Research Journal of Applied Sciences, Engineering and Technology 5(15): 3975-3977, 2013
ISSN: 2040-7459; e-ISSN: 2040-7467
© Maxwell Scientific Organization, 2013
Submitted: November 08, 2012
Accepted: December 06, 2012
Published: April 25, 2013
Neighborhoods of Certain Classes of Generalized Ruscheweyh Type Analytic Functions of
Complex Order
1
Jamal Kazemi Asl and 2Hesam Mahzoon
Payame Noor University, Bandar Imam Khomeini, Iran
2
Department of Mathematics, Firoozkooh Branch, Islamic Azad University, Firoozkooh, Iran
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Abstract: In this study, we introduce and study the (n, δ) neighborhoods of subclasses S n (γ, λ, β, η) R n (γ, λ, β, μ,
η) Sα n (γ, λ, β, μ, η) and Rα n (γ, λ, β, μ, η) of the class A(n) of normalized analytic functions in U = {z ∈ ℂ : |z|<1}
with negative coefficients, which are defined by using of the generalized Ruscheweyh derivative operator.
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Keywords: Analytic functions, convex functions, neighborhoods generalized ruscheweyh derivative, Star like
functions, univalent functions
INTRODUCTION
𝑅𝑅 �1 +
Let A(n) denote the class of functions f of the form:
1 𝑧𝑧𝑓𝑓 " (𝑧𝑧)
𝛾𝛾 𝑓𝑓′ (𝑧𝑧)
𝑓𝑓(𝑧𝑧) = −
{1, 2, ….} =)
(𝛼𝛼𝑘𝑘 ≥ 0, 𝑘𝑘 ∈ ℕ =
(1)
which are analytic in the open unit disk:
{
R
∞
∞
∑bz
∞
is defined as ( f * g )( z ) =
z + ∑ k =2 α k b k z k .
The main objective of this study is to use a recent
generalization of the Ruscheweyh derivative operator
(Al-Shaqsi and Darus, 2007) denoted Dλ η , define as
follows:
}
∞
and ∑ k α k − b k ≤ δ (2)
k
k
k=
k +1
k=
n +1
P
which is the (n, δ ) neighborhood of f.
For the identity function e(z) = z, we have:
{
N n ,δ ( f ) = g ∈ A ( n ) : g ( z ) = z −
∞
∑bz
∞
k
k
k=
k +1
and
∑kb
D ληf =
(z)
}
≤ δ (3)
k
k=
n +1
R
R
− 1]� > 0(𝑧𝑧 ∈ 𝑈𝑈, 𝛾𝛾 ∈ ℂ/{0})
A function f ∈ A (n) is said to be convex of
complex order γ(γ∈/{0})that is, f ∈ C n (γ) if:
R
R
z
(1 − z )
1+λ
PR
R
* Dηf ( z ) ( f ∈ A ( n ) , z ∈ U )
where, * stands for the Hadamard product of two power
series. Further, we have:
The concept of neighborhoods was first introduced
by Goodman (1983) and then generalized by
Ruscheweyh (1981).
A function f ∈ A is star like of complex order
γ(γ ∈ ℂ/{0})that is, f ∈ C n (γ) if:
𝑓𝑓(𝑧𝑧)
R
=
k 2=
k 2
N n,δ ( f ) = g ∈ A ( n ) : g ( z ) = z −
1 𝑧𝑧𝑓𝑓 ′ (𝑧𝑧)
R
∞
For any function f ∈ A(n) and δ ≥0, we define:
𝛾𝛾
PR
f ( z ) =z + ∑ α k z k and g ( z ) =z + ∑ b k z k
U = {z ∈ ℂ : |z|<1}
𝑅𝑅 �1 + [
(5)
The classes S* n (γ) and C n (γ) stem essentially form
the classes of starlike and convex functions of complex
orders which were introduced earlier by Nasr and Aouf
(1985). The Hadamard product of two power series:
P
𝑘𝑘
∑∞
𝑘𝑘=𝑛𝑛+1 𝛼𝛼𝑘𝑘 𝑧𝑧
� > 0(𝑧𝑧 ∈ 𝑈𝑈, 𝛾𝛾 ∈ ℂ/{0})
Dηf ( z )=
(1 − η) f ( z ) + ηzf ′ ( z ) , λ > −1, η ≥ 0,z ∈ U.
We can easily see that Dλ η admits a representation of
the form:
P
Dλη f ( z )= z −
(4)
∞
PR
R
 λ + k − 1
k
 α k z ( λ > −1,f ∈ A ( n ) , η ≥ 0 )
 k −1 
∑ 1 + ( k − 1) η 
k= k +1
(6)
Make use of the following standard notation:
Corresponding Author: Jamal Kazemi Asl, Payame Noor University, Bandar Imam Khomeini, Iran
3975
Res. J. Appl. Sci. Eng. Technol., 5(15): 3975-3977, 2013
𝐾𝐾
� �=
𝑘𝑘
𝐾𝐾(𝐾𝐾−1)(𝐾𝐾−2)…(𝐾𝐾−𝑘𝑘+1)
𝑘𝑘!
Hence, by the maximum modulus Theorem, we
have f ∈ R n (γ, λ, β, η) Similarly, we can prove the
following Lemma.
(𝐾𝐾 ∈ ℂ, 𝑘𝑘 ∈ ℕ)
A function f ∈ A (n) is said to belong to the class
S n (γ, λ, β, η) if:
'
λ

1  z ( D ηf ( z ) )

− 1 < β,
λ

D ηf
γ


(7)
Lemma 2: A function f ∈R n (γ, λ, β, μ, η) if and only
if:
(12)
 λ + k − 1
 µ ( k − 1) + 1 α k ≤ β γ .
 k −1 
∞
∑ 1 + ( k − 1) η 
k= n +1
where, 𝜆𝜆 ∈ ℂ / {0} , λ > −1,0 < β ≤ 1, η ≥ 0 and 𝑧𝑧 ∈ U.
A function f ∈ A (n) is said to belong to the class
R n (γ, λ, β, μ, η) if:
Theorem 1: if:
( n + 1) β γ
=
δ
Dλη f ( z )

'
1
+ µ ( Dλη f ( z ) ) − 1 < β,
 (1 − µ )
γ
z

λ + n

 n 
(β γ + n ) (1 + nη) 
(8)
Then, S n (γ, λ, β, η) ∩ N n,δ (e)
where, 𝜆𝜆 ∈ ℂ / {0} ,0 < β ≤ 1, η ≥ 0, λ > −1 and z ∈ U.
Proof: Let f ∈S n (γ, λ, β, η). By lemma 2, we have,
Neighborhoods for classes: S n (γ, λ, β, η) and R n (γ,
λ, β, μ, η)
λ + n ∞
 ∑ αk ≤ β γ
 n  k= n +1
(β γ + n ) (1 + nη) 
Lemma 1: A function f ∈ Sα n (γ, λ, β,η) if and only if:
So,
 λ + k − 1
(9)
β γ + k − 1) α k ≤ β γ .
1 + ( k − 1) η 
(
∑

k= n +1
 k −1 
∞
∞
∑
k= n +1
Proof: Let f ∈S n (γ, λ, β, η) Then, by (6) we can write:
 z ( D λ f ( z ) ) '  
η

 − 1 > −β γ ( z ∈ U ) .
R  λ

D
f
z
(
)
η




(10)
αk ≤
λ + n

 n 
(β γ + n ) (1 + nη) 
Using (9) and (14), we have:
λ + n ∞
λ + n ∞
 ∑ kα k ≤ β γ + (1 − β γ ) (1 + nη)  n  ∑ α k
n
=
+
k
n
1
n +1



 k=
(1 + nη) 
λ + n
≤ β γ + (1 − β γ ) (1 + nη) 

 n 
βγ
λ + n
(β γ + n ) (1 + nη)  n 


(11)


 λ + k − 1
∞
k
 ∑ k= n +1 1 + ( k − 1) η 
 ( k − 1) α k z 
k
1
−


 > −β γ ( z ∈ U ) .
R −


 λ + k − 1
∞
k
z
1
k
1
z
−
+
−
η
α


(
)
∑ k= n +1 


 k −1  k




≤
We choose values of z on the real axis and let Z
→1- , through real values, the inequality (11) yields the
desired condition (9). Conversely, by applying the
hypothesis (2.1) and letting ‫׀‬Z‫= ׀‬1 we have:
Dληf ( z )
 λ + k − 1
k
1 + ( k − 1) η  k − 1  ( k − 1) α k z


−1 =
 λ + k − 1
∞
k
z − ∑ k= n +1 1 + ( k − 1) η 
 αkz
 k −1 
∑
∞
( n + 1) β γ
(β γ + n )
( γ < 1) ,
That is:
∞
∑ kα
k= n +1
′
k= n +1

 λ + k − 1 
∞
β γ 1 − ∑ k= n +1 1 + ( k − 1) η 
 αk 
 k −1  

≤
 λ + k − 1
∞
1 − ∑ k= n +1 1 + ( k − 1) η 
 αk
 k −1 
(14)
βγ
Equivalently:
z ( Dληf ( z ) )′
(13)
, ( γ < 1) ,
k
≤
( n + 1) β γ
λ + n
(β γ + n ) (1 + nη)  n 


=δ.
Thus, by the definition given by (3), f ∈N n,δ (e).
Theorem 2: if:
δ=
≤β γ
3976
( n + 1) β γ
λ + n

 n 
( µn + 1)(1 + nη) 
(15)
Res. J. Appl. Sci. Eng. Technol., 5(15): 3975-3977, 2013
Then, R n (γ, λ, β, μ, η) ⊂ N n,δ (e).
Proof: Let f ∈R n (γ, λ, β, μ, η). Then, by lemma 2, we
have:
λ + n

 n 
α = 1−


λ + n
( n + 1) (β γ + n ) (1 + nη) 
−β γ 
n




(β γ + n ) δ (1 + nη) 
λ + n ∞
 ∑ αk ≤ β γ ,
 n  k= n +1
Then N n ,δ ( g ) ⊂ Sαn ( γ, λ, β, µ, η) .
( µn + 1)(1 + nη) 
Proof: Let f ∈N n,s (g) then, by (2), we have:
Which yields the following coefficient inequality:
∞
∞
∑α
k= n +1
k
≤
∑kα
(16)
βγ
k= n +1
λ + n

 n 
( µn + 1)(1 + nη) 
k
− b k ≤ δ,
∑∞
𝑘𝑘=𝑛𝑛+1 |𝛼𝛼𝑘𝑘 − 𝑏𝑏𝑘𝑘 | ≤
𝛿𝛿
𝑛𝑛+1
(𝑛𝑛 ∈ ℕ)
Since g ∈Rα (γ, λ, β,μ, η) by (14), we have:
n
λ + n ∞
η+ n ∞
k
1
1
n
µ (1 + nη) 
α
≤
β
γ
+
µ
−
+
η
α
(
)(
)
∑
∑
k
k

 n 
n +1
n +1
 n  k=

 k=
∞
η+ n
bk ≤
∑
≤ β γ + ( µ − 1)(1 + nη) 

k= n +1
 n 
βγ
(20)
(21)
λ + n

 n 
(β γ + n ) (1 + nη) 
βγ
λ + n

 n 
( µn + 1)(1 + nη) 
,
So that:
f (z)
∑ α − bk
− 1 < k= n +1 ∞ k
g(z)
1 − ∑ k= n +1 b k
∞
That is:
∞
k= n +1
(19)
Which yields that coefficient inequality:
Using (12) and (16), we also have:
∑ kα
(18)
k
≤
( n + 1) β γ
λ + n
( µn + 1)(1 + nη) 

 n 
λ + n
(β γ + n ) (1 + nη)  n 
δ


≤

( n + 1) 
λ + n
−β γ 
( β γ + n ) (1 + nη) 

 n 


= 1 − α.
=δ.
Thus, by the definition given by (3), f ∈N n, δ (e)
Neighborhoods for classes: Sα n (γ, λ, β, η) and Rα n (γ,
λ, β, μ, η)
In this section, we define the neighborhood for
each of the classes Sα n (γ, λ, β, η) and Sα n (γ, λ, β,μ, η)
A function f ∈A(n) is said to be in the class Sα n (γ, λ, β,
η) if there exists a function g ∈S n (γ, λ, β, η) such that
f (z)
− 1 < 1 − α ( 0 ≤ α ≤ 1,z ∈ U ) .
g(z)
(17)
Similarly, a function f ∈A(n) is said to be in the
class Rα n (γ, λ, β,μ, η) if there exists a function
g∈Rα n (γ, λ, β, μ, η) such that the inequality (17) holds
true.
Thus, by definition, f ∈Sα n (γ, λ, β, η) for α given
by (18) Thus, the proof is complete.
REFERENCES
Al-Shaqsi, K. and M. Darus, 2007. On certain subclass
of analytic univalent functions with negative
coefficients. Appl. Math. Sci., 13: 1121-1128.
Goodman, A.W., 1983. Univalent Functions. Volume I
and II, Mariner Publishing Company Inc., Tampa,
Florida.
Nasr, M.A. and M.K. Aouf, 1985. Starlike function of
complex order. J. Natur. Sci. Math., 25: 1-12.
Ruscheweyh, S., 1981. Neighborhoods of univalent
functions. Proc. Amer. Math. Soc., 81: 521-527.
Theorem 3: If g ∈S n (γ, λ, β, μ, η) and:
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