A Different Approach to Normalized Analytic Functions
Through Meromorphic Functions Defined by Extended
Multiplier Transformations Operator
Maslina Darus and Imran Faisal
School of Mathematical Sciences Faculty of Science and Technology
Universiti Kebangsaan Malaysia Bangi 43600 Selangor D. Ehsan
Malaysia
maslina@ukm.my(corresponding author);faisalmath@gmail.com
ABSTRACT
Let A(n) denote the class of analytic functions f, in the open unit disk U = {z : |z| < 1}
normalized by f (0) = f
′
(0) −1=0, and let
∑
p,k
denote the class of Meromorphic functions
in U
⋆
= U\{0}· In this paper, we introduce and study the classes of functions z
p+1
f (z) ∈A,
such that f (z) ∈
∑
p,k
· We also introduce and study the class V
α,β
λ,k
(n, μ, δ, ξ, ǫ) of functions
f (z) ∈
∑
p,k
satisfying
D
n+1
λ
(α, β, μ)z
p+1
f (z)
z
z
D
n
λ
(α, β, μ)z
p+1
f (z)
δ
−
1
ξ
< 1 − ǫ·
where
δ ≥ 0,λ ≥ 0,ξ ∈ (0, 1],ǫ ∈ [0, 1) and D
n
λ
(α, β, μ)f (z): A→A, is the Extended Multiplier
Transformations Operator, newly defined as follows
D
n
λ
(α, β, μ)f (z)= z +
∞
k=2
α +(μ + λ)(k − 1) + β
α + β
n
a
k
z
k
·
Several inclusion, subordination, and superordination properties have been discussed for
the above said classes of normalized analytic and meromorphic functions ·
Keywords: analytic functions, normalize analytic functions, meromorphic functions, differential
subordination, differential Superordination.
2000 Mathematics Subject Classification: 30C45.
1 Introduction and preliminaries
Let H(U) be the class of analytic functions in the unit disk U = {z : |z| < 1} and let
A(n)= {f ∈H(U),f (z)= z + a
n+1
z
n+1
+ a
n+2
z
n+2
+ ··· ,z ∈ U}
with A(1) = A·
Let
∑
p,k
denote the class of functions in U
⋆
= U \{0} of the form
f (z)=
1
z
p
+ a
k
z
k
+ a
k+1
z
k+1
+ ··· (1.1)
International Journal of Applied Mathematics & Statistics,
Int. J. Appl. Math. Stat.; Vol. 23; Issue No. D11; Year 2011,
ISSN 0973-1377 (Print), ISSN 0973-7545 (Online)
Copyright © 2011 by IJAMAS, CESER Publications
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