ON CERTAIN PROPERTIES OF NEIGHBORHOODS
OF MULTIVALENT FUNCTIONS INVOLVING THE
GENERALIZED SAITOH OPERATOR
Generalized Saitoh Operator
Hesam Mahzoon and S. Latha
vol. 10, iss. 4, art. 112, 2009
HESAM MAHZOON
S. LATHA
Department of Studies in Mathematics
Manasagangotri University of Mysore
India
EMail: mahzoon_hesam@yahoo.com
Department of Mathematics
Yuvaraja’s College University of Mysore
India
EMail: drlatha@gmail.com
Received:
10 June, 2009
Accepted:
03 November, 2009
Communicated by:
S.S. Dragomir
2000 AMS Sub. Class.:
30C45.
Key words:
Coefficient bounds, (n, δ) - neighborhood and generalized Saitoh operator.
Abstract:
In this paper, we introduce the generalized Saitoh operator Lp (a, c, η)
p,b
and using this operator, the new subclasses Hn,m
(a, c, η), Lp,b
n,m (a, c, η; µ),
p,b,α
p,b,α
Hn,m (a, c, η) and 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.
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1
Introduction
3
2
Coefficient Bounds
6
3
Inclusion Relationships Involving (n, δ)-Neighborhoods
8
Generalized Saitoh Operator
4
Further Neighborhood Properties
10
Hesam Mahzoon and S. Latha
vol. 10, iss. 4, art. 112, 2009
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1.
Introduction
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)
f (z) = z p −
∞
X
Generalized Saitoh Operator
Hesam Mahzoon and S. Latha
vol. 10, iss. 4, art. 112, 2009
ak z k ,
(ak ≥ 0, n, p ∈ N),
k=n+p
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which are p-valent in U.
The Hadamard product of two power series
p
f (z) = z +
∞
X
ak z
k
and
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p
g(z) = z +
k=n+p
∞
X
bk z
k
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(f ∗ g)(z) = z p +
∞
X
ak b k z k .
Z−
0,
Z−
0
Definition 1.1. For a ∈ R, c ∈ R \
where
= {..., −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),
η
Dη f (z) = (1 − η)f (z) + zf 0 (z),
p
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k=n+p
where
II
k=n+p
is defined as
(1.3)
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(η ≥ 0, z ∈ U)
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and
∞
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
k
p
(1.4) Nn,δ (f ) = g ∈ Tp (n) : g(z) = z −
bk z and
k|ak −bk | ≤ δ .
k=n+p
k=n+p
Generalized Saitoh Operator
Hesam Mahzoon and S. Latha
vol. 10, iss. 4, art. 112, 2009
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p
For the function h(z) = z , (p ∈ N) we have,
(
)
∞
∞
X
X
(1.5)
Nn,δ (h) = g ∈ Tp (n) : g(z) = z p −
bk z k and
k|bk | ≤ δ .
k=n+p
k=n+p
The concept of neighborhoods was first introduced by Goodman [2] and then generalized by Ruscheweyh [6] .
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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
Generalized Saitoh Operator
Hesam Mahzoon and S. Latha
(1.7)
"
#
(m)
1
Lp (a, c, η)f (z)
(m+1)
+ µ (Lp (a, c, η)f (z))
− (p − m) p(1 − µ)
b
z
< p − m,
where p ∈ N, m ∈ N0 , a > 0, η ≥ 0, p > m, µ ≥ 0, b ∈ C \ {0} and z ∈ U.
vol. 10, iss. 4, art. 112, 2009
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2.
Coefficient Bounds
In this section, we determine the coefficient inequalities for functions to be in the
p,b
subclasses Hn,m
(a, c, η) and Lp,b
n,m (a, c, η; µ).
p,b
Theorem 2.1. Let f ∈ Tp (n). Then, f ∈ Hn,m
(a, c, η) if and only if
(2.1)
∞ X
(a)k−p k
p
k
(k + |b| − p) ak ≤ |b|
.
1+
−1 η
p
(c)
m
k−p m
k=n+p
p,b
Proof. Let f ∈ Hn,m
(a, c, η). Then, by (1.6) and (1.7) we can write,
 P

i
∞ h
(a)k−p k k
k−p 

1
+
−
1
η
(p
−
k)a
z


k
p
(c)k−p m


k=n+p
(2.2) <
> −|b|,
h
i
∞
P


(a)k−p k p
k
k−p


1 + p − 1 η (c)k−p m ak z
 m −

Generalized Saitoh Operator
Hesam Mahzoon and S. Latha
vol. 10, iss. 4, art. 112, 2009
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(z ∈ U).
k=n+p
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)
(Lp (a, c, η)f (z))(m)
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 − ∞
k=n+p 1 + p − 1 η (c)k−p m ak z
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i
i
(a)k−p k |b|
− k=n+p 1 +
− 1 η (c)k−p m ak
h
i
≤
P∞
(a)k−p k p
k
−
1
+
−
1
η
a
k=n+p
m
p
(c)k−p m k
h
p
m
P∞
h
k
p
= |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
(2.3)
∞ X
k
(a)k−p k − 1
1+
−1 η
[µ(k − 1) + 1] ak
p
(c)k−p
m
k=n+p
p
|b| − 1
.
≤ (p − m)
+
m!
m
Generalized Saitoh Operator
Hesam Mahzoon and S. Latha
vol. 10, iss. 4, art. 112, 2009
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3.
Inclusion Relationships Involving (n, δ)-Neighborhoods
In this section, we prove certain inclusion relationships for functions belonging to
p,b
the classes Hn,m
(a, c, η) and Lp,b
n,m (a, c, η; µ).
Theorem 3.1. If
(n + p)|b| mp
δ=
n
(n + |b|) 1 + np η (a)
(c)n
(3.1)
,
n+p
m
(p > |b|),
Generalized Saitoh Operator
Hesam Mahzoon and S. Latha
vol. 10, iss. 4, art. 112, 2009
then
p,b
Hn,m
(a, c, η)
⊂ Nn,δ (h).
p,b
Proof. Let f ∈ Hn,m
(a, c, η). By Theorem 2.1, we have,
∞
n
(a)n n + p X
p
(n + |b|) 1 + η
ak ≤ |b|
,
p
(c)n
m
m
k=n+p
which implies
(3.2)
∞
X
k=n+p
ak ≤
|b|
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p
m
(n + |b|) 1 + np η
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(a)n n+p
(c)n m
.
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Using (2.1) and (3.2), we have,
∞
n
(a)n n + p X
1+ η
kak
p
(c)n
m
k=n+p
∞
n
p
(a)n n + p X
≤ |b|
+ (p − |b|) 1 + η
ak
m
p
(c)n
m
k=n+p
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|b| mp
p
n
(a)n n + p
≤ |b|
+ (p − |b|) 1 + η
n
m
m
p
(c)n
(n + |b|) 1 + np η (a)
(c)n
p n+p
.
= |b|
m n + |b|
n+p
m
That is,
Generalized Saitoh Operator
∞
X
|b|(n + p) mp
n n+p
|b|) 1 + np η (a)
(c)n m
kak ≤
k=n+p
(n +
= δ,
(p > |b|).
Thus, by the definition given by (1.5), f ∈ Nn,δ (h).
Hesam Mahzoon and S. Latha
vol. 10, iss. 4, art. 112, 2009
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Similarly, we prove the following theorem.
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Theorem 3.2. If
h
(3.3)
i
(p − m)(n + p) |b|−1
+ mp
m!
δ=
n
[µ(n + p − 1) + 1] 1 + np η (a)
(c)n
,
n+p
m
(µ > 1)
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then
Lp,b
n,m (a, c, η; µ)
⊂ Nn,δ (h).
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4.
Further Neighborhood Properties
In this section, we determine the neighborhood properties of functions belonging to
p,b,α
the subclasses 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)
p,b
if there exists a function g ∈ Hn,m
(a, c, η) such that
f (z)
(4.1)
g(z) − 1 < p − α.
Generalized Saitoh Operator
Hesam Mahzoon and S. Latha
vol. 10, iss. 4, art. 112, 2009
Lp,b,α
n,m (a, c, η; µ)
For 0 ≤ α < p and z ∈ U, a function f is said to be in the class
if
p,b
there exists a function g ∈ Ln,m (a, c, η; µ) such that the inequality (4.1) holds true.
Theorem 4.1. If g ∈
p,b
Hn,m
(a, c, η)
and
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(4.2)
n
δ(n + |b|) 1 + np η (a)
(c)n m
h
α=p−
n n+p
(n + p) (n + |b|) 1 + np η (a)
− |b|
(c)n m
n+p
∞
X
k|ak − bk | ≤ δ,
which yields the coefficient inequality,
k=n+p
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k=n+p
∞
X
p
m
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Proof. Let f ∈ Nn,δ (g). Then,
(4.4)
i ,
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p,b,α
then Nn,δ (g) ⊂ Hn,m
(a, c, η).
(4.3)
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|ak − bk | ≤
δ
,
n+p
(n ∈ N).
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p,b
Since g ∈ Hn,m
(a, c, η), by (3.2) we have,
∞
X
(4.5)
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
Generalized Saitoh Operator
Hesam Mahzoon and S. Latha
vol. 10, iss. 4, art. 112, 2009
≤
n
η
p
(n + |b|) 1 +
δ
h
n + p (n + |b|) 1 + n η (a)n
p
(c)n
(a)n n+p
(c)n m
n+p
m
− |b|
i
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p
m
Contents
= 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.
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Theorem 4.2. If g ∈ Lp,b
n,m (a, c, η; µ) and
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(4.6) α = p −
1
(n + p)
n n+p−1
δ[µ(n + p − 1) + 1] 1 + np η (a)
(c)n
m
×h
|b|−1
n n+p−1
{µ(n + p − 1) + 1} 1 + np η (a)
−
(p
−
m)
+
(c)n
m
m!
then Nn,δ (g) ⊂ Lp,b,α
n,m (a, c, η; µ).
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i ,
p
m
References
[1] B.C. CARLSON AND D.B. SHAFFER, Starlike and prestarlike hypergeometric
functions, SIAM. J. Math. Anal.,15(4) (1984), 737–745.
[2] A.W. GOODMAN, Univalent functions and non-analytic curves, Proc. Amer.
Math. Soc.,8 (1957), 598–601.
[3] G. MURUGUSUNDARAMOORTHY AND H.M. SRIVASTAVA, Neighborhoods of certain classes of analytic functions of complex order, J. Inequal. Pure
and Appl. Math., 5(2) (2004), Art. 24. [ONLINE: http://jipam.vu.edu.
au/article.php?sid=374]
Generalized Saitoh Operator
[4] M.A. NASR AND M.K. AOUF, Starlike function of complex order, J. Natur. Sci.
Math., 25 (1985), 1–12.
Title Page
[5] R.K. RAINA AND H.M. SRIVASTAVA, Inclusion and neighborhood properties of some analytic and multivalent functions, J. Inequal. Pure and
Appl. Math.,7(1) (2006), Art. 5. [ONLINE: http://jipam.vu.edu.au/
article.php?sid=640]
[6] S. RUSCHEWEYH, Neighbohoods of univalent functions, Proc. Amer. Math.
Soc., 81 (1981), 521–527.
[7] H. SAITOH, A linear operator and its applications of first order differential subordinations, Math. Japan, 44 (1996), 31–38.
[8] H.M. SRIVASTAVA AND S. OWA, Current Topics in Analytic Function Theory,
World Scientific Publishing Company, 1992.
[9] P. WIATROWSKI, On the coefficients of some family of holomorphic functions,
Zeszyty Nauk. Uniw. Lodz Nauk. Mat. - Przyrod., (2), 39 (1970), 75–85.
Hesam Mahzoon and S. Latha
vol. 10, iss. 4, art. 112, 2009
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