International Journal of Mathematical Archive-3(5), 2012,
2058-2064Available online through
ISSN 2229 – 5046ON HARMONIC UNIFORMLY STARLIKE FUNCTIONS WITH OTHER POINTS
DEFINED BY AN INTEGRAL OPERATOR
*Olatunji S. O.
Department of Pure and Applied Mathematics, Ladoke Akintola University of Technology, Ogbomoso P. M. B. 4000, Ogbomoso, Nigeria
(Received on: 25-01-12; Revised & Accepted on: 04-05-12)
ABSTRACT
The authors apply integral operator on
−
+
∑
∞=−
2
(
)
)
(
k
k k
p
z
a
z
ω
ω
to define a certain class of harmonic functions. They obtained coefficient inequalities, extreme points and distortion bounds for the functions in this class.Keywords: Harmonic function, uniformly harmonic starlike and integral operator.
2000 Mathematics Subject Classification: Primary 30C45.
1. INTRODUCTION
For an arbitrary fixed positive
ω
in U, we denote byH
p(
ω
)
the set of all harmonic multivalent functionsg
h
f
=
+
which are sense-preserving in the open unit diskU
=
{
z
:
z
<
1
}
where h and g are of the form2 2
( )
(
)
p k(
) ,
k( )
k(
) ,
k k1
.
k k
h z
= −
z
ω
+
∑
∞=a z
−
ω
g z
=
∑
∞=b z
−
ω
b
<
(1)
The Integral operator In was introduced by Salagean [7]. For fixed positive integer n and for
f
=
h
+
g
given by (1) we have(i)
I
0f
(
z
)
=
f
(
z
)
(ii)
I
f
z
If
z
z0f
t
t
1dt
1)
(
)
(
)
(
=
=
∫
−(iii)
I
nf
(
z
)
=
I
(
I
n−1f
(
z
)),
n
∈
N
=
1
,
2
,
3
...
and
f
∈
A
)
(
)
1
(
)
(
)
(
z
I
h
z
I
g
z
f
I
n=
n+
−
n nRecently, Oladipo and Breaz [6], Clunie and Sheil-Small [2], Ahuja and Jahangiri [1], El-Ashwah and Aouf [4], Cotirla [3], Guney and Sakar [5] extended the operator studied in [6] and good results were obtained.
Here
,
( , ) ( )
,( ) ( 1)
,( , ) ( )
n n n n
p p p
I
ωλ
l f z
=
I
ωh z
+ −
I
ωλ
l g Z
(2) where
∑
∞+ =
−
+
−
+
+
−
+
+
−
=
1
,
(
)
,
)
1
(
1
)
1
(
1
)
(
)
(
)
,
(
p k
k k
n p
n
p
a
z
and
l
k
l
p
z
z
h
l
I
ω
λ
λ
ω
λ
ω
,
1
1
(
1)
( , ) ( )
(
) ,
1.
1
(
1)
n
n k
p k k
k p
p
l
I
l g z
b z
b
k
l
ω
λ
λ
ω
λ
∞= +
+
− +
=
−
<
+
− +
∑
*Olatunji S. O. / ON HARMONIC UNIFORMLY STARLIKE FUNCTIONS WITH OTHER POINTS…. / IJMA- 3(5), May-2012, 2058-2064 P. M. B. 4000, Ogbomoso, Nigeria
For
0
≤
α
<
1
,
n
∈
N
,
λ
≥
0
,
l
≥
0
,
s
≥
0
,
θ
∈
R
.For the purpose of our result, the following definition shall be necessary.
Definition A: Let
R
sω,q(
λ
,
l
,
α
,
n
,
p
)
the family of harmonic function f of the form (2) such that,
,
( , ) ( )
Re (1
)
.
( , ) ( )
n p
i i
n q p
I
l f z
se
se
I
l f z
ω
θ θ
ω
λ
α
λ
+
+
−
≥
(3)For particular cases s and q, especially for
s
=
0
,
p
=
1
,
λ
=
1
,
l
=
0
andq
=
1
,
ω
=
0
, we can write)
,
(
)
,
,
0
,
1
,
1
(
1 ,
0
n
R
n
R
α
=
α
which is the class studied by Cotirla in [3].
Definition B: Let denote the subclass
R
s,q(
λ
,
l
,
α
,
n
,
p
)
ω
consists of harmonic functions
f
n=
h
+
g
n in)
,
,
,
,
(
,
l
n
p
R
sωqλ
α
so that h and gn are of the form,
1
1
(
1)
( , ) ( )
(
)
(
) ,
1
(
1)
n
n p k
p k
k p
p
l
I
l h z
z
a
z
k
l
ω
λ
λ
ω
ω
λ
∞= +
+
− +
= −
−
−
+
− +
∑
(4) and
,
1
1
(
1)
( , ) ( )
(
) ,
1.
1
(
1)
n
n k
p k k
k p
p
l
I
l g z
b
z
b
k
l
ω
λ
λ
ω
λ
∞= +
+
− +
=
−
<
+
− +
∑
In this work, we investigate the coefficient bounds, extreme points, distortion bounds and the convexity properties of the functions are also obtained.
2. MAIN RESULT
We prove the following sufficient conditions for harmonic functions in
R
s,q(
λ
,
l
,
α
,
n
,
p
)
.Theorem 1: Let
f
=
h
+
g
be such that h and g are given by (1). If2 1
[
(1
)
(
)]
[
(1
) ( 1)
(
)]
1.
1
1
n n q n q n q
k k
k k
s
s
s
s
a
b
γ
γ
α
γ
γ
α
α
α
+ +
∞ ∞
= =
+ −
+
+ − −
+
+
≤
−
−
∑
∑
(5)
where
+
−
+
+
−
+
=
l
k
l
p
)
1
(
1
)
1
(
1
λ
λ
γ
thenf
∈
R
s qω,( , , , , ).
λ α
l
n p
Proof: Suppose that (5) holds. Using the fact that
Re
ω
≥
ρ
if and only if1
−
ρ
+
ω
≥
1
−
ρ
+
ω
, it suffices to show that(1
−
α
)
I
ωn q+( , ) ( )
λ
l f z
+
I
ωn q+( , ) ( )(1
λ
l f z
+
se
iθ)
−
se I
iθ ωn q+( , ) ( )
λ
l f z
(1
α
)
I
ωn q+( , ) ( )
λ
l f z
I
ωn q+( , ) ( )(1
λ
l f z
se
iθ)
se I
iθ ωn q+( , ) ( )
λ
l f z
0,
− −
−
+
+
≥
substituting for
I
ωn(
λ
,
l
)
f
(
z
)
andI
ωn+q(
λ
,
l
)
f
(
z
)
in (6) yields
−
+
−
+
+
−
+
−
+
+
−
+
+
−
+
−
+
+
−
+
−
+
+
−
−
−
∑
∑
∑
∑
∞
=
+ ∞
= +
∞
=
+ +
∞
= +
1 2
1 2
)
(
)
1
(
1
)
1
(
1
)
1
(
)
(
)
1
(
)
(
)
1
(
1
)
1
(
1
)
1
(
)
(
)
1
(
k
k k
q n n
k
k q n p
i
k
k k
q n q
n k
k q n p
i
z
b
l
k
l
p
a
z
se
z
b
l
k
l
p
a
z
se
ω
λ
λ
γ
ω
ω
λ
λ
γ
ω
α
θ θ
2 1
1 1
(1
) (
)
( 1)
(
)
(1
) (
)
( 1)
(
)
i p n q n q n q k
k k
k k p
i p n q n n q k
k k
k k p
se
z
a
b z
se
z
a
b z
θ
θ
α
ω
γ
γ
ω
ω
γ
γ
ω
∞ ∞
+ + +
= = +
∞ ∞
+ +
= = +
− −
−
+
+ −
−
−
+ +
−
+
+ −
−
∑
∑
∑
∑
2
1
(2
)(
)
(1
(1
))
(
)
( 1)
(1
( 1)
(1
)) (
)
p n i q i k
k k
n n i q q i k k
k
z
se
se
a z
se
se
b z
θ θ
θ θ
α
ω
γ
γ
α
ω
γ
γ
α
ω
∞
= ∞
=
−
−
+
+
+
− −
−
=
+ −
+
+ −
− −
−
∑
∑
∑
∑
∞
= ∞
=
−
+
+
−
−
+
−
−
−
+
+
+
−
−
+
−
−
1 2
)
(
))
_
1
(
)
1
(
1
(
)
1
(
)
(
))
1
(
1
(
)
(
k
k k
i q
q i
n n k
k k
i q
i n
p
z
b
se
se
z
a
se
se
z
ω
α
γ
γ
ω
α
γ
γ
ω
α
θ θ
θ θ
k k
k q
q n
k k
k q
n p
z
b
s
s
z
a
s
s
z
)
2
[
1
)
(
)]
(
)
2
[
1
)
(
1
)
(
)]
(
)
(
)
1
(
2
1 2
ω
α
γ
γ
ω
α
γ
γ
ω
α
−
−
+
−
+
−
−
+
−
−
+
−
−
≥
∑
∑
∞= ∞
=
2 1
[1 ) ( )] [1 ) ( 1) ( )]
21 () ( ) ` ( ) 2 ( ) .
1 1
n q n q q
p k p k p
k k
k k
s s s s
z
γ
γ α
a zγ
γ α
b zα
ω
ω
ω
α
α
∞ ∞
− −
= =
+ − + + − − +
= − − − − − −
− −
∑
∑
This last expression is non-negative by hypothesis, and so the proof is complete.
The functions
,
2 1
1 1
( , ) ( ) ( ) ( ) ( ) .
[ (1 ) ( )] [ (1 ) (1 ) ( )]
n p k k
s q n n q k n q n q k
k k
I l f z z x z y z
r r r r
α α
λ ω ω ω
γ γ α γ γ α
∞ ∞
+ +
= =
− −
= − + − + −
+ − + + − − +
∑
∑
(7)
shows that the coefficient bound given by (5) is sharp where
n
∈
N
,
r
≥
0
,
q
∈
N
,
λ
≥
0
,
l
≥
0
and.
1
12
=
+
∑
∑
∞= ∞
= k
k k
k
y
x
Theorem 2: Let
f
n=
h
+
g
n be given by (4). Thenf
n∈
R
sω,q(
λ
,
l
,
α
,
n
,
p
)
. If and only if2 2
[
(1
)
(
)]
[
(1
) ( 1)
(
)]
1,
1
1
n n q n q n q
k k
k k
s
s
s
s
a
b
γ
γ
α
γ
γ
α
α
α
+ +
∞ ∞
= =
+ −
+
+
+ − −
+
≤
−
−
∑
∑
(8)
where
+
−
+
+
−
+
=
l
k
l
p
)
1
(
1
)
1
(
1
λ
λ
*Olatunji S. O. / ON HARMONIC UNIFORMLY STARLIKE FUNCTIONS WITH OTHER POINTS…. / IJMA- 3(5), May-2012, 2058-2064
Proof: Since
R
sω,q(
λ
,
l
,
α
,
n
,
p
)
⊂
R
sω,q(
λ
,
l
,
α
,
n
,
p
)
we only need to prove the ‘only if’ part of the theorem. For function fn of the form (4), we note that conditionα
λ
λ
θ ω ω θ≥
−
+
+ in q n n n i
se
z
f
l
I
z
f
l
I
se
)
(
)
,
(
)
(
)
,
(
)
1
(
Re
is equivalent to
+
−
+
+ +)
(
)
,
(
)
)(
(
)
,
(
)
(
)
,
(
)
1
(
Re
z
f
l
I
se
z
f
l
I
z
f
l
I
se
n q n i n q n n n iλ
α
λ
λ
ω θ ω ω θ[
]
[
]
−
−
+
−
−
−
−
−
+
−
−
−
+
−
−
−
+
−
−
−
−
−
+
−
−
−
+
=
∑
∑
∑
∑
∑
∑
∑
∑
∞ = + − + ∞ = + ∞ = + − + ∞ = + ∞ = + − + ∞ = + ∞ = − ∞ = 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2)
(
)
1
(
)
(
)
(
)
(
)
1
(
)
(
)
(
)
(
)
(
)
1
(
)
(
)
(
)
(
)
1
(
)
(
)
(
)
1
(
Re
k k k q n q n k k k q n p k k k q n q n k k k q n p i k k k q n q n k k k q n p k k k n n k k k n p iz
b
z
a
z
z
b
z
a
z
se
z
b
z
a
z
z
a
z
a
z
se
ω
γ
ω
γ
ω
ω
γ
ω
γ
ω
α
ω
γ
ω
γ
ω
ω
γ
ω
γ
ω
θ θ[
]
[
]
−
−
+
−
−
−
−
+
−
−
+
−
+
−
−
+
−
−
−
−
+
−
+
−
−
−
=
∑
∑
∑
∑
∑
∑
∞ = + − + ∞ = + ∞ = + − ∞ = + − + ∞ = + ∞ = + 1 1 2 2 2 1 2 1 1 2 2 2)
(
)
1
(
)
(
)
(
)
(
)
(
)
1
(
)
1
(
)
1
(
)
(
)
1
(
)
(
)
(
)
(
)
(
)
1
(
)
)(
1
(
Re
k k k q n q n k k k q n p k k k i q n q i n n k k k q n q n k k k q n p k k k i q n i n pz
b
z
a
z
z
b
se
se
z
b
z
a
z
z
a
se
se
z
ω
γ
ω
γ
ω
ω
α
γ
γ
ω
γ
ω
γ
ω
ω
α
γ
γ
ω
α
θ θ θ θ 2 2 1 2 1( ) 2 1
( ) 2
2 1
2
(1
)
(1
)
(
)
(
)
1
(
)
( 1)
(
)
Re
( 1)
(1
)
(
)
(
)
1
(
)
( 1)
p
n i n q i k p k
k
n q k p n q n q k
k k
k k
z n n i n q i k k
z k
n q k p n q n k
k
se
se
a
z
a
z
b
z
se
se
b
z
a
z
θ θ
ω θ θ
ω
α
γ
γ
α
ω
γ
ω
γ
ω
γ
γ
α
ω
γ
ω
γ
∞ + − = ∞ + − + − ∞ + = = ∞ − − + − = ∞ + − + − + =
−
−
+
−
+
−
−
−
+ −
−
=
−
+
−
+
−
+
−
−
+ −
∑
∑
∑
∑
∑
10.
(
)
q k kk
b
z
ω
∞ =
≥
−
∑
(9)Upon choosing the value of
(
z
−
ω
)
on the positive real axis and usingRe(
−
e
iθ)
≥
−
e
iθ=
−
1
where1
)
(
)
(
0
≤
z
−
ω
=
r
+
d
<
, the above inequalities reduces to2
2 1
2 1
(1
)
(1
)
(
)
(
)
1
(
)
( 1)
(
)
n n q k p k
k
n q k p n q n q k p
k k
k k
s
s
a
r
d
a
z
b
r
d
α
γ
γ
α
γ
ω
γ
∞ + − = ∞ + − + − ∞ + − = =
−
−
+ −
+
+
=
−
−
+ −
+
∑
∑
∑
2 2 1(1
)
(
)
(
)
0.
1
(
)
( 1)
(
)
n i n q i k p k
k
n q k p q n q k p
k k
k k
se
se
b
r
d
a
r
d
b
r
d
θ θ
γ
γ
α
γ
γ
∞ + − = ∞ + − ∞ + − = =
+
−
+
+
−
≥
−
+
− −
+
∑
∑
∑
(10) If the condition (8) does not hold, then the numerator (10) is negative for sufficiently close to 1.Thus there exist
z
0−
ω
=
r
0+
d
in (0, 1) for which the quotient in (10) is negative. This contradicts the conditionConvex hull of
R
sω,q(
λ
,
l
,
α
,
n
,
p
).
Theorem 3: Let fn be given by (2) if and only if
[
]
∑
∞=
+
=
1
)
(
)
(
)
(
k
nk k k
k
n
z
X
h
z
Y
g
z
f
where
,...)
3
,
2
(
)
(
)
(
)
1
(
1
)
(
)
(
)
(
)
(
−
=
+
−
+
−
−
−
=
−
=
+z
k
s
s
z
z
h
z
z
h
kq n n
p k
p
ω
α
γ
γ
α
ω
ω
and
+
−
+
+
−
+
=
l
k
l
p
)
1
(
1
)
1
(
1
λ
λ
γ
,...)
3
,
2
,
1
(
)
(
)
(
)
1
(
1
)
1
(
)
(
)
(
1,
−
=
+
−
+
−
−
−
+
−
=
− +k
z
s
s
z
z
g
nk p nγ
nγ
n qα
ω
kα
ω
1
1
(
1)
0,
0,
(
) 1.
1
(
1)
k k k k kp
l
X
Y
X
Y
k
l
λ
γ
λ
∞ =
+
− +
=
≥
≥
+
=
+
− +
∑
In particular, the extreme points of
R
sω,q(
λ
,
l
,
α
,
n
)
are hk and gn,k.Proof: For function fn of the form (2), we have
[
]
∑
∞=
+
=
1
)
(
)
(
)
(
k
nk k k
k
n
z
X
h
z
Y
g
z
f
[
]
2 1
1
(
)
(
)
(1
)
(
)
p k
k k n n q k
k k
X
Y
z
X
z
s
s
α
ω
ω
γ
γ
α
∞ ∞
+ =
=
−
+
−
−
−
=
∑
∑
+ −
+
1
1 ,
1
( 1)
(
)
(1
)
(
)
n k
k n n q
k
Y z
s
s
α
ω
γ
γ
α
∞ −
+ =
−
+ −
−
+ −
+
∑
Then
2 1
1
1 2
(1
)
(
)
1
(1
)
(
)
1
(
)
(
)
1
(1
)
(
)
1
(1
)
(
)
1
1.
n n q n n q
k k
k k
n n q n n q
k k
k k k k
s
s
s
s
X z
Y z
s
s
s
s
X
Y
X
γ
γ
α
α
ω
γ
γ
α
α
ω
α
γ
γ
α
α
γ
γ
α
+ +
∞ ∞
+ +
= =
∞ ∞
= =
+ −
+
−
−
+
+ −
+
−
−
−
+ −
+
−
+ −
+
=
+
= −
≤
∑
∑
∑
∑
and so
f
nR
s,q(
λ
,
l
,
α
,
n
,
p
)
ω
∈
.Conversely, suppose
f
nR
s,q(
λ
,
l
,
α
,
n
,
p
)
ω
∈
. Letting∑
∞∑
=
∞
=
+
−
=
1 2
1
1
,
k k k k
Y
X
X
...)
3
,
2
(
1
)
(
)
1
(
=
−
+
−
+
=
s
+s
a
k
X
kq n n
k
α
α
γ
γ
and
)
(
)
1
(
)
1
(
+
s
−
−
q n+q+
s
n
γ
α
*Olatunji S. O. / ON HARMONIC UNIFORMLY STARLIKE FUNCTIONS WITH OTHER POINTS…. / IJMA- 3(5), May-2012, 2058-2064
We obtained the required representation, since
1
1 1
( )
(
)
p(
)
k( 1)
n(
)
kn k k
k k
f z
z
ω
a
z
ω
b
z
ω
∞ ∞
−
= =
= −
−
∑
−
+ −
∑
−
1
2 1
2 1
2 1 2 1
1
1
(
)
(
)
( 1)
(
)
(1
)
(
)
(1
)
(
)
(
)
(
)
( )]
(
)
( )]
1
(
)
( )
( )
[
( )
p k n k
k k
n n q n n q
k k
p p p
k k nk k
k k
p
k k k k k nk
k k k k
k k k
z
X
z
Y z
s
s
s
s
z
z
h z X
z
g
z Y
X
Y
z
X h z
Y g
z
X h z
Y
α
α
ω
ω
ω
γ
γ
α
γ
γ
α
ω
ω
ω
ω
∞ ∞
−
+ +
= =
∞ ∞
= =
∞ ∞ ∞ ∞
= = = =
−
−
= −
−
−
+ −
−
+ −
+
+ −
+
= −
−
−
−
−
−
= −
−
−
+
−
=
+
∑
∑
∑
∑
∑
∑
∑
∑
2
( )].
k
gnk z
∞
=
∑
Distortion bounds for functions in
R
s,q(
λ
,
l
,
α
,
n
,
p
)
ω
is our next result.
Theorem 4: Let
f
nR
s,q(
λ
,
l
,
α
,
n
,
p
)
ω
∈
. Then forz
−
ω
= + <
r
d
1,
we have2 1
1
(
)
)
(
)
1
(
)
(
)
1
(
)
1
(
)
(
)
1
(
1
)
)(
1
(
)
(
b
r
d
s
s
s
s
s
s
d
r
b
z
f
n n qq q
n n
p
n
+
+
−
+
+
−
−
+
−
+
−
+
−
+
+
+
≤
+ +α
γ
γ
α
α
γ
γ
α
and
2 1
1
(
)
)
(
)
1
(
)
(
)
1
(
)
1
(
)
(
)
1
(
1
)
)(
1
(
)
(
b
r
d
s
s
s
s
s
s
d
r
b
z
f
n n qq q
n n
p
n
+
+
−
+
+
−
−
+
−
+
−
+
−
−
+
+
≤
+ +α
γ
γ
α
α
γ
γ
α
where q is a an odd positive integer.
Proof: We prove the right hand inequality. Let
f
nR
s,q(
λ
,
l
,
α
,
n
,
p
)
ω
∈
. Taking the absolute value of fn,… we obtain1
2
( )
(1
)(
)
p(
)(
)
kn k k
k
f z
b
r
d
a
b
r
d
∞
=
≤ +
+
+
∑
+
+
2 1
2
(1
)(
)
p(
k k)(
)
k
b
r
d
a
b
r
d
∞
=
≤ +
+
+
∑
+
+
2
1 1
2
1
(1
)
(
)
(1
)(
)
(
)(
)
(1
)
(
)
1
n n q p
k n n q
k
s
s
b
r
d
a
b
r
d
s
s
α
γ
γ
α
γ
γ
α
α
+ ∞
+
=
−
+ −
+
≤ +
+
+
−
+
+
+ −
+
∑
−
∑
∞=
+ +
+ +
−
+ −
+ + −
+ −
+ −
+ −
+ − +
+ + ≤
2
2
1 ( )
1
) ( ) 1 ( 1
) ( )
1 ( )
( ) 1 (
1 )
)( 1 (
k
k q
n n
k q
n n
q n n
p
d r b s s
a s s
s s
d r b
α
α
γ
γ
α
α
γ
γ
α
γ
γ
α
2 1
1
2 1
1
) ( ) ( )
1 (
) ( ) 1 ( ) 1 ( ) ( )
1 (
1 )
)( 1 (
) ( 1
) ( ) 1 ( ) 1 ( 1 ) ( )
1 (
1 )
)( 1 (
d r b s s
s s
s s
d r b
d r b s s
s s
d r b
q n n
q q
n n
p
q q
n n
p
+
+ −
+
+ − − + − + −
+ − +
+ + ≤
+
−
+ − − + − + −
+ − +
+ + ≤
+ +
+
α
γ
γ
α
α
γ
γ
α
α
α
α
γ
γ
α
Convex combination for the function
R
sω,q(
λ
,
l
,
α
,
n
,
p
)
:Theorem 5: The family
R
ωs,q(
λ
,
l
,
α
,
n
,
p
)
is closed under convex combination.Proof: For i=1, 2,…, suppose that
f
ni∈
R
ωs,q(
λ
,
l
,
α
,
n
,
p
)
, where∑
∑
∞= ∞
=
−
−
−
+
−
−
−
=
1 2
1
)
(
)
1
(
)
(
)
(
)
(
k
k i
k k
n k
i k p
i
n
z
z
a
z
b
z
f
ω
ω
ω
then by Theorem 2,
∑
∑
∞=
+ ∞
=
+
≤
−
+
−
−
+
+
−
+
−
+
1 2
)
(
2
1
)
(
)
1
(
)
1
(
1
)
(
)
1
(
k
i k q
n q n
k
i k q
n n
x
b
s
s
a
s
s
α
α
γ
γ
α
α
γ
γ
for
∑
∞==
≤
≤
1
1
,
0
1
i
t
it
i , theconvex combination of
f
ni may be written as1
1 2 1 2 1
( )
(
)
(
)
( 1)
(
) .
i p i k n i k
i n i k i k
i k i k i
t f
z
z
ω
t a
z
ω
t b
z
ω
∞ ∞ ∞ ∞ ∞
−
= = = = =
= −
−
−
+ −
−
∑
∑ ∑
∑ ∑
Then by (x)
∑
∑
∑
∑
∞=
∞
= +
∞
=
∞
= +
−
+
−
−
+
+
−
+
−
+
1 1
2 1
1
)
(
)
1
(
)
1
(
1
)
(
)
1
(
k i
i k i q
n q n
k i
i k i q
n n
b
t
s
s
a
t
s
s
α
α
γ
γ
α
α
γ
γ
2
2
1
)
(
)
1
(
)
1
(
1
)
(
)
1
(
1 1
2 1
=
≤
−
+
−
−
+
+
−
+
−
+
∑
∑
∑
∑
∞= ∞
=
+ ∞
=
+ ∞
= i
i k
i k q
n q n
k
i k q
n n
i
i
b
t
s
s
a
s
s
t
α
α
γ
γ
α
α
γ
γ
and therefore ` ,
1
( )
( , , , , ).
i
i n s q
i
t f
z
R
l
n p
ω
λ α
∞=
∈
∑
REFERENCES
[1] Om. P. Ahuja and J. M. Jahangiri: Multivalent harmonic starlike functions, Ann. Univ. Marie Cruie-Sklodowska Sect. A, LV 1(2001), 1-13.
[2] J. Clunie and T. Sheil-Small: Harmonic univalent functions, Ann. Acad. Sci. Fenn. Ser. A. I. Math. 9(1984), 3-25. [3] L. I. Cotrla: Harmonic univalent functions defined by an integral operator, Acta Universitatis Apulensis,
17(2009), 95-105.
[4] R. M. El-Ashwah and M. K. Aouf: Differential subordination and super ordination for certain subclasses of analytic functions involving an extended integral operator.
[5] H. Ozlem Guney and F. Muge Sakar: On Harmonic uniformly starlike functions defined by an integral operator, Acta Universitatis Apulensis, 28(2011), 293-301.
[6] A. T. Oladipo and D. Breaz: On a new class of harmonic multivalent functions defined by extended integral operator. (Submitted).
[7] G. S. Salagean: Subclass of univalent functions, Lecture notes in Math. Springer-Verlag, 1013 (1983), 362-372.
