TITLE MARCH 2012 - Pakistan Academy of Sciences

58 M.K. Aouf et al
I , p
( a 1,
b,
c)
g(
z)
h(
z)
. Then g( z)
S
, p(
a 1,
b,
c;
)
and
z(
I
pI
z(
I
pI
,
p
,
p
( a 1,
b,
c)
f ( z))
( z)
( z U) . (2.4)
( a 1,
b,
c)
g(
z)
Now let
z
I
,
p
,
p
( a 1,
b,
c)
f ( z))
q(
z),
( a 1,
b,
c)
g(
z)
(2.5)
2
where q ( z)
1
q1z
a2z
...
is analytic in
U and q ( z)
0 for all z U . Thus by using the
identity (1.14), we have
z(
I
pI
,
p
,
p
( )
I
,
( a 1, b,
c)
zf
p
( a 1, b,
c)
f ( z))
p z
( a 1,
b,
c)
g(
z)
I ( a 1,
b,
c)
g(
z)
,
p
( )
( )
,
( , , )
zf
p z
z I a b c
,
( , , )
zf
p z
p
I a b c
p
( a p)
I,
p
( a,
b,
c)
g(
z)
I,
p
( a,
b,
c)
g(
z)
.
z(
I,
p
( a,
b,
c)
g(
z))
( a p)
I,
p
( a,
b,
c)
g(
z)
(2.6)
Since g z)
S
( a 1,
b,
c;
)
S ( a,
b,
c;
)
( )
(
, p
,
p
S ,
by Theorem 1, we set
z(
I
,
( a,
b,
c)
g(
z))
p
G(
z)
,
pI ( a,
b,
c)
g(
z)
,
p
,
p
( )
( a,
b,
c)
zf
p z
( a p)
I
z
( )
( a,
b,
c)
zf
p z
I
( a,
b,
c)
g(
z)
( a p)
I ( a,
b,
c)
g(
z)
,
p
,
p
,
p
where G( z)
(
z)
( z U)
for S.
Then, by
virture of (2.5) and (2.6), we observe that
zf (
z)
I
,
( a,
b,
c)
p
q(
z)
I,
p(
a,
b,
c)
g(
z)
(2.7)
p
and
( )
,
( , , )
zf
p z
zI
a b c
p
z(
I ( a 1,
b,
c)
f ( z))
( a p)
q(
z)
, p
I,
p
( a,
b,
c)
g(
z)
. (2.8)
pI ( a 1,
b,
c)
g(
z)
pG(
z)
a p
,
p
Differentiating both sides of (2.7) with respect
to z , we obtain
( )
z I
,
( a,
b,
c)
zf
p p z
I ( a,
b,
c)
g(
z)
z(
I
pI
,
p
pG(
z)
q(
z)
zq(
z)
.
Making use of (2.4), (2.8) and (2.9), we get
,
p
,
p
(2.9)
( a 1,
b,
c)
f ( z))
pG(
z)
q(
z)
zq(
z)
( a p)
q(
z)
( a 1,
b,
c)
g(
z)
pG(
z)
a p
zq(
z)
q( z)
( z)
( z U) . (2.10)
pG(
z)
a p
Since
a p, p
and G( z)
(
z)
( z U)
,
pG(
z)
a p 0 ( zU)
.
Re
Hence, by taking
1
Q(
z)
pG(
z)
a p
in (2.10), and applying Lemma 2, we can show
that
p( z)
( z)
( zU) ,
so that
f ( z)
C
, p(
a,
b,
c;
,
) ( ,
S) .
For the second part, by using arguments
similar to those detailed above with the identity
(1.13), we obtain:
C a,
b,
c;
,
) C
( a,
b,
c;
,
) ( ,
) .
, p( 1,
p
S
The proof of Theorem 3 is thus completed.
3. INCLUSION PROPERTIES INVOLVING
J , p
In this section, we consider the generalized
Bernardi-Libera-Livingston integral operator
J , p
( p)
defined by (see [24],[25],and [26]).
p
z
1
J , p(
f )( z)
t
f ( t)
dt ( f A(
p);
p) . (3.1)
z 0
Theorem 4. Let p, p,
a p and
p . If f ( z)
S
, p(
a,
b,
c;
)
( S)
, then