TITLE MARCH 2012 - Pakistan Academy of Sciences

Inclusion Properties of p-Valent Meromorphic Functions 47
, A ; , B ; z
by
, A ; , B ; z
p, q, s i i 1, q i i 1, s
p, q, s i i 1, q i i 1, s
p, q, s i , Ai ; , ;
1, i B
q i z
1, s
1
0.
p
z 1
z
p, q, s
i i 1, q i i 1, s
defined by (1.6), we define the linear operator
, , , ; , :
p q s
i Ai
1, q i
B
i
1, s p
as follows:
p
p, q, s i , Ai ;
, ( )
1, i B
i f z
q 1, s
p, q, s i , Ai ; , ; ( )
1, q i Bi
z
f z
1, s
s
q
k p i
k p
Bi i (1.8)
z
Analogous to , A ;
, B
k
1 p
p
i1 i1
q
s
i k
p
Ai i
i1 i1
f p ; 0;z U .
For convenience, we write
1, A1 ,..., q, A ;
q
p, q, s 1, A1 , B1 f z
p, q,
s
f z.
1, B1 ,..., s,
Bs
One can easily verify from 1.8 that
1, ,
1 p, q, s 1 1 1
zA A B f z
, A , B f z
1 p, q, s 1 1 1
( pA )
1 1 p, q,
s
and
1, A , B f z ( A 0)
1 1 1 1
a z
1
p, q, s
1, 1, 1
p, q, s 1, 1,
1
z A B f z A B f z
p, q, s 1 1 1
k
k
(1.9)
( p) , A , B f z ( 0). (1.10)
Specializing the parameters p, q, s, A ( i 1,..., q),
B ( i 1,..., s)
and in (1.8) we have:
i
(i) For A 1( i 1,..., q),
B 1( i 1,..., s)
and
i
p p, p , we have
,1,1 f z M ( ) f ( z),
p
p, q, s 1 p, q, s 1
where the operator M p,q,s 1 was introduced
by Patel and Patil [21] and Mostafa [17];
(ii) For A i = 1,…,q), B i = 1(I = 1,…,s),
q 2, s 1,
n p n p, p
1
i
i
and 2 1 ( 0) , we have
p,2,1 n p, ; f z Inp
1,
f ( z),
where the operator I np1, was introduced by
Aouf and Xu [5] which for p 1 reduces to
I n, n 1, 0 , where the operator I n,
was introduced by Yuan et al. [28];
(iii) For A 1( i 1,..., q), B 1( i 1,..., s),
i
n p n p,p N, q 2, s 1 and
n
p
we have [1,1;1] ( )
,2,1
f z
D
1 2 1
1,
np1
f ( z),
where the operator
i
p
n p 1
D
introduced by Yang [26] and Aouf ([1] and [2]);
was
(iv) For p 1, we have 1, qs ,
1, A1 , B1
f z
, qs ,
(
1, A1 , B1
) f z,
where the operator
( , A, B)
was introduced by Aouf et al.
, qs , 1 1 1
[4];
(v) For A i 1 i 1,...,q,B i 1 i 1,...,s
and p 1 , we have
,1,1
1, qs , 1
f z
H, qs ,( ) f z , where the operator H ,q,s
was introduced by Cho and Kim [9], Muhamad
[18] and Noor and Muhamad [19].
Also, we note that:
(i) For 1 , then the operator
1
p, q, s 1 1 1
, A,
B reduces to the operator
p,q,s 1 ,A 1 ,B 1 , defined by:
, A , B f ( z)
z
p, q, s 1 1 1
s
p
k
p
B
i i i
i1 i1
q
s
k
1 p
i i i
i1 i1
k
p
A
q
az
(ii) For A i 1 i 1,...,q,B i 1 i 1,...,s
1
and 1 , then the operator
reduces to the operator
N p,q,s 1 fz z p
k1p
k
k
,
,
p, q, s 1
p q s
, , 1 ,1,1
N defined by:
1 kp ... s kp
1 kp ... q kp
a k z k