ON CERTAIN PROPERTIES OF NEIGHBORHOODS OF MULTIVALENT

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Volume 10 (2009), Issue 4, Article 112, 6 pp.
ON CERTAIN PROPERTIES OF NEIGHBORHOODS OF MULTIVALENT
FUNCTIONS INVOLVING THE GENERALIZED SAITOH OPERATOR
HESAM MAHZOON AND S. LATHA
D EPARTMENT OF S TUDIES IN M ATHEMATICS
M ANASAGANGOTRI U NIVERSITY OF M YSORE - INDIA.
[email protected]
D EPARTMENT OF M ATHEMATICS
Y UVARAJA’ S C OLLEGE U NIVERSITY OF M YSORE -INDIA.
[email protected]
Received 10 June, 2009; accepted 03 November, 2009
Communicated by S.S. Dragomir
A BSTRACT. In this paper, we introduce the generalized Saitoh operator Lp (a, c, η) and using
p,b
p,b,α
this operator, the new subclasses Hn,m
(a, c, η), Lp,b
and
n,m (a, c, η; µ), Hn,m (a, c, η)
p,b,α
Ln,m (a, c, η; µ) of the class of multivalent functions denoted by Ap (n) are defined. Further
for functions belonging to these classes, certain properties of neighborhoods are studied.
Key words and phrases: Coefficient bounds, (n, δ) - neighborhood and generalized Saitoh operator.
2000 Mathematics Subject Classification. 30C45.
1. I NTRODUCTION
Let Ap (n) be the class of normalized functions f of the form
(1.1)
∞
X
p
f (z) = z +
ak z k ,
(n, p ∈ N),
k=n+p
which are analytic and p-valent in the open unit disc U = {z ∈ C : |z| < 1}.
Let Tp (n) be the subclass of Ap (n), consisting of functions f of the form
(1.2)
p
f (z) = z −
∞
X
ak z k ,
(ak ≥ 0, n, p ∈ N),
k=n+p
which are p-valent in U.
The Hadamard product of two power series
p
f (z) = z +
∞
X
k=n+p
155-09
ak z
k
and
p
g(z) = z +
∞
X
k=n+p
bk z k
2
H ESAM M AHZOON AND S. L ATHA
is defined as
(f ∗ g)(z) = z p +
∞
X
ak b k z k .
k=n+p
Z−
0,
Definition 1.1. For a ∈ R, c ∈ R \
where Z−
0 = {..., −2, −1, 0} and η ∈ R (η ≥ 0), the
operator Lp (a, c, η) : Ap (n) → Ap (n), is defined as
Lp (a, c, η)f (z) = φp (a, c, z) ∗ Dη f (z),
(1.3)
where
η
Dη f (z) = (1 − η)f (z) + zf 0 (z),
p
and
(η ≥ 0, z ∈ U)
∞
X
(a)k−p k
φp (a, c, z) = z +
z ,
(c)k−p
k=n+p
p
z∈U
and (x)k denotes the Pochammer symbol given by
(
1
if k = 0,
(x)k =
x(x + 1) · · · (x + k − 1) if k ∈ N = {1, 2, 3, ...}.
In particular, we have, LP
1 (a, c, η) ≡ L(a, c, η).
k
Further, if f (z) = z p + ∞
k=n+p ak z , then
∞ X
(a)k−p
k
p
Lp (a, c, η)f (z) = z +
−1 η
ak z k .
1+
p
(c)
k−p
k=n+p
Remark 1. For η = 0 and n = 1, we obtain the Saitoh operator [7] which yields the Carlson Shaffer operator [1] for η = 0 and n = p = 1.
For any function f ∈ Tp (n) and δ ≥ 0, the (n, δ)-neighborhood of f is defined as,
(
)
∞
∞
X
X
(1.4) Nn,δ (f ) = g ∈ Tp (n) : g(z) = z p −
bk z k and
k|ak − bk | ≤ δ .
k=n+p
k=n+p
p
For the function h(z) = z , (p ∈ N) we have,
(
(1.5)
Nn,δ (h) =
g ∈ Tp (n) : g(z) = z p −
∞
X
k=n+p
bk z k
and
∞
X
)
k|bk | ≤ δ
.
k=n+p
The concept of neighborhoods was first introduced by Goodman [2] and then generalized by
Ruscheweyh [6] .
p,b
Definition 1.2. A function f ∈ Tp (n) is said to be in the class Hn,m
(a, c, η) if
!
1 z (L (a, c, η)f (z))(m+1)
p
(1.6)
−
(p
−
m)
< 1,
(m)
b
(Lp (a, c, η)f (z))
where p ∈ N, m ∈ N0 , a > 0, η ≥ 0, p > m, b ∈ C \ {0} and z ∈ U.
Definition 1.3. A function f ∈ Tp (n) is said to be in the class Lp,b
n,m (a, c, η; µ) if
"
#
(m)
1
L
(a,
c,
η)f
(z)
p
(m+1)
(1.7) p(1 − µ)
+ µ (Lp (a, c, η)f (z))
− (p − m) < p−m,
b
z
where p ∈ N, m ∈ N0 , a > 0, η ≥ 0, p > m, µ ≥ 0, b ∈ C \ {0} and z ∈ U.
J. Inequal. Pure and Appl. Math., 10(4) (2009), Art. 112, 6 pp.
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G ENERALIZED S AITOH O PERATOR
3
2. C OEFFICIENT B OUNDS
In this section, we determine the coefficient inequalities for functions to be in the subclasses
and Lp,b
n,m (a, c, η; µ).
p,b
Hn,m
(a, c, η)
p,b
Theorem 2.1. Let f ∈ Tp (n). Then, f ∈ Hn,m
(a, c, η) if and only if
∞ X
(a)k−p k
p
k
.
1+
−1 η
(k + |b| − p) ak ≤ |b|
m
p
(c)
k−p m
k=n+p
(2.1)
p,b
Proof. Let f ∈ Hn,m
(a, c, η). Then, by (1.6) and (1.7) we can write,
h
i
P

(a)k−p k k
k−p 
 ∞
(p
−
k)a
z
1
+
−
1
η
k
k=n+p
p
(c)k−p m
h
i
(2.2)
<
> −|b|,
P
(a)k−p k
k
 p − ∞
k−p 
1
+
−
1
η
a
z
k=n+p
m
p
(c)k−p m k
(z ∈ U).
Taking z = r, (0 ≤ r < 1) in (2.2), we see that the expression in the denominator on the left
hand side of (2.2), is positive for r = 0 and for all r, 0 ≤ r < 1. Hence, by letting r 7→ 1−
through real values, expression (2.2) yields the desired assertion (2.1).
Conversely, by applying the hypothesis (2.1) and letting |z| = 1, we obtain,
z (L (a, c, η)f (z))(m+1)
p
− (p − m)
(m)
(Lp (a, c, η)f (z))
h
i
P
(a)k−p k ∞
k
k−m
1
+
−
1
η
(p
−
k)a
z
k
k=n+p
p
(c)k−p m
h
i
= P
(a)k−p k
k
k−m mp z p−m − ∞
1
+
−
1
η
a
z
k=n+p
p
(c)k−p m k
h P
h
i
i
(a)k−p k k
|b| mp − ∞
1
+
−
1
η
a
k
k=n+p
p
(c)k−p m
h
i
≤
P
(a)k−p k p
k
− ∞
k=n+p 1 + p − 1 η (c)k−p m ak
m
= |b|.
p,b
Hence, by the maximum modulus theorem, we have f ∈ Hn,m
(a, c, η).
On similar lines, we can prove the following theorem.
Theorem 2.2. A function f ∈ Lp,b
n,m (a, c, η; µ) if and only if
∞ X
k
(a)k−p k − 1
(2.3)
1+
−1 η
[µ(k − 1) + 1] ak
p
(c)
m
k−p
k=n+p
|b| − 1
p
≤ (p − m)
+
.
m!
m
3. I NCLUSION R ELATIONSHIPS I NVOLVING (n, δ)-N EIGHBORHOODS
In this section, we prove certain inclusion relationships for functions belonging to the classes
and Lp,b
n,m (a, c, η; µ).
p,b
Hn,m
(a, c, η)
Theorem 3.1. If
(3.1)
(n + p)|b| mp
δ=
n
n
(n + |b|) 1 + p η (a)
(c)n
J. Inequal. Pure and Appl. Math., 10(4) (2009), Art. 112, 6 pp.
,
n+p
m
(p > |b|),
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4
H ESAM M AHZOON AND S. L ATHA
p,b
then Hn,m
(a, c, η) ⊂ Nn,δ (h).
p,b
Proof. Let f ∈ Hn,m
(a, c, η). By Theorem 2.1, we have,
∞
(a)n n + p X
p
n
ak ≤ |b|
,
(n + |b|) 1 + η
m
p
(c)n
m
k=n+p
which implies
∞
X
(3.2)
k=n+p
ak ≤
|b|
p
m
n
(n + |b|) 1 + np η (a)
(c)n
.
n+p
m
Using (2.1) and (3.2), we have,
∞
n
(a)n n + p X
1+ η
kak
p
(c)n
m
k=n+p
∞
p
n
(a)n n + p X
≤ |b|
+ (p − |b|) 1 + η
ak
m
p
(c)n
m
k=n+p
|b| mp
p
n
(a)n n + p
≤ |b|
+ (p − |b|) 1 + η
n
m
p
(c)n
m
(n + |b|) 1 + np η (a)
(c)n
p n+p
.
= |b|
m n + |b|
That is,
∞
X
|b|(n + p) mp
kak ≤
n
(n + |b|) 1 + np η (a)
k=n+p
(c)n
= δ,
n+p
m
n+p
m
(p > |b|).
Thus, by the definition given by (1.5), f ∈ Nn,δ (h).
Similarly, we prove the following theorem.
Theorem 3.2. If
(3.3)
h
i
p
(p − m)(n + p) |b|−1
+
m!
m
δ=
,
n n+p
[µ(n + p − 1) + 1] 1 + np η (a)
(c)n m
(µ > 1)
then Lp,b
n,m (a, c, η; µ) ⊂ Nn,δ (h).
4. F URTHER N EIGHBORHOOD P ROPERTIES
In this section, we determine the neighborhood properties of functions belonging to the subp,b,α
classes Hn,m
(a, c, η) and Lp,b,α
n,m (a, c, η; µ).
p,b,α
For 0 ≤ α < p and z ∈ U, a function f is said to be in the class Hn,m
(a, c, η) if there
p,b
exists a function g ∈ Hn,m
(a, c, η) such that
f (z)
< p − α.
(4.1)
−
1
g(z)
For 0 ≤ α < p and z ∈ U, a function f is said to be in the class Lp,b,α
n,m (a, c, η; µ) if there
p,b
exists a function g ∈ Ln,m (a, c, η; µ) such that the inequality (4.1) holds true.
J. Inequal. Pure and Appl. Math., 10(4) (2009), Art. 112, 6 pp.
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G ENERALIZED S AITOH O PERATOR
5
p,b
Theorem 4.1. If g ∈ Hn,m
(a, c, η) and
(4.2)
n n+p
δ(n + |b|) 1 + np η (a)
(c)n m
h
α=p−
n
n n+p
(n + p) (n + |b|) 1 + p η (a)
− |b|
(c)n m
i ,
p
m
p,b,α
then Nn,δ (g) ⊂ Hn,m
(a, c, η).
Proof. Let f ∈ Nn,δ (g). Then,
∞
X
(4.3)
k|ak − bk | ≤ δ,
k=n+p
which yields the coefficient inequality,
∞
X
(4.4)
|ak − bk | ≤
k=n+p
δ
,
n+p
(n ∈ N).
p,b
Since g ∈ Hn,m
(a, c, η), by (3.2) we have,
(4.5)
∞
X
bk ≤
k=n+p
|b|
p
m
n
(n + |b|) 1 + np η (a)
(c)n
n+p
m
so that,
P∞
f (z)
|ak − bk |
< k=n+p
P
−
1
∞
g(z)
1 − k=n+p bk
(a)n n+p
n
(n
+
|b|)
1
+
η
p
(c)n m
δ
h
≤
n + p (n + |b|) 1 + n η (a)n n+p − |b|
p
(c)n m
i
p
m
= p − α.
p,b,α
Thus, by definition, f ∈ Hn,m
(a, c, η) for α given by (4.2).
On similar lines, we prove the following theorem.
Theorem 4.2. If g ∈ Lp,b
n,m (a, c, η; µ) and
(4.6)
n n+p−1
δ[µ(n + p − 1) + 1] 1 + np η (a)
(c)n
m
h
α=p−
(a)n n+p−1
n
+
(n + p) {µ(n + p − 1) + 1} 1 + p η (c)n m − (p − m) |b|−1
m!
i ,
p
m
then Nn,δ (g) ⊂ Lp,b,α
n,m (a, c, η; µ).
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H ESAM M AHZOON AND S. L ATHA
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