with β > 1. Then In (f1 , . . . , fm ) ∈ N (ρ)
with ρ = 1 + (β − 1) m
i=1 ki and ki > 0, (1 ≤ i ≤ m).
Corollary 3.3. Let f ∈ Mn (δ, β) with β > 1. Then the integral operator
k
Z z n
D f (t)
In (f )(z) =
dt ∈ N (1 + k(β − 1))
t
0
and k > 0.
J. Inequal. Pure and Appl. Math., 9(4) (2008), Art. 115, 5 pp.
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S OME P ROPERTIES FOR AN I NTEGRAL O PERATOR
5
Corollary 3.4. Let f ∈ Mn (δ, β) with β > 1. Then the integral operator
Z z n
D f (t)
In (f )(z) =
dt ∈ N (β).
t
0
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J. Inequal. Pure and Appl. Math., 9(4) (2008), Art. 115, 5 pp.
http://jipam.vu.edu.au/