TITLE MARCH 2012 - Pakistan Academy of Sciences

Inclusion Properties of Certain Operators 59
J
, p( f )( z)
S
,
p(
a,
b,
c;
)
( S) .
Proof . Let f ( z)
S
,
( a,
b,
c;
)
for S
, and
set
z(
I
pI
,
p
,
p
( a,
b,
c)
J
( a,
b,
c)
J
, p
, p
p
( f )( z))
q(
z)
,
( f )( z)
(3.2)
2
where q ( z)
1
q1z
q2z
...
is analytic in
U and q ( z)
0 for all z U . From (3.1), we
obtain
z( I, p ( a, b, c) J, p ( f )( z)) ( p) I,
p ( a, b, c) f ( z)
(3.3)
I ( a, b, c) J ( f )( z) ( z U) .
, p
,
p
By applying (3.2) and (3.3), we obtain
I,
p(
a,
b,
c)
f ( z)
( p)
pq(
z)
.
I ( a,
b,
c)
J ( f )( z)
,
p
, p
(3.4)
Differentiating (3.4) logarithmically with respect
to z , we obtain
z(
I
I
,
p
,
p
( a,
b,
c)
f ( z))
zq(
z)
q(
z)
.
( a,
b,
c)
f ( z)
pq(
z)
Since
from (3.5), we have
Re
(3.5)
p, ( z)
S
, and f ( z)
S
,
( )
,
zq(
z)
pq(
z)
p
p(
z)
0 and q(
z)
(
z)
( z U)
.
Hence, by virbure of Lemma 1, we conclude
that q( z)
(
z)
( zU)
,
which implies that
J
, p( f )( z)
S
,
p(
a,
b,
c;
)
( S) .
Next, we derive an inclusion property
involving , which is given by
J , p
Theorem 5. Let p, p,
a p and
p . If f ( z)
K
, p(
a,
b,
c;
)
( S)
, then
J
, p( f )( z)
K,
p(
a,
b,
c;
)
( S) .
Proof . By applying Theorem 4, it follows that
zf (
z)
f ( z)
K, p(
a,
b,
c;
)
S,
p(
a,
b,
c;
)
p
zf ( z)
J, p
S,
p( a, b, c; )
p
z
J , p( f )( z ) S
,
p( a , b , c ; )
p
J f )( z)
K
( a,
b,
c;
)
(
, p( ,
p
S
which proves Theorem 5.
Finally, we prove
Theorem 6. Let
) ,
p, p,
a p and
p . If f ( z)
C
, p(
a,
b,
c;
,
) ( ,
S)
, then
J
, p( f )( z)
C
, p(
a,
b,
c;
,
) ( ,
S) .
Proof. Let f ( z)
C
,
( a,
b,
c;
,
) for , S
.
p
Then, in view of (1.7), there exists a function
g( z)
S
,
( a,
b,
c;
)
such that
z(
I
pI
,
p
,
p
p
( a,
b,
c)
f ( z))
( z)
( z U) . (3.6)
( a,
b,
c)
g(
z)
Thus we set
z(
I ( a,
b,
c)
J
pI
,
p
,
p
( a,
b,
c)
J
, p
, p
( f )( z))
q(
z)
,
( f )( z)
2
where q ( z)
1
q1z
q2z
...
is analytic in
U and q ( z)
0 for all z U . Applying (3.3), we
get
( )
I
,
( a,
b,
c)
zf
p
z(
I
,
( a,
b,
c)
f ( z))
p z
p
pI ( a,
b,
c)
g(
z)
I ( a,
b,
c)
g(
z)
,
p
z
I
z
z
I
,
p
( a,
b,
c)
J
,
p
( a,
b,
c)
J
I
( a,
b,
c)
J ( g)(
z)
I
( a,
b,
c)
J ( g)(
z)
,
p
,
p
, p
zf
( )
p z
I
, p
( a,
b,
c)
J
, p
I ( a,
b,
c)
g
,
p
z I
,
I
,
p
,
p
, p
, p
zf (
)
p z
z f ( )
(
)
p z
I
,
( a,
b,
c)
J
,
zf
p
p
p z
( z)
I
,
p
( a,
b,
c)
J
, pg(
z)
p
( a,
b,
c)
J
, p
( g)(
z)
, p
( a,
b,
c)
J
, pg(
z)
(3.7)
Since g( z)
S
, p(
a,
b,
c;
)
( S)
, by virtue
of Theorem 4, we have J g)(
z)
S ( a,
b,
c;
) .
Let us now put
, p( ,
p
.