2_834_frasin.dvi


EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Vol. 4, No. 1, 2011, 14-19
ISSN 1307-5543 – www.ejpam.com

(α,β ,δ)−Neighborhood for Certain Analytic Functions with Neg-

ative Coefficients

B.A. Frasin

Faculty of Science, Department of Mathematics, Al al-Bayt University, P.O. Box: 130095 Mafraq,

Jordan

Abstract. In this paper, we introduce (α,β ,δ)−neighborhoods of analytic functions with negative
coefficients. Furthermore, we obtain some interesting results for functions belonging to this neighbor-
hoods.

2000 Mathematics Subject Classifications: 30C45

Key Words and Phrases: Analytic functions, Neighborhood

1. Introduction and definitions

Let T denote the class of functions of the form :

f (z) = z −
∞
∑

n=2

anzn, (an ≥ 0). (1)

which are analytic in the open unit disk U = {z : |z| < 1}.For a function f (z) ∈ T , we define

D0 f (z) = f (z),

D1 f (z) = D f (z) = z f ′(z),

and

Dk f (z) = D(Dk−1 f (z))

= z −
∞
∑

n=2

nkanzn (k ∈ N0 = N∪ {0}).

The differential operator Dk was introduced by Sălăgean [12].

Email address: bafrasin�yahoo.
om
http://www.ejpam.com 14 c© 2010 EJPAM All rights reserved.



B. Frasin / Eur. J. Pure Appl. Math, 4 (2011), 14-19 15

Following a recent investigation by Frasin and Darus [6] [see also 1], if f (z) ∈ T and
µ ≥ 0, then we define the (k,µ)-neighborhood for the function f (z) by

N k

µ
( f ) = {g ∈ T : g(z) = z −

∞
∑

n=2

bnzn,
∞
∑

n=2

nk+1
�

�an− bn

�

� ≤ µ}. (2)

In particular, for the identity function e(z) = z, we immediately have

N k

µ
(e) = {g ∈ T : g(z) = z −

∞
∑

n=2

bnzn,
∞
∑

n=2

nk+1
�

�bn

�

� ≤ µ}, (3)

We observe that N 0
µ
( f ) ≡ N

µ
( f ) and N 1

µ
( f ) ≡ M

µ
( f ), where N k

µ
( f ) and Mµ( f ) denote,

respectively, the µ-neighborhoods of f as defined by Ruscheweyh [11] and Silverman [13].
For further details about the neighborhood of analytic functions see (as examples) the papers
in [2, 3, 4, 7, 5, 8, 9].

Very recently, Orhan et al. [10], introduced new definition of (α,δ)-neighborhood for
analytic function f (z) in the form

f (z) = z +

∞
∑

n=2

anzn, (4)

In this paper, we introduce the following new definition of (α,β ,δ)-neighborhood for a func-
tion given by 1.

Definition 1. A function f (z) ∈ T is said to be (α,β ,δ)-neighborhood for

g(z) = z −
∞
∑

n=2
bnzn ∈ T if it satisfies

�

�eiα(Dk f (z))′ − eiβ (Dk g(z))′
�

� < δ (z ∈ U ) (5)

for some −π≤ α,β ≤ π and δ >
p

2(1− cos(α−β)).
We denote this neighborhood by (α,β ,δ)−N (g).

Now we show some results for functions belonging to (α,β ,δ)−N (g).

2. Main results

In our first theorem, we introduce a sufficient condition to be in (α,β ,δ)−N (g).

Theorem 1. If f (z) ∈ T satisfies

∞
∑

n=2

nk+1
�

�eiαan− eiβ bn

�

�≤ δ−
p

2(1− cos(α− β)) (6)

for some −π≤ α,β ≤ π and δ >
p

2(1− cos(α−β)) then f (z) ∈ (α,β ,δ)−N (g).



B. Frasin / Eur. J. Pure Appl. Math, 4 (2011), 14-19 16

Proof. We observe that

�

�eiα(Dk f (z))′ − eiβ(Dk g(z))′
�

� =

�

�

�

�

�

eiα − eiβ −
∞
∑

n=2

nk+1(eiαan − eiβ bn)z
n−1

�

�

�

�

�

≤
�

�eiα − eiβ
�

�+

∞
∑

n=2

nk+1
�

�eiαan − eiβ bn

�

� |z|n−1

≤
p

2(1− cos(α− β)) +
∞
∑

n=2

nk+1
�

�eiαan− eiβ bn

�

� .

If
∞
∑

n=2

nk+1
�

�eiαan − eiβ bn

�

�≤ δ−
p

2(1− cos(α−β)),

then we have
�

�eiα(Dk f (z))′ − eiβ(Dk g(z))′
�

�< δ (z ∈ U ). This shows that
f (z) ∈ (α,β ,δ)−N (g).

Corollary 1. Let f (z) ∈ T . Then for 0< µ ≤ δ, we have N k
µ (g)⊆ (α,α,δ)−N k

δ
(g).

Proof. Assuming that f (z) ∈ N k
µ (g). We find from the definition (2) that

∞
∑

n=2

nk+1
�

�an − bn

�

� ≤ µ.

Now

∞
∑

n=2

nk+1
�

�eiαan− eiαbn

�

� =

∞
∑

n=2

nk+1
�

�eiα
�

�

�

�an− bn

�

�

=

∞
∑

n=2

nk+1
�

�an − bn

�

�

≤ δ.

Thus by Theorem 1, we have f (z) ∈ (α,α,δ)−N (g).

Corollary 2. If f (z) ∈ T satisfies

∞
∑

n=2

nk+1
�

�

�

�an

�

�−
�

�bn

�

�

�

� ≤ δ−
p

2(1− cos(α− β)) (7)

for some −π≤ α,β ≤ π, δ >
p

2(1− cos(α− β)) and arg an− arg bn = β −α (n= 2,3,4, ...),
then f (z) ∈ (α,β ,δ)−N (g).



B. Frasin / Eur. J. Pure Appl. Math, 4 (2011), 14-19 17

Proof. Let arg an− arg bn = β −α and arg an = θ n. Then arg bn = θ n+α−β . Therefore,

eiαan − eiβ bn =
�

�an

�

� ei(α+θ n) −
�

�bn

�

� ei(α+θ n),

which implies
�

�eiαan− eiβ bn

�

�=
�

�

�

�an

�

�−
�

�bn

�

�

�

� . (8)

From the hypotheses (7) and (8), we get (6). Thus by Theorem 1, it follows that
f (z) ∈ (α,β ,δ)−N (g).

Furthermore, from Theorem 1, we easily get

Corollary 3. If f (z) ∈ T satisfies

∞
∑

n=2

nk+1(
�

�an

�

�+
�

�bn

�

�)≤ δ−
p

2(1− cos(α− β)) (9)

for some −π≤ α,β ≤ π and δ >
p

2(1− cos(α−β) then f (z) ∈ (α,β ,δ)−N (g).

Next, we prove

Theorem 2. If f (z) ∈ (α,β ,δ)−N (g) and arg(eiαan − eiβ bn) = (n− 1)ϕ (n = 2,3,4, . . .),
then

∞
∑

n=2

nk+1
�

�eiαan− eiβ bn

�

�> δ+ cosα− cosβ . (10)

Proof. Let f (z) ∈ (α,β ,δ)−N (g) and arg z = −ϕ. Then for all z ∈ U , we have

�

�eiα(Dk f (z))′− eiβ (Dk g(z))′
�

� =

�

�

�

�

�

(eiα− eiβ )−
∞
∑

n=2

nk+1(eiαan − eiβ bn)z
n−1

�

�

�

�

�

=

�

�

�

�

�

(eiα− eiβ )−
∞
∑

n=2

nk+1
�

�eiαan− eiβ bn

�

� ei(n−1)ϕ |z|n−1 e−i(n−1)ϕ

�

�

�

�

�

=

�

�

�

�

�

(eiα− eiβ )−
∞
∑

n=2

nk+1
�

�eiαan− eiβ bn

�

� |z|n−1

�

�

�

�

�

=

 

[(cosα− cosβ)−
∞
∑

n=2

nk+1
�

�eiαan− eiβ bn

�

� |z|n−1]2+

(sinα− sinβ)2
�1/2

< δ

for z ∈ U . This implies that

(cosα− cosβ)−
∞
∑

n=2

nk+1
�

�eiαan − eiβ bn

�

� |z|n−1 < δ



REFERENCES 18

for z ∈ U . Letting |z| → 1−, we have

∞
∑

n=2

nk+1
�

�eiαan− eiβ bn

�

�> δ+ cosα− cosβ .

Finally, we prove

Theorem 3. If f (z) ∈ T satisfies

∞
∑

n=2

nk+1
�

�eiαan− eiβ bn

�

� < µ−
p

2(1− cos(α−β)) (11)

for some −π≤ α,β ≤ π and µ >
p

2(1− cos(α− β)), then

Re

�

eiα(Dk f (z))′

eiβ (Dk g(z))′

�

> 0 (12)

where g(z) ∈ N k
1−µ(e).

Proof. Note that

�

�

�

�

�

eiα(Dk f (z))′

eiβ(Dk g(z))′
− 1

�

�

�

�

�

≤

p

2(1− cos(α− β)) +
∞
∑

n=2
nk+1

�

�eiαan − eiβ bn

�

�

1−
∞
∑

n=2
nk+1bn

≤

p

2(1− cos(α− β)) +
∞
∑

n=2
nk+1

�

�eiαan − eiβ bn

�

�

µ
.

Hence by the condition (11), we have
�

�

�

�

�

eiα(Dk f (z))′

eiβ (Dk g(z))′
− 1

�

�

�

�

�

< 1.

This evidently proves Theorem 3.

References

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derivatives, Math. Japonica 51. No.3, 487-492. 2000.

[2] O. Altintaş, Neighborhoods of certain p-valently analytic functions with negative coeffi-

cients, Appl. Math. Comp., 187, 47–53. 2007.



REFERENCES 19

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