6_xxx_dziok.dvi


EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Vol. 2, No. 4, 2009, (544-553)

ISSN 1307-5543 – www.ejpam.com

Inclusion and Neighborhood Properties of Certain

Subclasses of Analytic and Multivalent Functions

M. K. Aouf1 and J. Dziok2∗

1 Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516,

Egypt

2 Institute of Mathematics, University of Rzeszow, ul. Rejtana 16A, PL-35-310 Rzeszow,

Poland

Abstract. In the paper we introduce and investigate two new subclasses of multivalently an-

alytic functions defined by Dziok-Srivastava operator. In this paper we obtain the coefficient

estimates and the consequent inclusion relationships involving the neighborhoods of the ana-

lytic functions.

2000 Mathematics Subject Classifications: 30C45, 25A33.

Key Words and Phrases: Analytic functions, p-valent functions, the, Dziok-Srivastava opera-

tor, neighborhood.

∗Corresponding author.

Email addresses: mkaouf127�yahoo.
om (M. Aouf), jdziok�univ.rzeszow.pl (J. Dziok)

http://www.ejpam.com 544 c© 2009 EJPAM All rights reserved.



M. Aouf and J. Dziok / Eur. J. Pure Appl. Math, 2 (2009), (544-553) 545

1. Introduction

Let Ap(n) denote the class of functions of the form:

f (z) = zp +

∞
∑

k=n

akzk (p, n ∈ N = {1, 2, ....}, p < n), (1)

which are analytic in the open unit disc U = {z : |z| < 1}. If f (z) ∈ Ap(n) is given by

(1) and g(z) ∈ Ap(n) is given by

g(z) = zp +

∞
∑

k=n

bkzk (z ∈ U),

then the Hadamard product (or convolution ) ( f ∗ g)(z) of f (z) and g(z) is defined

by

( f ∗ g)(z) = zp +

∞
∑

k=n

ak bkzk.

For complex parameters α1...,αr and β1, ....,βs (β j ∈ C\{0,−1,−2, ...}; j = 1, ..., s), we

define the generalized hypergeometric function r Fs(α1...,αr;β1, ....,βs; z) by

r Fs(α1...,αr;β1, ....,βs; z) =

∞
∑

k=0

(α1)k......(αr)k

(β1)k.......(βs)k
.
zk

k!

(r ≤ s+ 1; r, s ∈ N0 = N ∪ {0} ; z ∈ U),

where (θ )k is the Pochhammer symbol defined, in terms of the Gamma function Γ ,

by

(θ )k =
Γ(θ + k)

Γ(θ )
=







1 (k = 0)

θ (θ + 1)....(θ + k− 1) (k ∈ N).

Corresponding to a function hp(α1, ....,αr;β1, ....,βs; z) defined by

hp(α1, ....,αr;β1, ....,βs; z) = zp
r Fs(α1, ....,αr;β1, ....,βs; z),

we consider a linear operator Hp(α1, ....,αr;β1, ....,βs) : Ap(n)→ Ap(n), defined by the

convolution

Hp(α1, ....,αr;β1, ....,βs) f (z) = hp(α1, ....,αr;β1, ....,βs; z) ∗ f (z).



M. Aouf and J. Dziok / Eur. J. Pure Appl. Math, 2 (2009), (544-553) 546

We observe that, for a function f (z) of the form (1), we have

Hp(α1, ....,αr;β1, ....,βs) f (z) = zp +

∞
∑

k=n

Γkakzk,

where

Γk =
(α1)k−p......(αr)k−p

(β1)k−p......(βs)k−p (k− p)!
. (2)

For convenience, we write

H p
r,s
= Hp(α1, .....,αr;β1, .....βs).

The linear operator H p
r,s

was introduced by Dziok and Srivastava [1].

We denote by Tp(n) the subclass of Ap(n) consisting of functions f (z) of the form:

f (z) = zp −
∞
∑

k=n

akzk (ak ≥ 0) . (3)

By using the linear operator H p
r,s

we introduce a new subclass S(p, n, q,λ,β) of the

class Tp(n), which consists of functions f (z) ∈ Tp(n) satisfying the inequality:
�

�

�

�

�

z(H p
r,s

f )(1+q)(z) + λz2(H p
r,s

f )(2+q)(z)

λz(H
p
r,s f )(1+q)(z) + (1−λ)(H p

r,s f )(q)(z)
− (p− q)

�

�

�

�

�

< β (4)

(z ∈ U ; p ∈ N ; q ∈ N0; q < k− 1; k ≥ n; 0 ≤ λ≤ 1;β > 0).

Also, let P(p, n, q,λ,β) denote the subclass of Tp(n) consisting of functions f (z)which

satisfy the inequality:
�

�

�

�

�

(1−λ)
(H p

r,s
f )(q)(z)

zp−q
+λ
(H p

r,s
f )(1+q)(z)

(p− q)zp−q−1
− (p− q+ 1)q

�

�

�

�

�

< β (5)

(z ∈ U ; p ∈ N ; q ∈ N0; q < k− 1; k ≥ n;λ ≥ 0;β > 0) .

Now we define two classes related to the classes S(p, n, q,λ,β) and P(p, n, q,λ,β).

A function f (z) ∈ Tp(n) is said to be in the class Sγ(p, n, q,λ,β) if there exists a

function g(z) ∈ S(p, n, q,λ,β) such that
�

�

�

�

f (z)

g(z)
− 1

�

�

�

�

< γ (z ∈ U ; γ > 0). (6)



M. Aouf and J. Dziok / Eur. J. Pure Appl. Math, 2 (2009), (544-553) 547

Analogously, a function f (z) ∈ Tp(n) is said to be in the class Pγ(p, n, q,λ,β) if

there exists a function g(z) ∈ P(p, n, q,λ,β) such that the inequality (6) holds true.

We note that for suitable chosen parameters the classes were investigated by

(among others) Srivastava et al. ( [2] and [3]). Also, following the earlier inves-

tigation by Goodman [4], Ruscheweyh [5], and others we define the (n,δ)− neigh-

borhood of a function f (z) of the form (3) by

Nn,δ( f ) =

(

g(z) = zp −
∞
∑

k=n

bkzk ∈ Tp(n) :

∞
∑

k=n

k
�

�ak − bk

�

� ≤ δ

)

. (7)

In particular, if

h(z) = zp (p ∈ N),

we immediately have

Nn,δ(h) =

(

g(z) = zp −
∞
∑

k=n

bkzk ∈ Tp(n) :

∞
∑

k=n

k
�

�bk

�

�≤ δ

)

. (8)

The neighborhoods of function was studied among others by Altintas et al. ( [6],

[7] and [8]), Srivastava et al. ( [2], [3], [9]and [10]) and Aouf [11] (see also

Prajapart and Raina [12]). In this paper we obtain the coefficient estimates and

the consequent inclusion relationships involving the neighborhoods of some analytic

functions.

2. Coefficient Estimates

In our investigation of the inclusion relations involving Nn,δ(h), we shall require

Theorems 1 and 2 below.

Theorem 1. Let the function f (z) ∈ Tp(n) be defined by (3). Then f (z) is in the class

S(p, n, q,λ,β) if and only if

∞
∑

k=n

(k+ β − p)Ckak ≤ βCp , (9)



M. Aouf and J. Dziok / Eur. J. Pure Appl. Math, 2 (2009), (544-553) 548

where

Ck = [1+λ(k− q− 1)]
�

k− q+ 1
�

q Γk (10)

and Γk is given by (2).

Proof. Let a function f (z) of the form (3) belong to the class S(p, n, q,λ,β). Then,

in view of (3) and (4), we obtain the following inequality:

Re

(

z(H p
r,s

f )(1+q)(z) +λz2(H p
r,s

f )(2+q)(z)

λz(H
p
r,s f )(1+q)(z) + (1−λ)(H p

r,s f )(q)(z)
− (p− q)

)

> −β (z ∈ U),

or, equivalently,

Re











−
∞
∑

k=n

(k− p)Ckakzk−p

Cp −
∞
∑

k=n

Ckakzk−p











> −β (z ∈ U).

Setting z = r (0≤ r < 1) we obtain

∞
∑

k=n

(k− p)Ckak rk−p

Cp −
∞
∑

k=n

Ckak rk−p

< β (0≤ r < 1).

We observe that the expression in the denominator of the left-hand side of is positive

for r = 0 and also for 0 < r < 1. Thus we have

∞
∑

k=n

(k+ β − p)Ckak rk−p ≤ βCp,

and, by letting r → 1− through real values, we obtain the desired assertion of Theo-

rem 1. Conversely, by applying the hypothesis (9) and letting |z| = 1, we find from

(3) that

�

�

�

�

�

z(H p
r,s

f )(1+q)(z) +λz2(H p
r,s

f )(2+q)(z)

λz(H
p
r,s f )(1+q)(z) + (1−λ)(H p

r,s f )(q)(z)
− (p− q)

�

�

�

�

�

=

�

�

�

�

�

�

�

�

�

∞
∑

k=n

(k− p)Ckakzk−p

Cp −
∞
∑

k=n

Ckakzk−p

�

�

�

�

�

�

�

�

�



M. Aouf and J. Dziok / Eur. J. Pure Appl. Math, 2 (2009), (544-553) 549

≤

∞
∑

k=n

(k− p)Ckak

Cp −
∞
∑

k=n

Ckak

≤ β

Cp −
∞
∑

k=n

Ckak

Cp −
∞
∑

k=n

Ckak

= β .

Hence, by the maximum modulus theorem, we have f (z) ∈ S(p, n, q,λ,β), which

evidently completes the proof of Theorem 1.

Similarly, we can prove the following theorem.

Theorem 2. Let the function f (z) ∈ Tp(n) be given by (3). Then f (z) ∈ P(p, n, q,λ,β)

if and only if

∞
∑

k=n

�

p− q+λ
�

k− p
��

(k− q+ 1)qΓkak ≤ β
�

p− q
�

. (11)

where Γk is given by (2).

Using Theorems 1 and 2 we obtain following two corollaries.

Corollary 1. If the function f (z) given by (3) belongs to the class P(p, n, q,λ,β), then

ak ≤
βCp

(k+ β − p)Ck

, (k = n, n+ 1, ...) ,

where Ck is given by (10). The result is sharp.

Corollary 2. If the function f (z) given by (3) belongs to the class S(p, n, q,λ,β), then

ak ≤
β
�

p− q
�

�

p− q+λ
�

k− p
��

(k− q+ 1)qΓk

, (k = n, n+ 1, ...) ,

where Γk is given by (2). The result is sharp.

3. Neighborhoods Properties

Our first inclusion relation Nn,δ(h) is given in the following theorem.



M. Aouf and J. Dziok / Eur. J. Pure Appl. Math, 2 (2009), (544-553) 550

Theorem 3. If

Cn ≤ Ck (k = n, n+ 1, ...) , (12)

then

S(p, n, q,λ,β) ⊂ Nn,δ(h), (13)

where Ck is given by (10) and

δ =
nβCp

(n+ β − p)Cn

�

p ≥ β
�

.

Proof. Let f (z) ∈ S(p, n, q,λ,β). Using Theorem 1, by (12), we have

(n+ β − p)Cn

∞
∑

k=n

ak ≤
∞
∑

k=n

(k+ β − p)Ckak ≤ βCp,

which readily yields
∞
∑

k=n

ak ≤
βCp

(n+ β − p)Cn

. (14)

Making use of (9) again, in conjunction with (12) and (14), we get

Cn

∞
∑

k=n

kak ≤ βCp + (p− β)Cn

∞
∑

k=n

ak ≤ βCp +
(p− β)βCp

n+ β − p
=

nβCp

n+ β − p

Hence
∞
∑

k=n

kak ≤
nβCp

(n+ β − p)Cn

= δ,

which, by means of the definition (8), establishes the inclusion relation (13) asserted

by Theorem 1.

Remark 1. Putting λ = 0,β = |b| , b ∈ C\{0}, replacing n by n+ p (p, n ∈ N) and

taking r = 2; s = 1;α1 = µ+ p (µ > −p; p ∈ N);α2 = β1 = 1 in Theorem 3, we obtain

the result obtained by Raina and Srivastava [9].

In a similar manner, by applying the assertion (11) of Theorem 2 instead of the

assertion (9) of Theorem 1 to functions in the class P(p, n, q,λ,β) we can prove the

following inclusion relationship.



M. Aouf and J. Dziok / Eur. J. Pure Appl. Math, 2 (2009), (544-553) 551

Theorem 4. If

(n− q+ 1)qΓn ≤ (k− q+ 1)qΓk (k = n, n+ 1, ...) , (15)

then

P(p, n, q,λ,β) ⊂ Nn,δ(h),

where

δ =
nβ
�

p− q
�

�

p− q+λ
�

n− p
��

(n− q+ 1)qΓn)
(q+λp ≥ p).

Theorem 5. Let g(z) ∈ S(p, n, q,λ,β). If Ck given by (10) satisfies (12) and

γ >
δ

n

(n+ β − p)Cn

(n+ β − p)Cn − βCp

(δ > 0) , (16)

then

Nn,δ(g) ⊂ Sγ(p, n, q,λ,β).

Proof. Suppose that f (z) ∈ Nn,δ(k). We find from (8) that

∞
∑

k=n

k
�

�ak − bk

�

� ≤ δ,

which readily implies that
∞
∑

k=n

�

�ak − bk

�

�≤
δ

n
.

Next, since g(z) ∈ S(p, n, q,λ,β), we have [c.f. equation (14)] that

∞
∑

k=n

bk ≤
βCp

(n+ β − p)Cn

,

so that

�

�

�

�

f (z)

g(z)
− 1

�

�

�

�

≤

∞
∑

k=n

�

�ak − bk

�

�

1−
∞
∑

k=n

bk

≤
δ

n

(n+ β − p)Cn

(n+ β − p)Cn − βCp

Thus, by (16) we have. f (z) ∈ Sγ(p, n, q,λ,β). This evidently proves Theorem 5.

The proof of Theorem 6 below is similar to that of Theorem 5 above therefore, we

omit the details involved



REFERENCES 552

Theorem 6. Let g(z) ∈ P(p, n, q,λ,β). If the condition (15) holds true and

γ >
δ

n

�

p− q+λ
�

n− p
��

(n− q+ 1)qΓk
�

p− q+λ
�

n− p
��

(n− q+ 1)qΓn− β
�

p− q
� (δ > 0) ,

then

Nn,δ(g)⊂ Pγ(p, n, q,λ,β).

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