TITLE MARCH 2012 - Pakistan Academy of Sciences

Inclusion Properties of p-Valent Meromorphic Functions 49
Proof. To prove the first part, let
1
f MS ( , A , B ; ; )
and set
p, q, s 1 1 1
1 z p, q, s
1, A1 , B1
f ( z)
q( z)
( z U),
p
A B f z
p, q, s
1, 1, 1
( )
(2.1)
where q is analytic in U with q (0) 1.
Applying (1.10) in (2.1), we obtain
1
1 z p, q, s
1, A1 , B1
f ( z)
p
A B f z
1
p, q, s
1, 1, 1
( )
zq( z)
q( z) ( z U
).
( p ) q( z)
p
Since
z
p
maxRe ( ) , we see that
zU
p
(2.2)
Rep qz p 0 z U.
Applying Lemma 1 to (2.2), it follows that
q( z) ( z)
, that is f ( z) MS
p, q, s( 1, A1 , B1
, ; )
. Moreover, by using the arguments similar to
those detailed above with (1.9), we can prove the
second part. Therefore the proof is completed.
Theorem 2. Let S with
maxRe ( z)
zU
1
p
1 1
min , A p
( , 0,0 p).
p
p
A1
Then
MK
MK
1
p, q, s 1 A1 B1
( , , ; ; )
p, q, s 1 A1 B1
( , , ; ; )
p, q, s 1 A1 B1
MK ( 1, , ; ; ).
Proof. Applying (1.11) and using Theorem 1, we
observe that
1
p, q, s 1 1 1
f ( z) MK ( , A , B ; ; )
zf ()
z 1
MS p, q, s ( 1, A1 , B1
; ; )
p
zf ()
z
MS p, q, s ( 1, A1 , B1
; ; )
p
p, q, s 1 1 1
f ( z) MK ( , A , B ; ; ).
Also
p, q, s 1 1 1
f ( z) MK ( , A , B ; ; )
zf ()
z
MS p, q, s ( 1, A1 , B1
; ; )
p
zf ()
z
MS p, q, s ( 1 1, A1 , B1
; ; )
p
p, q, s 1 1 1
f ( z) MK ( 1, A , B ; ; ),
which evidently proves Theorem 2.
Taking
1
Az
(z) 1 B A 1 ,
1
Bz
in Theorem 1 and Theorem 2, we have
Corollary 1. Let
1
1
p 1
1
min , A p
A
B
p p
1
( , A
0,0
p, 1 B A 1).
1
Then
1
p, q, s 1 1 1
MS ( , A , B ; ; A, B)
p, q, s 1 1 1
MS ( , A , B ; ; A, B)
p, q, s 1 1 1
MS ( 1, A , B ; ; A, B),
and
1
p, q, s 1 1 1
MK ( , A , B ; ; A, B)
p, q, s 1 1 1
MK ( , A , B ; ; A, B)
p, q, s 1 1 1
MK ( 1, A , B ; ; A, B).
Next, by using Lemma 2, we obtain the following
inclusion relations for the class MC p,q,s 1 ,A 1 ,
B 1 ;,;,.
Theorem 3. Let , S with maxRe{ ( z)}
1
1
p
p
min , A p
MC
p
1
p, q, s 1 A1 B1
( , , ; , ; , )
p, q, s 1 A1 B1
MC ( , , ; , ; , )
p, q, s 1 A1 B1
MC ( 1, , ; , ; , ).
zU
1
( , A
0, 0 , p).
1
Then