International Journal of Mathematical Archive-2(2), Feb. - 2011, Page: 272-279
Available online through
www.ijma.info
A NOTE ON S L GEAN CARLSON-SHAFFER OPERATOR
L. Dileep* and S. Latha**
*Department of mathematics Vidyavardhaka College of Engineering Mysore Email: [email protected]
**Department of mathematics Yuvaraja’s College University of Mysore, Mysore Email: [email protected]
(Received on: 27-01-11; 08-02-11)
---
ABSTRACT
I
n the present work, using S l gean and Carlson-Shafer operator we introduce a linear operatorSL
λ.The objective is to define the classes α
(
,
,
,
β
)
λa
c
n
VS
andVS
λα(
a
,
c
,
n
)
using the above linear operator and for functions belonging to these classes we obtain coefficient estimates and many more properties like extreme points, integral means, unified radii results etc.2000 Mathematics Subject Classification: 30 C 45
Key words and phrases: Univalent functions, S l gean operator, Carlson-Shaffer operator, Integral means.
---
1. INTRODUCTION:
Let
U
=
{
z
∈
C
:
z
<
1
}
be the open unit disk andA
denote the class of functions normalized by
∞
=
+
=
2
)
(
m
m m
z
a
z
z
f
(1.1)which are analytic in the open unit disk U satisfying the conditions
f
(
0
)
=
f
'
(
0
)
−
1
=
0
.
The class
A
is closed under the convolution or Hadamard product
∞
=
+
=
∗
2
)
)(
(
m
m m m
b
z
a
z
z
g
f
,z
∈
U
,
(1.2)where
f
is given by (1.1) and∞
=
+
=
2
.
)
(
m m m
z
b
z
z
g
For
n
∈
N
0,
λ
≥
0
,
a
,
c
∈
R
\
Z
,
we introduce a linear operatorSL
λ:
A
→
A
defined by
SL
λf
(
z
)
=
(
1
−
λ
)[(
k
∗
k
∗
∗
k
)
∗
f
](
z
)
+
λ
[
φ
(
a
,
c
)
∗
f
](
z
),
z
∈
U
,
(1.3)where
k
(
z
)
=
z
(
1
−
z
)
−2 is Koebe function and
∞
= −
−
<
≠
−
−
=
2 1
1
,
2
,
1
,
0
,
,
1
,
)
(
)
(
)
;
,
(
m
m
m m
c
a
z
z
c
a
z
c
a
φ
Page: 272-279
For functions
f
∈
A
of the form (1.1), we have
∞
=
+
=
2
,
)
,
,
,
(
)
(
m
m m
z
a
n
m
c
a
B
z
z
f
SL
λ λ (1.4)where
.
)
(
)
(
)
1
(
)
,
,
,
(
1 1
+
−
=
− −
m m n
c
a
m
n
m
c
a
B
λλ
λ
(1.5)Here
(
a
)
m is the Pochhammer symbol defined in terms of the Gamma function by
∈
−
+
+
+
=
=
Γ
+
Γ
=
.
),
1
(
)
2
)(
1
(
0
,
1
)
(
)
(
)
(
N
m
for
m
a
a
a
a
m
for
a
m
a
a
mNow using the linear operator
SL
λwe define the classSL
αλ(
a
,
c
,
n
)
consisting functions of the form (1.1) satisfying the condition
1
.
)
(
))'
(
(
)
(
))'
(
(
−
>
−
z
f
SL
z
f
SL
z
z
f
SL
z
f
SL
z
R
λ λ
λ
λ
α
(1.6)
Silverman [9] defined the class
V
(
θ
m)
as the class of all functions inA
such thatarg
a
m=
θ
mfor all.
m
If further there exists a real numbert
such thatθ
m+
(
m
−
1
)
t
=
π
(mod
2
π
)
, thenf
is said to be in the classV
(
θ
m,
t
).
The union ofV
(
θ
m,
t
)
taken overall possible sequences{
θ
m}
and all possible real numberst
is denoted byV
.
Further, we define
VS
(
a
,
c
,
n
,
β
)
=
S
α(
a
,
c
,
n
,
β
)
∩
V
.
λα λ
Definition1.1: A function
f
∈
V
of the form (1.1) is in α(
,
,
,
β
)
λa
c
n
VS
iff
satisfies the analytic condition
1
,
)
(
))'
(
(
)
(
))'
(
(
α
β
λ λ
λ
λ
>
−
+
z
f
SL
z
f
SL
z
z
f
SL
z
f
SL
z
R
(1.7)where
α
,
β
≥
0
andz
∈
U
.
These classes stem essentially from the classes studied earlier by Vijaya and Murugusundaramoorthy [10].
2. MAIN RESULTS
Theorem 2.1: A function
f
of the form (1.1) is inVS
λα(
a
,
c
,
n
)
if and only if
(
2
1
)
(
,
,
,
)
1
.
2
α
α
λ≤
−
−
−
∞
=
m
m
a
n
m
c
a
B
m
(2.1)Proof: From (1.6), it suffices to show that
.
)
(
))'
(
(
1
)
(
))'
(
(
−
≤
−
α
λ λ
λ λ
z
f
SL
z
f
SL
z
R
z
f
SL
z
f
SL
z
Page: 272-279
.
|
|
|
|
)
,
,
,
(
1
|
||
|
)
,
,
,
(
)
1
(
2
1
)
(
))'
(
(
2
1
)
(
))'
(
(
1
)
(
))'
(
(
2 1 2 1 ∞ = − ∞ = −−
−
≤
−
≤
−
−
−
m m m m m mz
a
n
m
c
a
B
z
a
n
m
c
a
B
m
z
f
SL
z
f
SL
z
z
f
SL
z
f
SL
z
R
z
f
SL
z
f
SL
z
λ λ λ λ λ λ λ λNow the last expression is bounded by
(
1
−
α
)
if
(
2
1
)
(
,
,
,
)
1
.
2
α
α
λ≤
−
−
−
∞ = m ma
n
m
c
a
B
m
Conversely, if
f
∈
VS
λα(
a
,
c
,
n
)
then by definition
.
)
,
,
,
(
)
,
,
,
(
1
)
,
,
,
(
)
,
,
,
(
2 2 2 2−
+
+
≤
−
+
+
∞ = ∞ = ∞ = ∞ =α
λ λ λ λ m m m m m m m m m m m mz
a
n
m
c
a
B
z
z
a
n
m
c
a
mB
z
R
z
a
n
m
c
a
B
z
z
a
n
m
c
a
mB
z
That is+
−
+
−
≤
+
−
∞ = − ∞ = − − ∞ = ∞ = − 2 1 2 1 1 2 2 1)
,
,
,
(
1
)
,
,
,
(
)
(
)
1
(
)
,
,
,
(
1
)
,
,
,
(
)
1
(
m m m m m m m m m m m mz
a
n
m
c
a
B
z
a
n
m
c
a
B
m
R
z
a
n
m
c
a
B
z
a
n
m
c
a
B
m
λ λ λλ
α
α
Since
f
∈
V
andf
lies inV
(
θ
m,
t
)
for some sequenceθ
m and a real numbert
such that)
2
(mod
)
1
(
π
π
θ
m+
m
−
t
≡
setz
=
re
itin the above inequality
.
)
,
,
,
(
1
)
,
,
,
(
)
(
)
1
(
)
,
,
,
(
1
)
,
,
,
(
)
1
(
2 1 2 1 1 2 2 1+
−
+
−
≤
+
−
∞ = − ∞ = − − ∞ = ∞ = − m m m m m m m m m m m mr
a
n
m
c
a
B
r
a
n
m
c
a
B
m
R
r
a
n
m
c
a
B
r
a
n
m
c
a
B
m
λ λ λλ
α
α
Letting
r
→
1
,
leads the desired inequality
(
2
1
)
(
,
,
,
)
1
.
2
α
α
λ≤
−
−
−
∞ = m ma
n
m
c
a
B
m
Corollary.2.2: If
f
∈
VS
λα(
a
,
c
,
n
)
then
,
2
.
)
,
,
,
(
)
1
2
(
1
|
|
≥
−
−
−
≤
for
m
n
m
c
a
B
m
a
m λα
α
Page: 272-279
∞
=
∈
≥
−
−
−
+
=
2
.
,
2
,
)
,
,
,
(
)
1
2
(
1
)
(
m
U
z
m
for
n
m
c
a
B
m
z
z
f
λ
α
α
Similar to the proof of Theorem 2.1 we get the following result:
Theorem.2.3: A function
f
of the form (1.1) is inVS
λα(
a
,
c
,
n
,
β
)
if and only if
(
,
,
,
)
1
,
2
α
λ
≤
−
∞
=
m
m
m
B
a
c
m
n
a
E
(2.2)where
E
m=
m
(
β
+
1
)
−
(
α
+
β
).
The result obtained in our next Theorem unifies the radii results concerning close-to-convexity, starlikeness etc.
Theorem.2.4: Let
f
∈
VS
λα(
a
,
c
,
n
,
β
).
Then−
1
<
1
−
δ
,
Ψ
∗
Φ
∗
f
f
in
|
z
|
<
r
with∞
=
+
=
Φ
2
,
)
(
m
m m
z
z
z
γ
and∞
=
+
=
Ψ
2
,
)
(
m
m m
z
z
z
µ
are analytic inU
with the conditionsγ
m,
µ
m≥
0
,
γ
m≥
µ
m,
form
≥
2
and,
0
≠
Ψ
∗
f
where
,
2
.
)]
1
(
)
)[(
1
(
)
1
)(
,
,
,
(
inf
1 1
≥
−
+
−
−
−
=
−
m
n
m
c
a
B
E
r
m
m m m m
m
α
λ
µ
µ
δ
δ
λ (2.3)
Proof: Consider,
1
1
2 2
−
−
−
=
−
Ψ
∗
Φ
∗
∞
= ∞
=
m
m m m m
m m m
z
a
z
z
a
z
f
f
µ
γ
∞
= ∞
=
∞
=
−
+
−
−
≤
2
2 2
m
m m m
m m
m m m m
m m
z
a
z
z
a
z
z
a
z
µ
µ
γ
1
.
|
|
1
|
|
]
[
2
1 1
2
δ
µ
µ
γ
−
<
−
−
≤
∞=
− − ∞
=
m
m m m
m
m
m m m
z
a
z
a
[(
)
(
1
)
]
1
,
(|
|
,
0
1
),
2
<
≤
<
−
≤
−
+
−
∞
=
δ
δ
µ
δ
µ
γ
z
r
a
mm m
m
m (2.4)
where
r
is given by (2.3). From Theorem 2.3, (2.4) will be true if,,
]
)
1
(
)
)[(
1
(
)
1
)(
,
,
,
(
|
|
1
]
)
1
(
)
[(
1m m
m m m
m m
m
E
B
a
c
m
n
z
µ
δ
µ
γ
α
δ
δ
µ
δ
µ
γ
λ−
+
−
−
−
≤
−
−
+
Page: 272-279
that is, if
.
]
)
1
(
)
)[(
1
(
)
1
)(
,
,
,
(
|
|
1 1
−
−
+
−
−
−
=
m
m m
m
m
B
a
c
m
n
E
z
µ
δ
µ
γ
α
δ
λ (2.5)
As corollaries to the above Theorem we get the following result:
By choosing
2
)
1
(
)
(
z
z
z
−
=
Φ
andΨ
(
z
)
=
z
, we haveCorollary 2.5: Let the function
f
defined by (1.1) belongs to α(
,
,
,
β
).
λa
c
n
VS
Thenf
is close-to-convex of orderδ
(
0
≤
δ
<
1
),
hence univalent in the disc|
z
|
<
r
1,
where
,
2
.
)
1
(
)
1
)(
,
,
,
(
inf
1 1
1
≥
−
−
=
−
m
m
n
m
c
a
B
E
r
m m
m
α
δ
λ
(2.6)
The result is sharp.
For 2
)
1
(
)
(
z
z
z
−
=
Φ
and,
1
)
(
z
z
z
−
=
Ψ
we haveCorollary 2.6: Let the function
f
defined by (1.1) belongs toVS
λα(
a
,
c
,
n
,
β
).
Thenf
is starlike of orderδ
(
0
≤
δ
<
1
),
hence univalent in the disc|
z
|
<
r
2,
where
,
2
.
)
)(
1
(
)
1
)(
,
,
,
(
inf
11
2
≥
−
−
−
=
−m
m
n
m
c
a
B
E
r
m m
m
α
δ
δ
λ
(2.7)
The result is sharp.
If 3
2
)
1
(
)
(
z
z
z
z
−
+
=
Φ
and 2)
1
(
)
(
z
z
z
−
=
Ψ
, we haveCorollary 2.7: Let the function
f
defined by (1.1) belongs toVS
λα(
a
,
c
,
n
,
β
).
Thenf
is convex of orderδ
(
0
≤
δ
<
1
),
hence univalent in the disc|
z
|
<
r
3,
where
,
2
.
)
)(
1
(
)
1
)(
,
,
,
(
inf
11
\
3
≥
−
−
−
=
−m
m
m
n
m
c
a
B
E
r
m m
m
α
δ
δ
λ (2.8)
The result is sharp.
Using the coefficient inequality proved above we can easily prove the following growth and distortion theorem.
Theorem 2.8: Let
f
of the form (1.1) to be in α(
,
,
,
β
).
λa
c
n
VS
then2
2 2
2
(
,
,
2
,
)
1
|
)
(
|
)
,
2
,
,
(
1
r
n
c
a
B
E
r
z
f
r
n
c
a
B
E
r
λ λ
α
α
−
+
≤
≤
−
−
and
r
n
c
a
B
E
z
f
r
n
c
a
B
E
(
,
,
2
,
)
)
1
(
2
1
|
)
(
'
|
)
,
2
,
,
(
)
1
(
2
1
2
2 λ λ
α
α
−
+
≤
≤
−
Page: 272-279
The result is sharp.
Proof: Let
f
of the form (1.1) belongs to α(
,
,
,
β
).
λa
c
n
VS
|
(
)
|
|
|
|
|
|
,|
2 22
∞
= ∞
=
+
≤
+
=
m m m
m
m
z
z
z
a
a
z
z
f
since α
(
,
,
,
β
)
λa
c
n
VS
f
∈
and by Theorem 2.3, we have
(
,
,
2
,
)
|
|
(
,
,
,
)
|
|
1
.
2 2
2 λ
≤
λ≤
−
α
∞
= ∞
= m
m m
m
m
E
B
a
c
m
n
a
a
n
c
a
B
E
Thus
|
|
.
)
,
2
,
,
(
1
|
|
|
)
(
|
22
z
n
c
a
B
E
z
z
f
λ
α
−
+
≤
That is
,
)
,
2
,
,
(
1
|
)
(
|
22
r
n
c
a
B
E
r
z
f
λ
α
−
+
≤
similarly, we get
.
)
,
2
,
,
(
1
|
)
(
|
22
r
n
c
a
B
E
r
z
f
λ
α
−
−
≤
On the other hand
∞
=
−
+
=
2
1
,
1
)
(
'
m
m m
z
ma
z
f
and
∞
= −
∞
=
+
≤
+
=
2 1
2
|,
|
|
|
1
|
|
|
|
1
|
)
(
'
|
m
m m
m
m
z
z
m
a
a
m
z
f
since α
(
,
,
,
β
)
λa
c
n
VS
f
∈
.Then by Theorem 2.3 w have
.
)
,
2
,
,
(
)
1
(
2
|
|
2
2
E
B
a
c
n
a
m
m
m
λ
α
−
≤
∞
=
Thus
.
)
,
2
,
,
(
)
1
(
2
1
|
)
(
'
|
2
r
n
c
a
B
E
z
f
λ
α
−
+
≤
Similarly we get
.
)
,
2
,
,
(
)
1
(
2
1
|
)
(
'
|
2
r
n
c
a
B
E
z
f
λ
α
−
−
≥
This completes the proof.
Theorem 2.9: A function
f
of the form (1.1) belongs to α(
,
,
,
β
)
λa
c
n
VS
, witharg
a
m=
θ
mwhere).
2
(mod
]
)
1
(
[
θ
m+
m
−
t
=
π
π
Define andf
1(
z
)
=
z
and
,
2
,
.
)
,
,
,
(
1
)
(
e
z
m
z
U
n
m
c
a
B
E
z
z
f
i mm m
m
≥
∈
−
+
=
θλ
α
Then α
(
,
,
,
β
)
λa
c
n
VS
f
∈
if and only iff
expressed in the form∞
=
=
2)
(
)
(
mm
m
f
z
z
f
µ
where0
≥
m
µ
and∞
=
=
2.
1
mm
Page: 272-279 Proof: If
∞
=
=
2)
(
)
(
mm
m
f
z
z
f
µ
with∞
=
=
21
mm
µ
andµ
m≥
0
then
∞
= ∞
=
−
≥
−
−
=
−
=
−
2
1 2
.
1
)
1
)(
1
(
)
1
(
)
,
,
,
(
1
)
,
,
,
(
m m
m m
m m
n
m
c
a
B
E
n
m
c
a
B
E
α
α
µ
α
µ
µ
α
λ λ
Hence
f
∈
VS
λα(
a
,
c
,
n
,
β
)
.Conversely, let the function
f
defined by (1.1) be in the class α(
,
,
,
β
)
λa
c
n
VS
, since
,
2
,
3
,
.
)
,
,
,
(
1
|
|
≤
−
m
=
n
m
c
a
B
E
a
m m
λ
α
We may set
,
2
1
|
|
)
,
,
,
(
≥
−
=
E
mB
a
c
m
n
a
mm
m
α
µ
λ and∞
=
−
=
2
1
1
.
m m
µ
µ
Then
(
)
(
)
,
1
∞
=
=
m
m
m
f
z
z
f
µ
this completes the proof.Lemma 2.10: [3] If for the functions
f
andg
are analytic inU
withg
f
,
then fork
>
0
and0
<
r
<
1
|
(
)
|
2|
(
)
|
.
0 2
0
θ
θ
π θ
π θ
d
re
f
d
re
g
k i k
i
≤
In [6] Silverman found that the function
2
)
(
2 2
z
z
z
f
=
−
is often extremal over the familyT
.
He applied this function toresolve the integral means inequality, conjectured in [7] and [8], such that
|
(
)
|
2|
(
)
|
,
0 2
2
0
θ
θ
π θ
π θ
d
re
f
d
re
f
k i k
i
≤
for allf
∈
V
,
k
>
0
and
0
<
r
<
1
.
In [8] he also proved his conjecture for the subclasses ∗
(β
)
T
andC
(
β
)
of
T
.
Theorem 2.11: Let
f
of the form (1.1) belongs toVS
λα(
a
,
c
,
n
,
β
)
andf
2 is defined by2
2 2
)
,
2
,
,
(
1
)
(
z
n
c
a
B
E
z
z
f
λ
α
−
−
=
then forz
=
re
iθ,
0
<
r
<
1
,
we have
|
(
)
|
2|
(
)
|
.
0 2
2
0
θ
θ
π π
d
z
f
d
z
f
k k
≤
Proof: For
∞
=
−
=
2
,
|
|
)
(
m
m
m
z
a
z
z
f
(2.9) is equivalent to prove that
.
)
,
2
,
,
(
1
1
|
|
1
2
0 2
2
0
1
2
θ
α
θ
π
λ π
d
z
n
c
a
B
E
d
z
a
k k
m
m m
−
−
≤
−
−∞
=
Page: 272-279
z
n
c
a
B
E
z
a
mm m
)
,
2
,
,
(
1
1
|
|
1
2 1
2 λ
α
−
−
−
−∞
=
setting
(
)
)
,
2
,
,
(
1
1
|
|
1
2 1
2
z
n
c
a
B
E
z
a
mm
m
ω
α
λ
−
−
=
−
−∞
=
And using (2.2) we obtain
.
|
|
|
|
1
)
,
,
,
(
|
|
|
|
1
)
,
,
,
(
)
(
2
1
2
z
a
n
m
c
a
B
E
z
z
a
n
m
c
a
B
E
z
m m
m
m
m
m m
≤
−
≤
−
=
∞
=
− ∞
=
α
α
ω
λ λ
This completes the proof.
In Theorem 2.4, 2.8, 2.9 and 2.11 if we substitute
β
=
1
we get the result for the classVS
λα(
a
,
c
,
n
).
References:
[1] H S Al-Amiri, On Ruscheweyh derivatives, Ann. Polon. Math. 38 (1980). 87-94
[2] B C Carlson-Shaffer, Starlike and prestarlike hypergeometric functions, SIAM J. on Math. Analysis 15 (1984), 737-745. [3] J E Littlewood, On inequalities in theory of functions, Proc. London Math. Soc., 23 (1925), 481-519.
[4] St. Ruscheweyh, New criteria for univalent functions, Proc. Ann. Math. Soc., Volume 6, 1975, 109-115.
[5] G S S l gean, Subclasses of univalent functions, Lectures Notes in Math. Springer- Verlag, 2013 (1983), 362-372. [6] H Silverman, Univalent functions with negative coefficients, Proc. Amer. Math. Soc., 51 (1975), 109-116.
[7] H Silverman, A survey with open problem on univalent functions whose coefficients are negative, Rocky Mountain J. Math., 21 (3) (1991), 1099-1125.
[8] H Silverman, integral means for univalent functions with negative coefficients, Houston J. Math., 23 (1), (1997), 169-174.
[9] H Silverman, Univalent functions with varying arguments, Houston J. Math., Vol. 7, No. 2 (1981).
[10] K Vijaya and G Murugusundaramoorthy, Some uniformly starlike functions with varying arguments, Tamkang Journal of Mathematics, Vol. 35, No.1, ASpring 2004.
