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A NEW CLASS OF HARMONIC UNIVALENT FUNCTIONS ASSOCIATED WITH
GENERALIZED-DERIVATIVE OPERATOR
A. L. Pathak
1, a,
K. K. Dixit
2and R. Agarwal
3*1,3
Department of Mathematics, Brahmanand College, The Mall, Kanpur, (U.P.), India - 208004 2
Department of Mathematics, G. I. I. T., Gwalior, (M.P.), India-474015 E-mail: [email protected], [email protected],[email protected]
(Received on: 01-12-11; Accepted on: 19-12-11)
________________________________________________________________________________________________ ABSTRACT
I
n this paper, we study a new class of harmonic univalent functions associated with modified generalized – derivative operator in the open unit disk. We obtain numerous sharp results including coefficient conditions, extreme points, distortion bounds, convolution properties and convex combinations for the above class of harmonic univalent functions. The results obtained for the class reduce to the corresponding results for various well-known classes in the literature.AMS 2010 Mathematics Subject Classification: 30C45, 30C50.
Keywords: Harmonic, Univalent, Starlike, Extreme points, Generalized-derivative operator.
________________________________________________________________________________________________ 1. INTRODUCTION:
A continuous complex valued function
f
=
u
+
iv
defined in a simply connected domain D is said to be harmonic inD if both
u
andv
are harmonic in D. In any simply connected domain D we can writef
=
h
+
g
,
whereh
andg
are analytic in D. A necessary and sufficient condition forf
to be locally univalent and sense preserving in D is that,
,
)
(
)
(
z
g
z
z
D
h
′
>
′
∈
see [4].Denote by SH the class of functions
f
=
h
+
g
that are harmonic univalent and sense preserving in the unit disk{
:
<
1
}
=
z
z
U
for whichf
(
0
)
=
f
z(
0
)
−
1
=
0
.
Then forf
=
h
+
g
∈
S
H,
we may express the analytic functionh
andg
as
∞ =
∞ =
=
+
=
2 1
,
)
(
,
)
(
k k
k k k
k
z
g
z
b
z
a
z
z
h
b
1<
1
.
(1.1)Note that the class SHreduces to the class S of normalized analytic univalent functions if the co-analytic part of
g
h
f
=
+
is identically zero.In 1984, Clunie and Sheil-Small [4] investigated the class SH and studied some coefficient bounds. Since then, there have been published several related papers on SH and its subclasses.
In fact by introducing new subclasses Sheil-Small [14] Silverman [15], Silverman and Silvia [16] and Jahangiri [9] presented a systematic and unified study of harmonic univalent functions. Furthermore we refer to Ahuja [1], Duren [7] Ponnusamy and Rasila [11] and references therein for basic results on the subjects.
To study our work systematically first we give the class introduced by Gencel and Yalcin [8]
( $) * $& '$& ' $+# * , - ./ * 0 .1 234* 256
Let n( , )
α β
H
S denote the family of harmonic functions f of the form (1.1) such that
( )
( )
( )
( )
1 1
Re
(1 2 )α β,
+ +
−
< + −
n n
n n
D f z D f z
D f z D f z
for 0≤
α
<1, 0<β
≤1,n N0 and z U.It should be noted that the differential operatorn
D was introduced by Salagean
and D fn
( )
z = D h zn( )
+ −( 1)nD g zn ( ) .
Forn m N, 0,N0 =N∪
{ }
0 ,λ>0,and f =h+ggiven by (1.1), we define the modified generalized-derivative operatorof f as
Dmn,λf( )z =Dnm,λh( )z + −( 1)nDmn,λg( ) ,z (1.2)
where ( )
[
( )]
2
,λ 1 1 λ ( , ) ,
∞ =
= + + − n
m
k
n k
k
D h z z k C m k a z
, ( )
[
( )]
1
1 1 ( , )
λ λ
= ∞
= + − n
k
n k
m k
D g z k C m k b z
and ( ) ( )
( ) 1
.
( )
,
1
+ − +
= =
+
k m k m
k m
m C m k
Remark: 1.1 It is worthy to note that the generalized derivative operatorDnm,λ was introduced by Darus and Shaqsi
Remark: 1.2 Some special cases of this operator includes the following derivative operator ( ,1) ≡ ([3],[10]),
n n
m m
D D the
Salagean derivative operator 0,1≡ ([6],[13]),
n n
D D the generalized Salagean derivative operator 0,λ≡ λ [2],
n n
D D the
Ruscheweyh derivative operatorDm0,1≡Dm
[
12 ,]
and the generalized Ruscheweyh derivative operator[
]
1
,λ≡ ,λ 17 .
m m
D D
Darus and Shaqsi gave the following inclusion relations:
( ) (
)
( )
( )
1
,λ 1
λ
,λλ
( ,λ ) '+ = − +
n n n
m m m
D f z D f z z D f z
( )
(
)
( )
( )
, 1, ,
( mnλ ) '= 1+λ nm+ λ −λ nmλ
z D f z D f z D f z
0
1,0 ( )= ( ).
D f z f z
For a1=1, ,n m∈N0,
λ
>0 ,0≤α
<1, 0<β
≤1 and z∈U,we let n(
α β λ
, , , ,
)
HS
m k
denote the family ofharmonic functions
f
of the form (1.1) such that
( )
( )
( )
( )
1
, ,
1
, ,
R
1 e
( 2 ) ,
λ λ
λ α λ
β
+
+
−
< + −
n n
m m
n n
m m
D f z D f z
D f z D f z (1.3)
,λ
n m
D f is defined by (1.2).
We let the subclass
S
Hn(
α
,
β
,
λ
,
m
,
k
)
consist of harmonic functionf
n=
h
+
g
ninS
(
m
k
)
nH
α
,
β
,
λ
,
,
so that h and gn are of the form( )
∞ =
∞ =
−
=
−
=
2 1
.
1
)
(
,
)
(
k k
k k n n
k
k
z
g
z
b
z
a
z
z
h
a
k,
b
k≥
0
and (1.4)In this paper, the coefficient condition given in [8] for the class is extended to the class
S
Hn(
α
,
β
,
λ
,
m
,
k
)
(
m
k
)
2. MAIN RESULT:
In our first theorem, we introduce a sufficient bound for harmonic functions in
S
Hn(
α
,
β
,
λ
,
m
,
k
)
Theorem: Letf
(
z
)
=
h
(
z
)
+
g
(
z
)
be given by (1.1). Furthermore, let(
)
[
]
{
[
(
)
(
(
)
)
]
(
)
(
(
)
[
(
)
]
) (
)
}
(
α
)
β
α
λ
β
λ
α
λ
β
λ
λ
−
≤
+
−
+
−
+
+
−
−
+
+
−
−
+
∞=
1
4
,
2
1
1
2
,
2
1
2
1
1
1
1
k
k k
n
b
k
m
C
k
k
a
k
m
C
k
k
k
(2.1) where
a
1=
1
,
n
,
m
∈
N
0,
λ
>
0
,
0
≤
α
<
1
,
0
<
β
≤
1
.
Then
f
(
z
)
is harmonic univalent, sense preserving in U andf
∈
S
Hn(
α
,
β
,
λ
,
m
,
k
)
.
Proof: For
z
1<
z
2<
1
, we have
)
(
)
(
)
(
)
(
1
)
(
)
(
)
(
)
(
2 1
2 1
2 1
2 1
z
h
z
h
z
g
z
g
z
h
z
h
z
f
z
f
−
−
−
≥
−
−
(
)
(
)
∞(
)
= ∞
=
−
−
−
−
−
≥
2
2 1 2
1 1
2 1
1
k
k k k k
k k k
z
z
a
z
z
z
z
b
∞ = ∞
=
−
−
>
2 1
1
1
k k k
k
a
k
b
k
(
)
[
]
(
(
)
[
(
)
]
) (
)
(
)
[
] (
[
)
(
(
)
)
] (
)
∞ = ∞
=
−
−
+
+
−
−
+
−
+
−
+
−
+
−
+
−
>
2 1
,
2
1
2
1
1
1
1
,
2
1
1
2
1
1
1
k
k n
k
k n
a
k
m
C
k
k
k
b
k
m
C
k
k
k
α
λ
β
λ
λ
α
λ
β
λ
λ
≥
0
Hence,
f
( )
z
is univalent in U. Note thatf
( )
z
is sense preserving in U.This is because
∞ =
−
−
≥
′
2
1
1
)
(
k
k k
z
a
k
z
h
∞ =
−
>
2
1
kk
a
k
[
(
)
] (
[
)
(
(
)
)
] (
)
∞=
−
−
+
+
−
−
+
−
>
2
,
2
1
2
1
1
1
1
k
k n
a
k
m
C
k
k
k
λ
λ
β
λ
α
[
(
)
]
[
(
)
(
(
)
)
] (
)
1 1,
2
1
1
2
1
1
−∞ =
+
−
+
−
+
−
+
>
kk
k n
z
b
k
m
C
k
k
k
λ
λ
β
λ
α
1
1
− ∞
=
≥
kk
k
z
b
k
( $) * $& '$& ' $+# * , - ./ * 0 .1 234* 256
It remains to show that
f
( )
z
∈
S
Hn(
α
,
β
,
λ
,
m
,
k
)
.Suppose that the inequality (2.1) holds true and let:
and
1
z
∈ ∂
U
=
z z
∈
C
z
=
. Then we find from definition (1.2) that
(
)
[
]
(
)(
)
( )
[
(
)
]
(
)
[
(
)
]
(
)
[
(
)
]
(
)
[
(
)
]
( )
∞[
(
)
]
(
) (
[
)
]
= ∞
= ∞ =
∞ =
+
−
−
+
−
−
−
−
+
−
+
+
−
−
+
−
+
−
−
−
−
+
=
1 2
2 1
2
1
,
1
1
1
2
1
2
,
1
1
1
2
1
2
,
1
1
1
1
,
1
1
k
k k n
n k
k k
n
k k
k k n
n k
k n
z
b
k
k
m
C
k
z
a
k
k
m
C
k
z
z
b
k
k
m
C
k
z
a
k
k
m
C
k
α
λ
λ
α
λ
λ
α
λ
λ
λ
λ
(
)
(
)(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
) (
)
2 1
2 1
1
1
,
1
1
1
,
2
1
2 1
1
1
,
2
1
2
1
1
,
1
2
n k n k
k k
k k
n k n k
k k
k k
k
C m k k
a z
k
C m k
k
b z
z
k
C m k
k
a z
k
C m k
k
b z
λ
λ
λ
λ
α
λ
λ
α
λ
λ
α
∞ ∞
= =
∞ ∞
= =
+
−
−
+
+
−
+
−
≤
−
−
+
−
+
−
−
−
+
−
−
+
β
≤
provided that the inequality (2.1) is satisfied . Hence, by the maximum modulus theorem, we have
(
m
k
)
S
f
∈
Hnα
,
β
,
λ
,
,
. The harmonic mappings( )
(
)
(
)
[
]
(
) (
[
)
(
(
)
)
]
(
)
(
)
[
]
(
)
(
(
)
[
(
)
]
)
∞
= + − + − + − +
− +
∞
= + − − + + − −
− +
=
11 1 , 2 1 1 2
1 2
2 1 1 , 1 2 1 2
1 2
k
k z k y k
k k m C n k
k
k z k x k
k k m C n k
z z f
α
λ
β
λ
λ
α
β
α
λ
β
λ
λ
α
β
(2.2)
where
1
1 2
=
+
∞
= ∞
= k
k k
k
y
x
, show that the coefficient bound given by (2.1) is sharp. The functions of the form (2.2) are inS
Hn(
α
,
β
,
λ
,
m
,
k
)
because(
)
[
]
(
) (
[
)
(
(
)
)
]
(
)
(
)
[
]
(
)
(
(
)
[
(
)
]
)
(
)
∞
=
−
+
−
+
−
+
−
+
+
∞
=
−
−
−
+
+
−
−
+
1
2
1
2
1
1
2
,
1
1
2
2
1
2
1
2
1
,
1
1
k
k
b
k
k
k
m
C
n
k
k
k
a
k
k
k
m
C
n
k
α
β
α
λ
β
λ
λ
α
β
α
λ
β
λ
λ
1
2
1 2
=
+
+
=
∞ = ∞
= k
k k
k
y
x
Theorem: 2 Let
f
n=
h
+
g
nbe given by(1.4). Thenf
n∈
S
Hn(
α
,
β
,
λ
,
m
,
k
)
if and only if(
)
{
(
)
(
(
)
)
(
)
(
(
)
(
)
)
(
)
}
1
1 1 n 1 2 1 2 , k 2 1 1 2 , k
k
k
λ
kλ β
kλ
α
C m k a kλ β
kλ
α
C m k b∞
=
+ − − + + − − + + − + − +
(
)
4
β
1
α
≤
−
(2.3)Proof: Since
S
(
m
k
)
S
Hn(
m
k
)
n
H
α
,
β
,
λ
,
,
⊂
α
,
β
,
λ
,
,
, we only need to prove the “only if” part of the theorem . To this end, for functionsf
n of the form (1.4), we notice that the conditionRe is equivalent to
(
)
(
)(
)
( )
(
)
(
)
(
)
(
)
(
)
(
)
(
)
( )
(
)
(
) (
)
2
2 1
2
2 1
1
1
,
1
1
1
1
,
2
1
Re
2 1
1
1
,
2
1
2
1
1
1
,
1
2
.
n k n n k
k k
k k
n k n n k
k k
k k
k
C m k k
a z
k
C m k
k
b z
z
k
C m k
k
a z
k
C m k
k
b z
λ
λ
λ
λ
α
λ
λ α
λ
λ α
β
∞ ∞
= =
∞ ∞
= =
−
+ −
−
− −
+ −
+ −
−
−
+ −
+ −
−
− −
+ −
−
+
>−
If we choose z on the real axis and
z
→
1
− we get(
)
[
] (
)(
)
[
(
)
] (
)
[
(
)
]
(
)
[
(
)
] (
)
[
(
)
]
[
(
)
] (
) (
[
)
]
β
α
λ
λ
α
λ
λ
α
λ
λ
λ
λ
< +
− −
+ − −
− + −
+ − −
− + −
+ + − −
+
∞ = ∞
= ∞ =
∞ =
1 2
2 1
2 1 ,
1 1 2
1 2 , 1
1 1
2
1 2 , 1
1 1
, 1
1
k
k n
k k
n
k k
k n
k n
b k
k m C k
a k
k m C k
b k k m C k
a k k m C k
whence
(
)
[
]
(
)(
)
[
(
)
]
(
)
[
(
)
]
(
)
[
(
)
]
(
)
[
(
)
]
∞[
(
)
]
(
) (
[
)
]
= ∞
= ∞
=
∞ =
+ − −
+ −
− − + −
+ −
− <
− + −
+ + − −
+
1 2
2 1
2 1 ,
1 1 2
1 2
, 1
1 1
2
1 2 , 1
1 1
, 1
1
k
k n
k k
n
k k
k n
k n
b k
k m C k
a k
k m C k
b k k m C k
a k k m C k
α
λ
λ
β
α
λ
λ
β
α
β
λ
λ
λ
λ
and so
(
)
[
]
[
(
)
(
(
)
)
]
(
)
[
(
)
]
[
(
)
(
(
)
)
]
(
)
(
α
)
β
α
λ
β
λ
λ
α
λ
β
λ
λ
− <
+ − + − + −
+ + −
− + + − −
+
∞ =
∞ =
1 2
, 2 1 1
2 1 1 ,
2 1 2 1
1 1
2 1
k k
k n
k n
b k m C k
k k
a k m C k
k k
which is equivalent to (2.3).
Next, we determine the extreme points of the closed convex hulls of
S
Hn(
α
,
β
,
λ
,
m
,
k
)
denoted by(
,
,
,
m
,
k
)
.
S
clco
Hnα
β
λ
Theorem: 3 Let
f
n=
h
+
g
n be given by (1.4). Thenf
clco
S
(
m
k
)
nH
n
∈
α
,
β
,
λ
,
,
if and only if( )
(
( )
( )
)
∞ =
+
=
1
,
knk k k
k
n
z
x
h
z
y
g
z
f
where
( $) * $& '$& ' $+# * , - ./ * 0 .1 234* 256
( )
(
)
(
)
[
1
1
]
[
(
1
)
(
2
(
1
)
2
)
]
(
,
)
,
1
2
k n kz
k
m
C
k
k
k
z
z
h
α
λ
β
λ
λ
α
β
−
−
+
+
−
−
+
−
−
=
(
k
=
2, 3,
……
)
and
( )
( )
(
)
(
)
[
1
1
]
[
2
(
1
)
(
(
1
)
2
)
] (
,
)
,
1
2
1
k n n nkz
k
m
C
k
k
k
z
z
g
α
λ
β
λ
λ
α
β
+
−
+
−
+
−
+
−
−
+
=
(
k
=
1, 2, 3,
…
)
(
)
∞ ==
+
1,
1
k k ky
x
x
k≥
0
andy
k≥
0
.
In particular, the extreme points of
S
Hn(
α
,
β
,
λ
,
m
,
k
)
are{ }
h
k and{ }
g
nk.
Proof: For the functions
f
( )
z
of the form (2.6),we have
( )
(
)
∞ =+
=
1 k k k k kh
y
g
x
z
f
(
)
(
)
(
)
(
)
(
(
)
)
(
)
( )
(
)
(
)
(
(
)
(
)
)
(
)
2 11 2 1 1 1 2 1 2
2 1
1 .
1 1 1 2 1 1 2
,
,
k
x y z x z
k k n k
k k k k k
n k
y z k n
k k k k
C m k
C m k
β
α
λ
λ β
λ
α
β
α
λ
λ β
λ
α
∞ ∞ − = + − = = + − − + + − − ∞ − + − = + − + − + − + Then
(
)
[
]
[
(
)
(
(
)
)
]
(
)
(
)
[
(
)
]
[
(
)
(
(
)
(
)
)
]
(
)
(
)
[
]
(
(
)
[
(
)
]
) (
)
(
)
(
)
(
)
[
]
(
(
)
[
(
)
]
)
(
)
∞ =+
+ − + − + − + − − + − + − + − + ∞ = + − − + + − − − − − − + + − − + 1,
,
2 1 1 2 1 1 1 2 1 2 2 1 1 2 1 12 1 1 1 2 1 2 ,
1 2 1 2 , 2 1 2 1 1 1 k k y k k n k k k n k k k x k m C k k n k k m C k k n k
k
m
C
k
m
C
α
λ
β
λ
λ
α
β
α
β
α
λ
β
λ
λ
α
λ
β
λ
λ
α
β
α
β
α
λ
β
λ
λ
∞ = ∞ =≤
−
=
+
=
2 1 1,
1
1
k k kk
y
x
x
and so
f
n( )
z
∈
S
Hn(
α
,
β
,
λ
,
m
,
k
)
,
conversely, iff
n( )
z
∈
clco
S
Hn(
α
,
β
,
λ
,
m
,
k
)
,
then(
)
(
)
[
1
1
] (
[
1
)
(
2
(
1
)
2
)
] (
,
)
,
1
2
k
m
C
k
k
k
a
k nα
λ
β
λ
λ
α
β
−
−
+
+
−
−
+
−
≤
and
(
)
(
)
[
1
1
]
[
2
(
1
)
(
(
1
)
2
)
] (
,
)
.
1
2
k
m
C
k
k
k
b
k nα
λ
β
λ
λ
α
β
+
−
+
−
+
−
+
−
≤
Set
[
(
)
]
[
(
)
(
(
)
)
]
(
)
(
1
)
,
2
,
2
1
2
1
1
1
k n ka
k
m
C
k
k
k
x
α
β
α
λ
β
λ
λ
−
−
−
+
+
−
−
+
=
(k=2, 3…)
[
(
)
]
[
(
)
(
(
)
)
]
(
)
(
1
)
.
2
,
2
1
1
2
1
1
k n kb
k
m
C
k
k
k
y
α
β
α
λ
β
λ
λ
−
+
−
+
−
+
−
+
=
(k=1, 2…)
We define
∞ =
∞ =
−
−
=
2 1
1
1
k k
k
k
y
x
x
and note that by Theorem 2,x
1≥
0
.
Consequently, we obtain
( )
(
)
∞ =
+
=
1
)
(
)
(
knk k k
k
n
z
x
h
z
y
g
z
f
as required.
The following theorem gives the distortion bounds for functions in
S
Hn(
α
,
β
,
λ
,
m
,
k
)
which yields a covering result for this class.Theorem: 4. Let
f
n( )
z
∈
S
Hn(
α
,
β
,
λ
,
m
,
k
)
.
Then forz
=
r
<
1
we have( )
(
)
[
]
(
)
(
) (
)
(
)
[
2
2
]
(
,
)
,
1
1
1
1
2
1
1C
m
k
b
1r
2m
r
b
z
f
n n−
+
+
+
−
−
+
+
+
+
≤
α
λ
β
λ
βα
α
β
λ
z
=
r
<
1
and
( )
(
)
[
]
(
)
(
) (
)
(
)
[
2
2
]
(
,
)
,
1
1
1
1
2
1
1C
m
k
b
1r
2m
r
b
z
f
n n−
+
+
+
−
−
+
+
−
−
≥
α
λ
β
λ
βα
α
β
λ
z
=
r
<
1
.
Proof: We only prove the right hand inequality. The proof for the left hand inequality is similar and will be omitted. Let
f
n∈
S
Hn(
α
,
β
,
λ
,
m
,
k
)
. Taking the absolute value off
n we have( )
(
)
∞(
)
=
+
+
+
≤
2 1
1
k
k k k
n
z
b
r
a
b
r
f
(
)
(
)
∞ =
+
+
+
≤
2
2 1
1
k
k
k
b
r
a
r
b
(
)
(
)
[
]
[
(
)
]
(
)
[
]
[
(
)
]
(
)
(
−
)
(
+
)
+
−
+
+
+
+
−
+
+
+
−
+
+
=
∞ =2
2 1
1
2
1
2
2
1
1
2
2
1
1
2
1
k
k k n
n
a
b
r
m
m
r
b
α
β
α
λ
β
λ
λ
α
λ
β
λ
λ
α
β
(
)
(
)
[
]
[
(
)
]
(
)
(
)
(
)
2 1
1
,
1
2
2
2
1
1
2
2
1
1
2
1
C
m
k
b
r
m
r
b
n−
+
−
+
−
+
+
+
−
+
+
≤
α
β
βα
α
λ
β
λ
λ
α
β
(
)
[
]
(
)
(
) (
)
(
)
[
]
(
)
2 1
1
,
2
2
1
1
1
1
2
1
C
m
k
b
r
m
r
b
n−
+
+
+
−
−
+
+
+
+
=
α
λ
β
λ
βα
α
β
λ
The following covering result follows from the left hand inequality in Theorem 4. Corollary: 1 If
f
n( )
z
∈
S
H(
α
,
β
,
λ
,
m
,
k
)
,
then[
]
[
(
)
](
)
(
)
[
]
[
(
)
](
)
[
]
[
(
)
](
)
(
)
[
]
[
(
)
](
)
( )
1
1
2
2
1
2
1
1
2
2
1
2 1
:
1
2
2
1
1
2
2
1
.
n n
n n
n
m
m
w w
b
m
m
f U
λ
λ β
λ
α
β
α
λ
λ β
λ
α
βα
λ
λ β
λ
α
λ
λ β
λ
α
+
+
+ −
+ −
−
+
+
+ −
+ −
+
<
−
+
+
+ −
+
+
+
+ −
+
⊂
For our next theorem, we need to define the convolution of two harmonic functions. For harmonic functions of the form
( )
(
)
2 1
1
n,
k k
n k k
k k
f
z
z
a z
b
z
∞ ∞
= =
=
−
+ −
and
( )
( )
∞ ∞
−
+
−
=
1
k,
k n k
k
n
z
z
A
z
B
z
( $) * $& '$& ' $+# * , - ./ * 0 .1 234* 256
we define the convolution of two harmonic functions
f
( )
z
andF z
( )
as( )
( )
(
)
2 1
*
k1
n k.
n n k k k k
k k
f
z
F
z
z
a
A z
b
B
z
∞ ∞
= =
=
−
+ −
(2.4) Using this definition, we show that the classS
Hn(
α
,
β
,
λ
,
m
,
k
)
is closed under convolution.Theorem: 5 For
0
≤
α
1≤
α
2<
1
,
letf
n( )
z
∈
S
Hn(
α
2,
β
,
λ
,
m
,
k
)
andF
( )
z
S
(
1,
,
,
m
,
k
)
.
n H
n
∈
α
β
λ
Then
*
(
2, , , ,
)
(
1, , , ,
)
.
n n
n n H H
f
F
∈
S
α β λ
m k
⊂
S
α β λ
m k
Proof: Let
( )
(
)
2 1
1
nk k
n k k
k k
f
z
z
a z
b
z
∞ ∞
= =
=
−
+ −
be inS
Hn(
α
2,
β
,
λ
,
m
,
k
)
,
and
( )
( )
∞ =
∞ =
−
+
−
=
2 1
1
kk
k k n k
k
n
z
z
A
z
B
z
F
be inS
(
1,
,
,
m
,
k
)
.
n
H
α
β
λ
Then the convolution
(
f
n*
F
n)
is given by (2.4). We wish to show that the coefficient of(
f
n*
F
n)
satisfies the required condition given in Theorem 2.For
F
( )
z
S
(
1,
,
,
m
,
k
)
,
n H
n
∈
α
β
λ
we note thatA
k<
1
andB
k<
1
. Now, for the convolution function(
f
n*
F
n)
, we obtain(
)
[
]
[
(
)
(
(
)
)
]
(
)
(
)
(
)
[
]
[
(
)
(
(
)
)
]
(
)
(
)
k kk
n
k k k
n
B
b
k
m
C
k
k
k
A
a
k
m
C
k
k
k
∞ = ∞
=
−
+
−
+
−
+
−
+
+
−
−
−
+
+
−
−
+
1 1
1
2 1
1
1
2
,
2
1
1
2
1
1
1
2
,
2
1
2
1
1
1
α
β
α
λ
β
λ
λ
α
β
α
λ
β
λ
λ
(
)
[
]
[
(
)
(
(
)
)
]
(
)
(
)
(
)
[
]
[
(
)
(
(
)
)
]
(
)
(
)
kk
n
k k
n
b
k
m
C
k
k
k
a
k
m
C
k
k
k
∞ = ∞
=
−
+
−
+
−
+
−
+
+
−
−
−
+
+
−
−
+
≤
1 1
1
2 1
1
1
2
,
2
1
1
2
1
1
1
2
,
2
1
2
1
1
1
α
β
α
λ
β
λ
λ
α
β
α
λ
β
λ
λ
(
)
[
]
[
(
)
(
(
)
)
]
(
)
(
)
(
)
[
]
[
(
)
(
(
)
)
]
(
)
(
)
kk
n
k k
n
b
k
m
C
k
k
k
a
k
m
C
k
k
k
∞ = ∞
=
−
+
−
+
−
+
−
+
+
−
−
−
+
+
−
−
+
≤
1 2
2
2 2
2
1
2
,
2
1
1
2
1
1
1
2
,
2
1
2
1
1
1
α
β
α
λ
β
λ
λ
α
β
α
λ
β
λ
λ
1
≤
Since
0
≤
α
1≤
α
2<
1
,
and( )
(
2, , , ,
)
.
n
n H
f
z
∈
S
α β λ
m k
Therefore
f
n*
F
n∈
S
Hn(
α
2,
β
,
λ
,
m
,
k
)
⊂
S
Hn(
α
1,
β
,
λ
,
m
,
k
)
.Now we show that
S
Hn(
α
,
β
,
λ
,
m
,
k
)
is closed under convex combination of its members.Theorem: 6 The class
S
Hn(
α
,
β
,
λ
,
m
,
k
)
is closed under convex combination. Proof: Fori
=1, 2…letf
n( )
z
S
Hn(
,
,
,
m
,
k
)
,
( )
( )
∞ =
∞ =
−
+
−
=
2 1
,
1
k
k
k k n k
k
n
z
z
a
z
b
z
f
i i
i
Then by (2.3),
(
)
[
]
(
) (
[
)
(
(
)
)
]
(
)
(
)
[
]
(
)
(
(
)
[
(
)
]
)
(
)
1
1
1
,
1
2
1
2
1
4.
1
1
,
2
1
1
2
1
i
i n
k
n k
k
k
C m k
k
k
a
k
C m k
k
k
b
λ
λ
β
λ
α
β
α
λ
λ
β
λ
α
β
α
∞ =
+
−
−
+
+
−
−
−
≤
+
−
+
−
+
−
+
+
−
(2.5)
For
1
,
0
1
,
1
≤
≤
=
∞ =
i i
i
t
t
the convex combination of i nf
may be written as
( )
( )
∞ =
∞ =
∞ = ∞
= ∞
=
−
+
−
=
2 1 1 1
1
,
1
k
k
k
k i
i n
k k i
i n
i
i
f
z
z
t
a
z
t
b
z
t
i i
i
Then by (2.5),
(
)
[
]
[
(
)
(
(
)
)
]
(
)
(
)
[
]
(
(
)
[
(
)
]
)
(
)
∞ =
∞ = ∞
=
−
+
−
+
−
+
−
+
+
−
−
−
+
+
−
−
+
1 1
1
12
1
1
2
1
1
1
2
1
2
1
1
1
kk i
i n
k i
i n
i
i
t
b
k
k
k
a
t
k
k
k
α
β
α
λ
β
λ
λ
α
β
α
λ
β
λ
λ
(
)
[
] (
) (
[
)
(
(
)
)
]
(
)
(
)
[
] (
)
(
(
)
[
(
)
]
)
(
−
)
+
−
+
−
+
−
+
+
−
−
−
+
+
−
−
+
=
∞ = ∞
=1 1
1
2
1
1
2
,
1
1
1
2
1
2
1
,
1
1
k
k n
k n
i i
i i
b
k
k
k
m
C
k
a
k
k
k
m
C
k
t
α
β
α
λ
β
λ
λ
α
β
α
λ
β
λ
λ
1
4
i4.
i
t
∞ =
≤
=
This is the condition required by (2.3) and so
( )
(
)
1
, , , ,
.
i
n
i n H
i
t f
z
S
α β λ
m k
∞ =
∈
ACKNOWLEDGEMENT:
The present research of the author is supported by the University Grants Commission, Government of India under Grant No. F.8-2(223)/2011(MRP/NRCB).
REFERENCES:
[1] Ahuja, O.P., Planar harmonic univalent and related mappings, J. Inequal. Pure Appl. Math., 6(4) (2005), Art.122, 1-18.
[2] Al-Oboudi, F. M., On univalent functions defined by a generalized Salagean operator, Int. J. Math. Math. Sci. 27, (2004), 1429-1436.
[3] Al-Shaqsi, K. and Darus, M., An operator defined by convolution involving the poly-logarithms functions, J. Math. Stat., 4 (1) 2008, 46-50.
[4] Clunie, J. and Sheil-Small, T., Harmonic univalent functions, Ann. Acad. Sci. Fen. Series AI Math., 9(3) (1984), 3-25.
[5] Darus, M. and Al-Shaqsi, K., Differential sandwich theorems with generalized derivative operator World Academy of Science, Engineering and Technology, 38 (2008), 11-14.
[6] Dixit, K.K., Pathak, A.L., Porwal, S. and Agarwal, R., A new subclass of harmonic univalent functions defined by Salagean operator, Int. J. Contemp. Math. Sciences, 4(8) (2009), 371-383.
( $) * $& '$& ' $+# * , - ./ * 0 .1 234* 256
[8] Gencel, T. and Yalcin, S., A special class of harmonic univalent functions, Sarajevo J. Math., 4 (17) (2008), 197-206.
[9] Jahangiri, J.M., Harmonic functions starlike in the unit disc, J. Math. Anal. Appl., 235, (1999), 470-477.
[10] Pathak, A.L., Joshi, S.B., Dwivedi P. and Agarwal, R., Subclass of harmonic univalent functions associated with derivative operator, Hacettepe J. Math. Stat., Accepted (2011).
[11] Ponnusamy, S. and Rasila, A. , Planar harmonic mappings, Mathematics Newsletter, 17(2) (2007), 40-57. [12] Ruscheweyh, St., New criteria for univalent functions, Proc. Amer. Math. Soc., 49, (1975), 109-115.
[13] V Salagean, G.S., Subclass of univalent functions, Complex Analysis-Fifth Romanian Finish Seminar, Bucharest, 1 (1983), 362-372.
[14] Sheil-Small, T., Constants for planar harmonic Mappings, J. Lon. Math. Soc., 42(2) (1990), 237-248.
[15] Silverman, H., Harmonic univalent functions with negative coefficients, J. Math. Anal. Appl., 220 (1998), 283-289.
[16] Silverman, H. and Silvia, E.M., Subclasses of harmonic univalent functions, New Zealand J. Math., 28 (1999), 275-284.
[17] Yalcin, S., Joshi, S.B. and Yasar, E., On certain subclass of harmonic univalent functions defined by a generalized Ruscheweyh derivative operator, Appl. Math. Sci., 4(7), (2010), 327 – 336.
