TITLE MARCH 2012 - Pakistan Academy of Sciences

2 Variability of Some Differential Operators Implying Starlikeness
Theorem 1.3. For a function f , the
differential inequality
zf ( z) zf ( z)
1 0, z E,
f ( z) f ( z)
ensures the membership for f in the class
In 2002, Li and Owa [3] proved the following
two results:
Theorem 1.4. If f satisfies
zf ( z) zf ( z)
1
, z E,
f ( z) f ( z)
2
for some , 0 , then
*
f .
Theorem 1.5. If f satisfies
2
zf ( z) zf ( z)
(1 )
1
, z E,
f ( z) f ( z)
4
for some , 0
2 , then
f * ( / 2) .
Later on Ravichandran et al. [10] proved the
following result:
Theorem 1.6. If f satisfies
zf ( z) zf ( z)
1
f ( z) f ( z)
1
, z E,
2
2
for some , , 0 , 1, then
* .
f * ( )
.
For more such results, we refer the readers to
[5, 7, 9]. Recently, Singh et al [11] proved the
following more general result for starlikeness
which unifies all the above mentioned results.
Theorem 1.7. Let , 0, ,0 1, and
,0
1, be given real numbers. Let
M ( , , ) [1 (1 )]
(1 )(1 ) (1 )
2 2
2
(1 ) ,
and
N( , , ) [1 (1 )]
2 (1 )(1 )
(1 )
2
[2 (1 2 )(1 )(3 2 )
2(1 )
2
(1 )(3 2 )].
(i) For 0
1/ 2 , let a function f A,
f( z)
0 in E, satisfy
z
(a)
zf ( z) zf ( z)
1
f ( z) f ( z)
M
( , , ),
zf ( z) zf ( z)
1
1
f ( z) f ( z)
whenever
3 4 3
(2 13 2 ) (3 2 ) 0, and
(b)
zf ( z) zf ( z)
1
f ( z) f ( z)
N( , , ),
zf ( z) zf ( z)
1
1
f ( z) f ( z)
whenever
3 4 3
(2 13 2 ) (3 2 ) 0.
Then
f * ( )
.
(ii) For 1/ 2
1, if a function f A,
f( z)
0 in E, satisfies
z
zf ( z) zf ( z)
1
f ( z) f ( z)
M
( , , ),
zf ( z) zf ( z)
1
1
f ( z) f ( z)
then
f * ( )
.
The main objective of this paper is to extend
the region of variability of above mentioned
differential operators implying starlikeness. The
extended regions are shown pictorially using
Mathematica 7.0.