TITLE MARCH 2012 - Pakistan Academy of Sciences

54 M.K. Aouf et al
function w(z)
in U with w( z)
z for all
z U , such that ( z)
f ( w(
z))
g , denoted
g f of g( z)
f ( z)
. In case f (z)
is univalent
in U we have that the subordination g( z)
f ( z)
is equivalent to g( 0) f (0)
and g( U)
f ( U)
(see [4]; see also [5],[6, p. 4]).
For the functions f j
( z)
( j 1,2)
defined by
p
k
p
f
j
( z)
z ak
p,
jz
( p)
(1.6)
k1
we denote the Hadamard product (or convolution)
of f 1(
z ) and f ( z)
by 2
p
k
p
f f )( z)
z a a z . (1.7)
(
1 2
k
p,1
k
p,
2
k1
Let M be the class of analytic functions
(z) in U normalized by ( 0) 1, and let S be
the subclass of M consisting of those functions
(z) which are univalent in U and for which
(U) is convex and Re ( z)
0 ( zU)
.
Making use of the principle of subordination
between analytic functions, we introduce the
subclasses S ( ),
K ( )
and C ( ,
) of the
p
p
class A ( p)
for , S , which are defined by
zf (
z)
S p
( )
f : f A(
p)
and (
z)
in U ,
pf ( z)
1 zf ( z)
K p
( )
f : f A(
p)
and 1
(
z)
in U ,
p f (
z)
f : f A( p) and h
Cp
( , ) f()
z
.
K p ( ) s. t. ( z) in U
h()
z
p
1
z
Kp
Kp
,
1
z
p ( p 2 )
z
K p
K p( ) (0 p),
1
z
1z
1z Cp
, Cp
,
1z
1z
p ( p 2 ) z p ( p 2 )
z
Cp
, Cp( , ) (0 , p).
1z
1z
Furthermore, for the function classes S p
[ A,
B,
]
and K p
[ A,
B,
]
investigated by Aouf ([9, 10], it is
easily seen that
1[
B(
AB)(1
)]
p
S
p
1
Bz S
p[
A,
B,
] ( 1
B A 1;0
p)
(see Aouf [9]),
And
1[
B(
AB)(1
)]
p
K
p
1
Bz K
p[
A,
B,
] ( 1
B A 1;0
p)
(see Aouf [10]).
For real or complex number a , b,
c other than
0,
1, 2,...
, the hypergeometric series is defined
by
( a)
k
( b)
k k
2
F1
( a,
b;
c;
z)
z ,
(1.8)
k0
( c)
(1)
k
where ( x)
k
is Pochhammer symbol defined by
k
(
x k)
x(
x 1)...(
x k 1)
( k N;
xC),
( x)
k
(
x)
1
( k 0; k C
\{0}).
We note that the series (1.8) converges
We note that for p 1, the classes
S
1
( )
S
( ),
K1(
)
K(
)
and
C1 ( ,
C(
,
) are investigated by Ma and
Minda [7] and Kim et al [8].
Obviously, for special choices for the
functions and involved in the above
definitions, we have the following relationships:
1
z
Sp
Sp
,
1
z
p ( p 2 )
z
S p
S p( ) (0 p),
1
z
absolutely for all z U so that it represents an
analytic function in U (see, for details, [11,
Chapter 14]).
Now we set
p
z
f , p(
z)
p
(1 z)
( p)
(1.9)
and define f , p(
z)
by means of the Hadamard
product
( 1)
p
f , p( z)
f,
p
( z)
z
2F1
( a,
b;
c;
z)
( zU)
, (1.10)
This leads us to a family of linear operators