TITLE MARCH 2012 - Pakistan Academy of Sciences

Inclusion Properties of Certain Operators 55
( 1)
, p
,
p
I ( a, b, c) f ( z) f ( z)
0
( a, b, c R \ Z , p, p , z U).
(1.11)
After some computations, we obtain
p ( a)
k
( b)
k
k
p
I , p(
a,
b,
c)
f ( z)
z
ak
pz
. (1.12)
k1
( c)
( p)
From (1.12), we deduce that
( a,
p,
a)
f ( z)
f ( z)
( p,
p
I
, p
and
zf (
z)
I1,
p
( p 1, p 1,
p)
f ( z)
,
p
z( I ( a, b, c) f ( z)) ( p) I ( a, b, c) f ( z)
1, p
,
p
I ( a, b, c) f ( z) ( p),
1,
p
and
z( I ( a, b, c) f ( z)) aI ( a 1, b, c) f ( z)
, p
,
p
( a p) I ( a, b, c) f ( z).
,
p
We note that;
k
k
)
(1.13)
(1.14)
(i) I a,
p 1,
a)
f ( z)
I ( n ) , where
n, p( n
p1
p
I
n p1
is the Noor integral operator of
( n p 1)
th
order (see Liu and Noor [12]
and Patel and Cho [13]);
n
p1
(ii) I ( p 1,
n p,1)
f ( z)
D f ( z)
( n ) ,
1,
p
p
1
where D n
p
f ( z)
is the ( n p 1)
th
order Ruscheweyh derivative of a function
f ( z)
A(
p)
(see Kumar and Shukla [14]);
(iii) I a,2,
a)
f ( z)
I f ( z)
( n 1)
( n , 1 n
is the Noor integral operator of
(see [15]);
( ,
p)
(iv) I ( a,
p 1,
a)
f ( z)
f ( )
1
,
p
z
z
( k p 1) ( p 1 )
z ak
pz
( p 1) ( k p 1 )
k1
z F (1, p 1; p 1 ; z) f ( z)
2 1
( p 1; z U
).
, where I
n
n th order
p k p
p
( ,
p)
The operator
z
was introduced and
studied by Patel and Mishra [16]:
(v)
I,
p ( p, p, p 1) f ( z)
,
J f ( z) ( p)
,
p
where J , p
is the generalized Bernardi-
Libera-Livingston operator defined by (3.1)
(see [17]);
I,1 ( , b, b) f ( z) I,
f ( z)
(vi)
,
( 1, 0, f ( z) A(1) A)
where I
,
is the Choi-Saigo-Srivastava
operator (see [17]).
We also note that:
p
I , p( ,
b,
b)
f ( z)
I,
f ( z)
( p,
0, f ( z)
A(
p))
,
p
where I ,
is the generalized Choi-Saigo-
Srivastava operator (see [17]) defined by
p
p ( )
k
k
p
I,
f ( z)
z ak
pz
( p;
0; z U) .
k1
( p)
k
Next, by using the general operator
( a,
b,
) , we introduce the following classes of
I , p
c
analytic
S
,
p
p valent functions for
f : f A( p) and
( a, b, c; )
,
I,
p( a, b, c) f ( z) S
p( )
f : f A( p) and
K,
p ( a, b, c; )
,
I,
p( a, b, c) f ( z) K
p( )
And
C
,
p
f : f A( p) and
( a, b, c; , )
.
I,
p( a, b, c) f ( z) Cp( , )
We also note that
zf (
z)
f ( z)
K, p(
a,
b,
c;
)
S,
p(
a,
b,
c;
).
(1.15)
p
In particular, we set
1
z
Sn, pa, p 1,
a;
Sn
p1
( n p),
1
z
S
1
Az
a
b,
c;
S
1
Bz
a,
b,
c;
A,
B ( 1
B A 1),
,
p
,
,
p