113 Awasthi, 2017
INTERNATIONAL JOURNAL OF APPLIED RESEARCH AND TECHNOLOGY
ISSN 2519-5115 RESEARCH ARTICLE
A Generalized Sub Class of Univalent Starlike Functions with a Linear
Operator
Jitendra Awasthi
Department of Mathematics
S.J.N.P.G. College, Lucknow,
226001
Corresponding author:
Dr. Jitendra Awasthi
[email protected]
Received: March 26, 2017
Revised: April 24, 2017
Published: April 30, 2017
ABSTRACT
This paper deals with a new class T* (α,β,λ) which is a subclass
of uniformly starlike functions involving a linear operator
L(a,b;c). Coefficients inequality, Distortion theorem, Extreme
points, Radius of starlikeness and radius of convexity for
functions belonging to this class are also obtained.
Keywords
-
Univalent, starlike, convex, analytic, linear
operator.
INTRODUCTION
Let S denote the class of functions of the form
2 , )
( ) 1 . 1 (
n n nz
a z z f
Which are analytic and univalentin the unit open disk ∆= {z: |z|<1}.
Let T be the subclass of S consisting of functions of the form
(1.2) ( ) , 0, 2. 2
n a
z a z z f
n
n n n
Which was introduced and studied by Silvarman (1975).
Now, we consider a function
(a,b;c;z) as.
,
1
,
,....,
1
,
0
)
(
)!
1
(
)
(
)
(
)
;
;
,
(
)
3
.
1
(
2 1
1
1
z
forc
a
b
z
c
n
b
a
z
z
c
b
a
n kn n
n n
Where
(
)
n is the Pochhammer symbol defined by(1.4)
.
),
1
)...(
1
(
0
,
1
)
(
)
(
)
(
N
n
n
n
n
n
Carlson and Shaffer (2002) introduced a linear operator L(a;c) which is defined as
L(a;c)f(z)
(a;c;z)*f(z)
z
z
a
c
a
z
n
n n n
n
,
)
(
)
(
2 1
1
Where * stands for the Hadamard product of two power series
2 ) (
n n nz
z
and
2 ) (
n n nz
z
defined by
n n
n n z
z
2 ) *
(
We note that L(a,1;c)=L(a;c).Also L(a,1;a)f(z)=f(z) L(2,1;1)f(z)=zf ( ) L(r+1,1;1)f(z)=Dr f(z)
where Dr f(z)is theRuscheweyh derivatives of f(z) defined by Ruscheweyh (1975) as
(1.6) * ( ), 1. )
1 ( )
( 1
f z r
z z z
f
Dr r
Which is equivalent to
)}
(
{
!
)
(
z
1f
z
dz
d
r
z
z
f
D
r rr r
For
0
1
,β≥0 and -1≤ α <1, we introduced a subclass T* (α,β,λ) of T consisting of functions f(z) of the form(1.2) and satisfying the condition.
,
1
)}
(
)
;
,
(
{
)
(
)
;
,
(
)
1
(
)}
(
)
;
,
(
{
)}
(
)
;
,
(
{
)
(
)
;
,
(
)
1
(
)}
(
)
;
,
(
{
Re
' '
' '
z
z
f
c
b
a
L
z
z
f
c
b
a
L
z
f
c
b
a
L
z
z
f
c
b
a
L
z
z
f
c
b
a
L
z
f
c
b
a
L
z
For b=1, λ=0, T *(α,β,λ) reduces to TS (α,β,) which was defined and studied by G. Murugusundaramoorthy (2004).
In this paper we will obtain a necessary and sufficient conditions for the functions f(z) T* (α,β,λ). Furthermore extreme points, distortion bounds, Closure properties, radius of starlikeness and convexity for f(z)T* (α,β,λ) are also obtained.
COEFFICIENTS INEQUALITY
Theorem2.1: A necessary and sufficient condition for f(z) of the form (1.2) to be in the classT *(α,β,λ) ,-1≤ α <1, β ≥ 0 is that
(2.1)
2 1
1
1
(
1
)
)
(
)!
1
(
)
(
)
(
)]
}(
)
1
(
1
{
)
1
(
[
n
n n n n
a
c
n
b
a
n
n
Proof: Let f (z) T *(α,β,λ),then it is sufficient to show that
.
1
1
)}
(
)
;
,
(
{
)
(
)
;
,
(
)
1
(
)}
(
)
;
,
(
{
Re
1
)}
(
)
;
,
(
{
)
(
)
;
,
(
)
1
(
)}
(
)
;
,
(
{
' '
' '
z
f
c
b
a
L
z
z
f
c
b
a
L
z
f
c
b
a
L
z
z
f
c
b
a
L
z
z
f
c
b
a
L
z
f
c
b
a
L
z
2 1
1
1
,
.
)
(
)!
1
(
)
(
)
(
)
(
)
;
,
(
)
5
.
1
(
n
n n n n
n
a
z
z
c
n
b
a
z
We have
1
)}
(
)
;
,
(
{
)
(
)
;
,
(
)
1
(
)}
(
)
;
,
(
{
Re
1
)}
(
)
;
,
(
{
)
(
)
;
,
(
)
1
(
)}
(
)
;
,
(
{
' ' ' 'z
f
c
b
a
L
z
z
f
c
b
a
L
z
f
c
b
a
L
z
z
f
c
b
a
L
z
z
f
c
b
a
L
z
f
c
b
a
L
z
1
)}
(
)
;
,
(
{
)
(
)
;
,
(
)
1
(
)}
(
)
;
,
(
{
)
1
(
' '
z
f
c
b
a
L
z
z
f
c
b
a
L
z
f
c
b
a
L
z
2 1 1 1 2 1 1 1 ) 1 ( ) ( )! 1 ( ) ( ) ( ) 1 )( 1 ( ) ( )! 1 ( ) ( ) ( ) 1 ( n n n n n n n n n n n n z a n c n b a z z a n c n b a
2 1 1 1 2 1 1 1)
1
(
)
(
)!
1
(
)
(
)
(
1
)
1
)(
1
(
)
(
)!
1
(
)
(
)
(
)
1
(
n n n n n n n n n na
n
c
n
b
a
a
n
c
n
b
a
This expression is bounded above by ( ) if
2 1 1 1).
1
(
)
(
)!
1
(
)
(
)
(
)]
}(
)
1
(
1
{
)
1
(
[
n n n n na
c
n
b
a
n
n
Conversely let (2.1) holds. Using the fact that Re (ω) > δ if and only if |ω-(1+δ)|<|ω+(1-δ)|,it is enough to show that
1
)}
(
)
;
,
(
{
)
(
)
;
,
(
)
1
(
)}
(
)
;
,
(
{
1
)}
(
)
;
,
(
{
)
(
)
;
,
(
)
1
(
)}
(
)
;
,
(
{
' ' ' '
z
f
c
b
a
L
z
z
f
c
b
a
L
z
f
c
b
a
L
z
z
f
c
b
a
L
z
z
f
c
b
a
L
z
f
c
b
a
L
z
E
Let
1
)
1
(
)
(
)!
1
(
)
(
)
(
)
(
)!
1
(
)
(
)
(
1
)
1
(
)
(
)!
1
(
)
(
)
(
)
(
)!
1
(
)
(
)
(
2 1 1 1 2 1 1 1 2 1 1 1 2 1 1 1 n n n n n n n n n n n n n n n n n n n n n n n nz
a
n
c
n
b
a
z
z
na
c
n
b
a
z
z
a
n
c
n
b
a
z
z
na
c
n
b
a
z
Putting ) ( )} ( ) ; , ( { ) ( ) ; , ( ) 1(
L a b c f z
z L a b c f z ' G z
2 2 1
1 1 1 1 1 2 1 1 1 ) 1 )( 1 ( ) ( )! 1 ( ) ( ) ( ) 1 ( ) ( )! 1 ( ) ( ) ( ) 1 ( ) 1 ( ) ( )! 1 ( ) ( ) ( ) ( 1 n n n n n n n n n n n n n n n n n n z a n c n b a z a n c n b a z z na c n b a z z G E
Thus
2 1 1 1)
(
)!
1
(
)
(
)
(
)
1
)(
1
(
)
1
)(
1
(
)
2
(
)
(
)
2
.
2
(
n n n n na
c
n
b
a
n
n
n
z
G
z
E
2 2 1
1 1 1 1 1 2 1 1 1 ) 1 )( 1 ( ) ( )! 1 ( ) ( ) ( ) 1 ( ) ( )! 1 ( ) ( ) ( ) 1 ( ) 1 ( ) ( )! 1 ( ) ( ) ( ) ( 1 n n n n n n n n n n n n n n n n n n z a n c n b a z a n c n b a z z na c n b a z z G
2 1 1 1 1 1 1 2)
(
)!
1
(
)
(
)
(
)
1
)(
1
(
)
(
)!
1
(
)
(
)
(
)}
1
)(
1
(
{
)
(
1
n n n n n n n n n n n nz
a
c
n
b
a
n
z
a
c
n
b
a
n
n
z
z
G
Thus
n n n n na
c
n
b
a
n
n
n
z
G
z
F
1 1 12
(
1
)!
(
)
)
(
)
(
)
1
)(
1
(
)
1
)(
1
(
)
(
)
3
.
2
(
Now, from (2.2), (2.3), it follows that (2.4)
n n n n na
c
n
b
a
n
n
z
G
z
F
E
1 1 12
(
1
)!
(
)
)
(
)
(
)
}(
)
1
(
1
{
)
1
(
)
1
(
)
(
2
)
4
.
2
(
Thus (2.1) proves the theorem.
The result is sharp. The extremal function being
,
2
.
)
(
)!
1
(
)
(
)
(
)
}(
)
1
(
1
{
)
1
(
)
1
(
)
(
)!
1
(
)
(
)
(
)
}(
)
1
(
1
{
)
1
(
)
(
)
5
.
2
(
1 1 1 1 1 1
n
c
n
b
a
n
n
z
z
c
n
b
a
n
n
z
f
n n n n n n n
DISTORTION THEOREMSTheorem 3.1: If f (z) T *(α,β,λ), then for |z|= r < 1
Proof: In view of inequality (2.1), it follows that
2 1
1
1
(
1
).
)
(
)!
1
(
)
(
)
(
)]
}(
)
1
(
1
{
)
1
(
[
n
n n n n
a
c
n
b
a
n
n
By the fact that
1 1 1
)
(
)
(
)
(
n n n
c
b
a
is non-decreasing for n ≥ 2.Then
2 1
1 1
2
(
1
)!
(
)
)
(
)
(
)]
}(
)
1
(
1
{
)
1
(
[
)}
)(
1
(
)
1
(
2
{
n
n n n n n
n
a
c
n
b
a
n
n
a
c
ab
).
1
(
or,
2 {2(1 ) (1 )( )} )
1 (
n n
ab c
a
Therefore
(3.3)
2)}
)(
1
(
)
1
(
2
{
)
1
(
)
(
r
ab
c
r
z
f
and
(3.4)
2)}
)(
1
(
)
1
(
2
{
)
1
(
)
(
r
ab
c
r
z
f
From (3.3) and (3.4) inequality (3.1) follows.
Further, for f (z) T *(α,β,λ),inequality (2.1) gives
2 1
1 1
).
1
(
)
(
)!
1
(
)
(
)
(
)}
)(
1
(
)
1
(
2
{
n
n n n n
a
c
n
b
a
Or,
2 1
1 1
)}
)(
1
(
)
1
(
2
{
)
1
(
)
(
)!
1
(
)
(
)
(
n
n n n n
a
c
n
b
a
Thus,
(3.5)
2)}
)(
1
(
)
1
(
2
{
)
1
(
)
(
)
;
,
(
a
b
c
f
z
r
r
L
(3.6)
2)}
)(
1
(
)
1
(
2
{
)
1
(
)
(
)
;
,
(
a
b
c
f
z
r
r
L
On using (3.5) and (3.6) inequality (3.2) follows.
Remark3.2: The bounds in (3.1) & (3.2) are sharp, since the inequalities are attained for the function.
(3.7)
,
)}
)(
1
(
)
1
(
2
{
)
1
(
)}
)(
1
(
)
1
(
2
{
)
(
2
ab
cz
abz
z
f
EXTREME POINTS
Theorem 4.1: Let
(4.1)
f
1(
z
)
z
and
n n n n
n
z
c
n
b
a
n
n
z
a
z
f
1 1 1 1
)
(
)!
1
(
)
(
)
(
)
}(
)
1
(
1
{
)
1
(
)
1
(
)
(
for n
2, then f (z) T *(α,β,λ), if and only if it can be expressed in the form(4.2)
1
) ( )
(
n n nf z
d z
f , where
d
n
0
and
1
. 1
n n
d
In particular the extreme points of T *(α,β,λ) are the functions given by (4.1).
Proof: Let f(z) be expressed in the form (4.1),then
1 2
1 1 1
)
(
)!
1
(
)
(
)
(
)
}(
)
1
(
1
{
)
1
(
)
1
(
)
(
)
(
n n
n n n n n
n
n
z
c
n
b
a
n
n
d
z
z
f
d
z
f
2
n
n n nt z
d z
Where
1 1 1
)
(
)!
1
(
)
(
)
(
)
}(
)
1
(
1
{
)
1
(
)
1
(
n n n n
c
n
b
a
n
n
t
Now, since
2 1 2
1
1
(
1
)
)
(
)!
1
(
)
(
)
(
)
}(
)
1
(
1
{
)
1
(
n
n n
n n n n
n
d
t
d
c
n
b
a
n
n
).
1
(
)
1
)(
1
(
Therefore, f (z) T *(α,β,λ).
Conversely, let f (z) T *(α,β,λ),then(2.1) yields
n n n n
n
z
c
n
b
a
n
n
a
1 1 1
)
(
)!
1
(
)
(
)
(
)
}(
)
1
(
1
{
)
1
(
)
1
(
for
n
2
.
Setting
nn
n n
n
a
c
n
b
a
n
n
d
)
1
(
)
(
)!
1
(
)
(
)
)](
}(
)
1
(
1
{
)
1
(
[
1
1 1
for
n
2
and
2 1 1
n n
d
d .
Then
2
1 1 1
)
(
)!
1
(
)
(
)
(
)
}(
)
1
(
1
{
)
1
(
)
1
(
)
(
n
n n n n n
z
d
c
n
b
a
n
n
z
z
f
2
)} ( {
n
n n z f z
d z
2 2 1
). ( )
( )
1 (
n n n
n n n
n
n d f z d f z
d z
This completes the proof.
RADIUS OF STARLIKENESS
THEOREM 5.1: Let f (z) T *(α,β,λ),then f(z) is starlike in
z
r
(
,
,
)
,where(5.1)
,
2
,
)
1
(
)
(
)!
1
(
)
(
)
)](
}(
)
1
(
1
{
)
1
(
[
inf
1 1
1
1
1
n
n
c
n
b
a
n
n
r
n n
n n
Proof: It suffices to show that
1
1
)
(
)
(
,
z
f
i.e.
1
1
)
1
(
1
)
(
)
(
2
1 1
2 ,
n
n n
n n
n
z
a
z
a
n
z
f
z
zf
(5.2) or
2
1 1
n
n n z
na
It is easily to see that (5.1) holds if
.
)
1
(
)
(
)!
1
(
)
(
)
)](
}(
)
1
(
1
{
)
1
(
[
1
1 1 1
n
c
n
b
a
n
n
z
n
n n n
This completes the proof.
RADIUS OF CONVEXITY
THEOREM 6.1: Let f (z) T *(α,β,λ),then f(z) is convex in
z
r
(
,
,
)
,where(6.1)
,
2
.
)
1
(
)
(
)!
1
(
)
(
)
)](
}(
)
1
(
1
{
)
1
(
[
inf
1 1
2 1
1
1
n
n
c
n
b
a
n
n
r
n n
n n
Proof: Upon noting the fact that f(z) is convex if and only if ( )is starlike, the Theorem(6.1) follows.
REFERANCES
H. Silverman (1975) Univalent functions with negative coefficients, Proc. Amer. Math. Soc., 51, 109-116.
B.C. Carlson and Shafeer (2002) Starlike and prestarlike hypergeometric functions, SIAM J. Math. Anal., 15, 737-745.
S.Ruscheweyh (1975) New criteria for Univalent Functions, Proc. Amer. Math., Soc., 49, 109-115.
