TITLE MARCH 2012 - Pakistan Academy of Sciences

Sukhwinder Singh Billing 5
Corollary 3.1. Let be a real number with
zf ()
z
0 . If f , 0, zE
, satisfies the
f()
z
differential subordination
zf ( z) zf ( z)
(1 ) 1
f ( z) f ( z)
1 (1 2 ) z 2 (1 )
z
,
1 z (1 z)(1 (1 2 ) z)
can vary over the portion of the plane right to the
curve h () 1
z for the same conclusion. Thus our
result extends the region of variability of this
operator for the same implication and the region
bounded by the dashing line and the curve is the
claimed extension as shown in Fig. 1.
then
f
* ( )
.
1 3
Remark 3.2. For and , Corollary
2 4
3.1 reduces to the following result.
zf ()
z
If f , 0, zE
, satisfies the
f()
z
condition
zf ( z) zf ( z) 2 z z
1 h1
( z),
f ( z) f ( z) 1 z (1 z)(2 z)
then
f * (3/ 4) .
Fig. 1.
Substituting the same values of and in
the result of Fukui [1] stated in Theorem 1.2, we
obtain the following result:
If f
, satisfies the condition
zf ( z) zf ( z) 4
1 , z E,
f ( z) f ( z) 3
then
f * (3/ 4) .
To compare both the results, we plot h ( ) 1
4
and the line ( z)
in Fig. 1.
3
We see that according to the result of Fukui
[1], for the starlikeness of order 3/ 4 of f()
z ,
zf ( z) zf ( z)
the differential operator 1
can
f ( z) f ( z)
vary in the complex plane on the right side of the
4
line ( z)
shown with dashes in Fig. 1
3
whereas according to our result, the same operator
Remark 3.3. When we select
1 (1 2 )
z
qz ( ) , 0 1. Then
1
z
zq
( z) 1 z zq
( z)
1 , i.e. 1 0,
q
( z) 1 z q( z)
and
zq( z) 1
1z
1 2 qz ( )
q( z) 1
z
1 (1 2 ) z 1
2 .
1
z
Therefore for 0
1, we have
zq( z) 1
1 2 qz ( )
0.
q()
z