Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 7, Issue 1, Article 5, 2006
INCLUSION AND NEIGHBORHOOD PROPERTIES OF SOME ANALYTIC AND
MULTIVALENT FUNCTIONS
R.K. RAINA AND H.M. SRIVASTAVA
D EPARTMENT OF M ATHEMATICS
C OLLEGE OF T ECHNOLOGY AND E NGINEERING
M AHARANA P RATAP U NIVERSITY OF AGRICULTURE AND T ECHNOLOGY
U DAIPUR 313001, R AJASTHAN , I NDIA
rainark_7@hotmail.com
D EPARTMENT OF M ATHEMATICS AND S TATISTICS
U NIVERSITY OF V ICTORIA
V ICTORIA , B RITISH C OLUMBIA V8W 3P4, C ANADA
harimsri@math.uvic.ca
Received 08 November, 2005; accepted 15 November, 2005
Communicated by Th.M. Rassias
A BSTRACT. By means of a certain extended derivative operator of Ruscheweyh type, the
authors introduce and investigate two new subclasses of p-valently analytic functions of
complex order. The various results obtained here for each of these function classes include
coefficient inequalities and the consequent inclusion relationships involving the neighborhoods
of the p-valently analytic functions.
Key words and phrases: Analytic functions, p-valent functions, Hadamard product (or convolution), Coefficient bounds,
Ruscheweyh derivative operator, Neighborhood of analytic functions.
2000 Mathematics Subject Classification. Primary 30C45.
1. I NTRODUCTION , D EFINITIONS AND P RELIMINARIES
Let Ap (n) denote the class of functions f (z) normalized by
(1.1)
f (z) = z p −
∞
X
ak z k
(ak = 0; n, p ∈ N := {1, 2, 3, ...}) ,
k=n+p
which are analytic and p-valent in the open unit disk
U = {z : z ∈ C
and
|z| < 1} .
ISSN (electronic): 1443-5756
c 2006 Victoria University. All rights reserved.
The present investigation was supported, in part, by AICTE-New Delhi (Government of India) and, in part, by the Natural Sciences and
Engineering Research Council of Canada under Grant OGP0007353.
333-05
2
R.K. R AINA AND H.M. S RIVASTAVA
The Hadamard product (or convolution) of the function f ∈ Ap (n) given by (1.1) and the
function g ∈ Ap (n) given by
∞
X
p
(1.2)
g(z) = z −
bk z k (bk = 0; n, p ∈ N)
k=n+p
is defined (as usual) by
(1.3)
p
(f ∗ g)(z) := z +
∞
X
ak bk z k =: (g ∗ f )(z).
k=n+p
We introduce here an extended linear derivative operator of Ruscheweyh type:
Dλ,p : Ap → Ap Ap := Ap (1) ,
which is defined by the following convolution:
zp
λ,p
(1.4)
D f (z) =
∗ f (z) (λ > −p; f ∈ Ap ) .
(1 − z)λ+p
In terms of the binomial coefficients, we can rewrite (1.4) as follows:
∞ X
λ+k−1
λ,p
p
(1.5)
D f (z) = z −
ak z k (λ > −p; f ∈ Ap ) .
k
−
p
k=1+p
In particular, when λ = n (n ∈ N), it is easily observed from (1.4) and (1.5) that
(n)
p n−p
z
z
f
(z)
(1.6)
Dn,p f (z) =
(n ∈ N0 := N ∪ {0}; p ∈ N) ,
n!
so that
D1,p f (z) = (1 − p)f (z) + zf 0 (z),
(1.7)
(1.8)
D2,p f (z) =
z2
(1 − p)(2 − p)
f (z) + (2 − p)zf 0 (z) + f 00 (z),
2!
2!
and so on.
By using the operator
Dλ,p f (z) (λ > −p; p ∈ N)
p
given by (1.5), we now introduce a new subclass Hn,m
(λ, b) of the p-valently analytic function
class Ap (n), which includes functions f (z) satisfying the following inequality:
!
1 z Dλ,p f (z)(m+1)
(1.9)
(m) − (p − m) < 1
b
Dλ,p f (z)
(z ∈ U; p ∈ N; m ∈ N0 ; λ ∈ R; p > max(m, −λ); b ∈ C \ {0}) .
Next, following the earlier investigations by Goodman [3], Ruscheweyh [5] and Altintaş et
al. [2] (see also [1], [4] and [6]), we define the (n, δ)-neighborhood of a function f (z) ∈ An (p)
by (see, for details, [2, p. 1668])
(
)
∞
∞
X
X
k |ak − bk | 5 δ .
(1.10) Nn,δ (f ) := g ∈ Ap (n) : g(z) = z p −
bk z k and
k=n+p
k=n+p
It follows from (1.10) that, if
(1.11)
J. Inequal. Pure and Appl. Math., 7(1) Art. 5, 2006
h(z) = z p
(p ∈ N),
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I NCLUSION AND N EIGHBORHOOD P ROPERTIES OF S OME A NALYTIC
AND
M ULTIVALENT F UNCTIONS
3
then
(
(1.12)
Nn,δ (h) =
p
g ∈ Ap (n) : g(z) = z −
∞
X
bk z
k
and
k=n+p
∞
X
)
k |bk | 5 δ
.
k=n+p
Finally, we denote by Lpn,m (λ, b; µ) the subclass of Ap (n) consisting of functions f (z) which
satisfy the inequality (1.13) below:
!
λ,p
(m)
1
D
f
(z)
(m+1)
(1.13)
p(1 − µ)
+ µ Dλ,p f (z)
− (p − m) < p − m
b
z
(z ∈ U; p ∈ N; m ∈ N0 ; λ ∈ R; p > max(m, −λ); µ = 0; b ∈ C \ {0}) .
The object of the present paper is to investigate the various properties and characteristics of
analytic p-valent functions belonging to the subclasses
p
Hn,m
(λ, b) and Lpn,m (λ, b; µ),
which we have introduced here. Apart from deriving a set of coefficient bounds for each
of these function classes, we establish several inclusion relationships involving the (n, δ)neighborhoods of analytic p-valent functions (with negative and missing coefficients) belonging
to these subclasses.
Our definitions of the function classes
p
Hn,m
(λ, b) and Lpn,m (λ, b; µ)
are motivated essentially by two earlier investigations [1] and [4], in each of which further
details and references to other closely-related subclasses can be found. In particular, in our
definition of the function class Lpn,m (λ, b; µ) involving the inequality (1.13), we have relaxed
the parametric constraint 0 5 µ 5 1, which was imposed earlier by Murugusundaramoorthy
and Srivastava [4, p. 3, Equation (1.14)] (see also Remark 3 below).
2. A S ET OF C OEFFICIENT B OUNDS
In this section, we prove the following results which yield the coefficient inequalities for
functions in the subclasses
p
Hn,m
(λ, b) and Lpn,m (λ, b; µ).
p
(λ, b) if and only if
Theorem 1. Let f (z) ∈ Ap (n) be given by (1.1). Then f (z) ∈ Hn,m
∞ X
λ+k−1
k
p
(2.1)
(k + |b| − p) ak 5 |b|
.
k−p
m
m
k=n+p
p
Proof. Let a function f (z) of the form (1.1) belong to the class Hn,m
(λ, b). Then, in view of
(1.5), (1.9) yields the following inequality:
!
k
P∞
λ+k−1
k−p
(p
−
k)z
k=n+p
k−p
m
P∞
k
> − |b| (z ∈ U) .
(2.2)
<
p
λ+k−1
k−p
−
z
k=n+p
m
k−p
m
Putting z = r (0 5 r < 1) in (2.2), we observe that the expression in the denominator on the
left-hand side of (2.2) is positive for r = 0 and also for all r (0 < r < 1). Thus, by letting
r → 1− through real values, (2.2) leads us to the desired assertion (2.1) of Theorem 1.
J. Inequal. Pure and Appl. Math., 7(1) Art. 5, 2006
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4
R.K. R AINA AND H.M. S RIVASTAVA
Conversely, by applying (2.1) and setting |z| = 1, we find by using (1.5) that
z Dλ,p f (z)(m+1)
−
(p
−
m)
(Dλ,p f (z))(m)
P∞
k
λ+k−1
k−m
(p
−
k)z
k=n+p
k−p
m
k
= p
P∞
λ+k−1
p−m
k−m
m z
− k=n+p k−p m z
i
h P
k
λ+k−1
|b| mp − ∞
a
k=n+p
k−p
m k
5
= |b| .
P
∞
p
λ+k−1
k
−
a
k
k=n+p
m
k−p
m
p
Hence, by the maximum modulus principle, we infer that f (z) ∈ Hn,m
(λ, b), which completes
the proof of Theorem 1.
Remark 1. In the special case when
m = 0, p = 1,
(2.3)
and
(0 < β 5 1; γ ∈ C \ {0}) ,
b = βγ
Theorem 1 corresponds to a result given earlier by Murugusundaramoorthy and Srivastava [4,
p. 3, Lemma 1].
By using the same arguments as in the proof of Theorem 1, we can establish Theorem 2
below.
Theorem 2. Let f (z) ∈ Ap (n) be given by (1.1). Then f (z) ∈ Lpn,m (λ, b; µ) if and only if
∞ X
λ+k−1 k−1
|b| − 1
p
(2.4)
[µ (k − 1) + 1] ak 5 (p − m)
+
.
k−p
m
m!
m
k=n+p
Remark 2. Making use of the same parametric substitutions as mentioned above in (2.3),
Theorem 2 yields another known result due to Murugusundaramoorthy and Srivastava
[4, p. 4, Lemma 2].
3. I NCLUSION R ELATIONSHIPS I NVOLVING (n, δ)-N EIGHBORHOODS
In this section, we establish several inclusion relationships for the function classes
p
Hn,m
(λ, b) and Lpn,m (λ, b; µ)
involving the (n, δ)-neighborhood defined by (1.12).
Theorem 3. If
(3.1)
δ=
p
m
λ+n+p−1 n+p
n
m
(n + p) |b|
(n + |b|)
(p > |b|) ,
then
(3.2)
p
Hn,m
(λ, b) ⊂ Nn,δ (h).
p
Proof. Let f (z) ∈ Hn,m
(λ, b). Then, in view of the assertion (2.1) of Theorem 1, we have
∞
λ+n+p−1 n+p X
p
(3.3)
(n + |b|)
ak 5 |b|
.
n
m
m
k=n+p
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I NCLUSION AND N EIGHBORHOOD P ROPERTIES OF S OME A NALYTIC
AND
M ULTIVALENT F UNCTIONS
5
This yields
∞
X
(3.4)
k=n+p
ak 5
p
m
.
λ+n+p−1 n+p
n
m
|b|
(n + |b|)
Applying the assertion (2.1) of Theorem 1 again, in conjunction with (3.4), we obtain
∞
λ+n+p−1 n+p X
kak
n
m
k=n+p
∞
p
λ+n+p−1 n+p X
5 |b|
+ (p − |b|)
ak
m
n
m
k=n+p
p
λ+n+p−1 n+p
5 |b|
+ (p − |b|)
m
n
m
p
|b| m
n+p
·
(n + |b|) λ+n+p−1
m
n
n+p
p
.
= |b|
m
n + |b|
Hence
(3.5)
∞
X
k=n+p
kak 5
p
m
λ+n+p−1 n+p
n
m
|b| (n + p)
(n + |b|)
=: δ
(p > |b|),
which, by virtue of (1.12), establishes the inclusion relation (3.2) of Theorem 3.
In an analogous manner, by applying the assertion (2.4) of Theorem 2 instead of the assertion
(2.1) of Theorem 1 to functions in the class Lpn,m (λ, b; µ), we can prove the following inclusion
relationship.
Theorem 4. If
(p − m)(n + p)
(3.6)
δ=
h
[µ (n + p − 1) + 1]
|b|−1
m!
+
λ+n+p−1
n
i
p
m
n+p
m
(µ > 1),
then
Lpn,m (λ, b; µ) ⊂ Nn,δ (h).
Remark 3. Applying the parametric substitutions listed in (2.3), Theorems 3 and 4 would yield
the known results due to Murugusundaramoorthy and Srivastava [4, p. 4, Theorem 1; p. 5,
Theorem 2]. Incidentally, just as we indicated in Section 2 above, the condition µ > 1 is needed
in the proof of one of these known results [4, p. 5, Theorem 2]. This implies that the constraint
0 5 µ 5 1 in [4, p. 3, Equation (1.14)] should be replaced by the less stringent constraint
µ = 0.
4. F URTHER N EIGHBORHOOD P ROPERTIES
In this last section, we determine the neighborhood properties for each of the following
(slightly modified) function classes:
p,α
Hn,m
(λ, b) and Lp,α
n,m (λ, b; µ).
J. Inequal. Pure and Appl. Math., 7(1) Art. 5, 2006
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6
R.K. R AINA AND H.M. S RIVASTAVA
p,α
Here the class Hn,m
(λ, b) consists of functions f (z) ∈ Ap (n) for which there exists another
p
function g(z) ∈ Hn,m (λ, b) such that
f (z)
< p − α (z ∈ U; 0 5 α < p) .
(4.1)
−
1
g(z)
Analogously, the class Lp,α
n,m (λ, b; µ) consists of functions f (z) ∈ Ap (n) for which there exists
another function g(z) ∈ Lpn,m (λ, b; µ) satisfying the inequality (4.1).
The proofs of the following results involving the neighborhood properties for the classes
p,α
Hn,m
(λ, b) and Lp,α
n,m (λ, b; µ)
are similar to those given in [1] and [4]. We, therefore, skip their proofs here.
p
Theorem 5. Let g(z) ∈ Hn,m
(λ, b). Suppose also that
(4.2)
δ (n + |b|) λ+n+p−1
n h
α=p−
λ+n+p−1
(n + p) (n + |b|)
n+p
n+p
m
n+p
−
m
|b|
p
m
i .
Then
(4.3)
p,α
Nn,δ (g) ⊂ Hn,m
(λ, b).
Theorem 6. Let g(z) ∈ Lpn,m (λ, b; µ). Suppose also that
δ [µ (n + p − 1) + 1]
h
(4.4) α = p −
(n + p) [µ (n + p − 1) + 1] λ+n+p−1
n
λ+n+p−1
n+p−1
n
m
n
n+p−1
− (p − m) |b|−1
m
m!
+
p
m
oi .
Then
(4.5)
Nn,δ (g) ⊂ Lp,α
n,m (λ, b; µ).
R EFERENCES
[1] O. ALTINTAŞ, Ö. ÖZKAN AND H.M. SRIVASTAVA, Neighborhoods of a class of analytic
functions with negative coefficients, Appl. Math. Lett., 13(3) (2000), 63–67.
[2] O. ALTINTAŞ, Ö. ÖZKAN AND H.M. SRIVASTAVA, Neighborhoods of a certain family of
multivalent functions with negative coefficients, Comput. Math. Appl., 47 (2004), 1667–1672.
[3] A.W. GOODMAN, Univalent functions and nonanalytic curves, Proc. Amer. Math. Soc., 8 (1957),
598–601.
[4] G. MURUGUSUNDARAMOORTHY AND H.M. SRIVASTAVA, Neighborhoods of certain classes
of analytic functions of complex order, J. Inequal. Pure Appl. Math., 5(2) (2004), Art. 24. 8 pp.
[ONLINE: http://jipam.vu.edu.au/article.php?sid=374].
[5] S. RUSCHEWEYH, Neighborhoods of univalent functions, Proc. Amer. Math. Soc., 81 (1981), 521–
527.
[6] H.M. SRIVASTAVA AND S. OWA (Eds.), Current Topics in Analytic Function Theory, World
Scientific Publishing Company, Singapore, New Jersey, London and Hong Kong, 1992.
J. Inequal. Pure and Appl. Math., 7(1) Art. 5, 2006
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