A NOTE ON GENERALIZED INTEGRAL OPERATOR
Lakshmi Narayana Swamy Dileep and Satyanarayana Latha
Abstract. In this paper, we define the subclass Sg(α) of analytic functions by using Hadamrad product. Also we investigate certain properties of the generalized integral operator Ig(f1, . . . , fm) for the functions belonging to the class Sg(α).
2000Mathematics Subject Classification: 30C45. 1.Introduction
Let A be the class of analytic functions f of the form
f(z) =z+
∞
X
j=2
ajzj, (1)
defined in the unit disc U = {z ∈ C : |z| < 1} and satisfying the normalization condition f(0) =f′
(0)−1 = 0.
A function f ∈ A is said to be starlike of order α if it satisfies the inequality ℜ
zf′
(z) f(z)
> α, (z∈ U) for some 0≤α <1 and it is denoted by S∗
(α) [6].
The class of convex functions of order α, denote by K(α) consists of function f ∈ A such that
ℜ
zf′′
(z) f′(z) + 1
> α, (z∈ U)
for some 0 ≤α <1 and it is denoted by K(α) [6]. Further, f ∈ K(α) if and only if zf′
∈ S∗
(α).
For any two functions f and g such that
f(z) =z+
∞
X
j=2
ajzj and g(z) =z+ ∞
X
j=2
the Hadamard product or convolution of f and g denoted by f∗g is given by (f ∗g)(z) =z+
∞
X
j=2
ajbjzj. (2)
For different choice of g(z), the Hadamrad product f∗g yields the following well known operators such as Ruscheweyh derivative operator [7], Sˇalˇagean operator [8], Carlson-Shaffer operator [2], Dziok-Srivasava operator [5] and Al-Oboudi differ-ential operator [1].
We designate Sg(α), as the class of functions f ∈ A which satisfy the condition
ℜ
z((f∗g)(z))′
(f∗g)(z)
> α, (z∈ U) for some 0≤α <1.
For g(z) =z+
∞
X
j=2
[1 + (j−1)δ]nzj, we get the class Sn(δ, α) introduced by Serap
Bulut [9].
For n, m∈N0, ki>0 and 1≤i≤m the generalized integral operator Ig(f1, . . . , fm) :An−→ A is defined as
Ig(f1, . . . , fm)(z) =
Z z
0
(f1∗g)(t) t
k1
, . . . ,
(fm∗g)(t)
t
km
dt, z∈ U (3) where fi ∈ A and ∗ denotes the Hadamard product.
2.Prime Results
In this section, we determine certain properties for functions belonging to the class Sg(α) using the generalized integral operator Ig(f1, . . . , fm).
Theorem 2.1 Let fi ∈ Sg(αi), for 1 ≤ i ≤ m, with 0 ≤ αi < 1. Let ki > 0, with 1≤i≤m. If
m
X
i=1
ki(1−αi)≤1, then Ig(f1, . . . , fm)∈ K(λ), where
λ= 1 +
m
X
i=1
ki(αi−1).
Proof. For 1≤i≤m, using ( 2) we can write
(fi∗g)(z)
z = 1 +
∞
X
j=2 ajbjzj
and
(fi∗g)(z)
z 6= 0, forall z∈ U. On the other hand, we have
Ig(f1, . . . , fm) ′
(z) =
(f1∗g)(z) z
k1
. . .
(fm∗g)(z)
z
km
dt, z∈ U
which implies,
lnIg(f1, . . . , fm)′(z) =k1ln
(f1∗g)(z)
z +· · ·+kmln
(fm∗g)(z)
z
or equivalently lnIg(f1, . . . , fm)
′
(z) =k1[ln(f1∗g)(z)−lnz] +· · ·+km[ln(fm∗g)(z)−lnz].
By differentiating the above equality, we get Ig(f1, . . . , fm)′′(z)
Ig(f1, . . . , fm)′(z)
= m X i=1 ki
((fi∗g)(z))′
(fi∗g)(z)
− 1 z
.
Thus,
zIg(f1, . . . , fm)′′(z)
Ig(f1, . . . , fm)′
+ 1 =
m
X
i=1 ki
z((fi∗g)(z))′
(fi∗g)(z)
−
m
X
i=1 ki+ 1
which is equivalent to
ℜ
zIg(f1, . . . , fm)′′(z)
Ig(f1, . . . , fm)′
+ 1 = m X i=1 kiℜ
z((fi∗g)(z))′
(fi∗g)(z)
−
m
X
i=1
ki+ 1.
Since fi ∈ Sg(αi), we get
ℜ
zIg(f1, . . . , fm)′′(z)
Ig(f1, . . . , fm)′
+ 1 > m X i=1 kiαi−
m
X
i=1
ki+ 1 = 1 + m
X
i=1
ki(αi−1).
Hence, the integral operator Ig(f1, . . . , fm) is convex of order λ, where
λ= 1 +
m
X
i=1
ki(αi−1).
For g(z) = z+
∞
X
m=2
Corollary 2.2 [9]:Let fi ∈ Sn(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
i=1 ki ≤
1
1−α, then In(f1, . . . , fm)∈ K(λ), where λ= 1 + (α−1)
m
X
i=1 ki.
Corollary 2.3[9]:Let f ∈ Sn(δ, α) with 0 ≤α < 1, δ ≥0 and n∈ N0. Also let 0< k≤ 1
1−α. Then, In(f)(z)∈ K(1 +k(α−1)), where
In(f)(z) =
Z z
0
Dnf(t)
t k
dt.
Corollary 2.4 [9]:Let f ∈ Sn(δ, α). Then, the integral operator I
n(f)(z) ∈ K(α), where
In(f)(z) =
Z z
0
Dnf(t) t
k dt.
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Department of Mathematics
Vidyavardhaka College of Engineering Gokolum 3rd stage
Mysore - 570002 India
email:[email protected]
Satyanarayana Latha Department of Mathematics Yuvaraja’s College
University of Mysore Mysore 570005 India
