General Mathematics Vol. 15, No. 4 (2007), 131-143
Carlson - Shaffer operator and their
applications to certain subclass of uniformly
convex functions1
G.Murugusundaramoorthy, T. Rosy and
K.Muthunagai
Abstract
Making use of Carlson - Shaffer operator,we define a new subclass of uniformly convex functions with negative coefficients and
obtain the coefficient bounds, extreme points and radius of starlikeness for functions belonging to the generalized class T S(λ, α, β). Furthermore, partial sums fk (z) of functions f (z) in the class S(λ, α, β)
are considered and sharp lower bounds for the ratios of real part of
f (z) to fk (z) and f ′ (z) to fk′ (z) are determined.
2000 Mathematical Subject Classification: 30C45.
Key words: Univalent, convex, starlike, uniformly convex.
1
Introduction
Let A denote the class of functions of the form
(1.1)
f (z) = z +
∞
X
an z n
n=2
1
Received 16 August, 2007
Accepted for publication (in revised form) 22 December, 2007
131
132
G.Murugusundaramoorthy, T. Rosy and K.Muthunagai
which are analytic and univalent in the open disc U = {z : z ∈ C |z| < 1}.
Also denote by T the subclass of A consisting of functions of the form
(1.2)
f (z) = z −
∞
X
an z n , (an ≥ 0)
n=2
Following Gooodman [3, 4], Rønning [5, 6] introduced and studied the following subclasses
(i) A function f ∈ A is said to be in the class Sp (α, β) of uniformly
β−starlike functions if it satisfies the condition
¯ ′
¯
¾
½ ′
¯ zf (z)
¯
zf (z)
− α > β ¯¯
− 1¯¯ , z ∈ U,
(1.3)
Re
f (z)
f (z)
−1 < α ≤ 1 and β ≥ 0.
(ii) A function f ∈ A is said to be in the class U CV (α, β) of uniformly
β−convex functions if it satisfies the condition
¯ ′′ ¯
½
¾
¯ zf (z) ¯
zf ′′ (z)
¯ , z ∈ U,
(1.4)
Re 1 + ′
− α > β ¯¯ ′
f (z)
f (z) ¯
and −1 < α ≤ 1 and β ≥ 0.
Indeed it follows from (1.3) and (1.4) that
(1.5)
f ∈ U CV (α, β) is equivalent with zf ′ ∈ Sp (α, β).
For functions f ∈ A given by (1.1) and g(z) ∈ A given by g(z) = z+
∞
P
n=2
we define the Hadamard product (or Convolution ) of f and g by
(1.6)
(f ∗ g)(z) = z +
∞
X
an bn z n , z ∈ U.
n=2
Let φ(a, c; z) be the incomplete beta function defined by
(1.7)
φ(a, c; z) = z +
∞
X
(a)n−1
n=2
(c)n−1
z n , c 6= 0, −1, −2, . . .
bn z n ,
Carlson - Shaffer operator...
133
where (x)n is the Pochhammer symbol defined interms of the Gamma functions, by
(
1
n=0
Γ(x + n)
=
(1.8) (x)n =
Γ(x)
x(x + 1)(x + 2) . . . (x + n − 1), n ∈ N
Further, for f ∈ A
(1.9)
L(a, c)f (z) = φ(a, c; z) ∗ f (z) = z +
∞
X
(a)n−1
n=2
(c)n−1
an z n ,
where L(a, c) is called Carlson - Shaffer operator [2] and the operator ∗
stands for the hadamard product (or convolution product) of two power
series as given by (1.6).
We notice that
L(a, a)f (z) = f (z), L(2, 1)f (z) = zf ′ (z).
For −1 ≤ α < 1 ,0 ≤ λ ≤ 1 and β ≥ 0, we let S(λ, α, β) be the subclass
of A consisting of functions of the form (1.1) and satisfying the analytic
criterion
½
¾
z(L(a, c)f (z))′ + λz 2 (L(a, c)f (z))′′
Re
−α
(1 − λ)L(a, c)f (z) + λ(zL(a, c)f (z))′
¯
¯
¯ z(L(a, c)f (z))′ + λz 2 (L(a, c)f (z))′′
¯
¯, z ∈ U
> β ¯¯
−
1
(1.10)
¯
(1 − λ)L(a, c)f (z) + λ(zL(a, c)f (z))′
where L(a, c)f (z) is given by (1.9). We also let T S(λ, α, β) = S(λ, α, β)∩T.
By suitably specializing the values of λ, (a) and (c), the class S(λ, α, β)
can be reduces to the class studied earlier by Rønning [5, 6]. Also choosing
α = 0 and β = 1 the class coincides with the classes studied in [10] and [11]
respectively.
The main object of this paper is to study the coefficient bounds, extreme
points and radius of starlikeness for functions belonging to the generalized
class T S(λ, α, β). Furthermore, partial sums fk (z) of functions f (z) in the
class S(λ, α, β) are considered and sharp lower bounds for the ratios of real
part of f (z) to fk (z) and f ′ (z) to fk′ (z) are determined.
134
2
G.Murugusundaramoorthy, T. Rosy and K.Muthunagai
Basic Properties
In this section we obtain a necessary and sufficient condition for functions
f (z) in the classes S(λ, α, β) and T S(λ, α, β).
Theorem 2.1. A function f (z) of the form (1.1) is in S(λ, α, β) if
(2.1)
∞
X
(a)n−1
|an | ≤ 1 − α,
(1 + λ(n − 1))[n(1 + β) − (α + β)]
(c)
n−1
n=2
−1 ≤ α < 1, 0 ≤ λ ≤ 1, β ≥ 0.
Proof. It sufficies to show that
¯
¯
¯ z(L(a, c)f (z))′ + λz 2 (L(a, c)f (z))′′
¯
β ¯¯
− 1¯¯
′
(1 − λ)L(a, c)f (z) + λ(zL(a, c)f (z))
¾
½
z(L(a, c)f (z))′ + λz 2 (L(a, c)f (z))′′
−1
−Re
(1 − λ)L(a, c)f (z) + λ(zL(a, c)f (z))′
≤1−α
We have
¯
¯
¯
¯ z(L(a, c)f (z))′ + λz 2 (L(a, c)f (z))′′
¯
−
1
β ¯¯
¯
(1 − λ)L(a, c)f (z) + λ(zL(a, c)f (z))′
−Re
½
¾
z(L(a, c)f (z))′ + λz 2 (L(a, c)f (z))′′
−1
(1 − λ)L(a, c)f (z) + λ(zL(a, c)f (z))′
¯
¯
¯ z(L(a, c)f (z))′ + λz 2 (L(a, c)f (z))′′
¯
¯
¯
≤ (1 + β) ¯
−
1
¯
(1 − λ)L(a, c)f (z) + λ(zL(a, c)f (z))′
(1 + β)
≤
∞
P
n−1
|a |
(n − 1)[1 + λ(n − 1)] (a)
(c)n−1 n
n=2
∞
P
1−
n=2
.
n−1
|a |
[1 + λ(n − 1)] (a)
(c)n−1 n
Carlson - Shaffer operator...
135
This last expression is bounded above by (1 − α) if
∞
X
[1 + λ(n − 1)][n(1 + β) − (α + β)]
n=2
(a)n−1
|an | ≤ 1 − α,
(c)n−1
and hence the proof is complete.
Theorem 2.2. A necessary and sufficient condition for f (z) of the form
(1.2) to be in the class T S(λ, α, β), −1 ≤ α < 1, 0 ≤ λ ≤ 1, β ≥ 0 is that
(2.2)
∞
X
(1 + λ(n − 1))[n(1 + β) − (α + β)]
n=2
(a)n−1
an ≤ 1 − α,
(c)n−1
Proof. In view of Theorem 1, we need only to prove the necessity. If
f ∈ T S(λ, α, β) and z is real then
1−
∞
P
n−1
n[1 + λ(n − 1)] (a)
a z n−1
(c)n−1 n
n=2
∞
P
1−
−α≥
[1 + λ(n −
n=2
n−1
1)] (a)
a z n−1
(c)n−1 n
¯
¯ ∞
¯
¯P
(a)n−1
¯ (n − 1)[1 + λ(n − 1)] (c) |an | ¯
n−1
¯
¯ n=2
¯
≥ β ¯¯
∞
¯
P
(a)
n−1
¯
¯ 1−
|a
|
[1
+
λ(n
−
1)]
n
¯
¯
(c)n−1
n=2
Letting z → 1 along the real axis, we obtain the desired inequality
∞
X
(1 + λ(n − 1))[n(1 + β) − (α + β)]
n=2
(a)n−1
an ≤ 1 − α.
(c)n−1
Theorem 2.3. Let f (z) defined by (1.2) and g(z) defined by g(z) = z −
∞
P
bn z n be in the class T S(λ, α, β). Then the function h(z) defined by
n=2
h(z) = (1 − µ)f (z) + µg(z) = z −
∞
X
qn z n ,
n=2
where qn = (1 − µ)an + µbn ,
0 ≤ µ < 1 is also in the class T S(λ, α, β).
136
G.Murugusundaramoorthy, T. Rosy and K.Muthunagai
Proof. Let the function
(2.3)
fj (z) = z −
∞
X
an, j z n ,
an, j ≥ 0,
j = 1, 2 ,
n=2
be in the class T S(λ, α, β). It is sufficient to show that the function g(z)
defined by
g(z) = µf1 (z) + (1 − µ)f2 (z),
0 ≤ µ ≤ 1,
is in the class T S(λ, α, β). Since
∞
X
[µan,1 + (1 − µ)an,2 ]z n ,
g(z) = z −
n=2
an easy computation with the aid of Theorem 2.2 gives,
∞
X
[1 + λ(n − 1)][n(β + 1) − (α + β)]
n=2
+
∞
X
(a)n−1
µan,1
(c)n−1
[1 + λ(n − 1)][n(β + 1) − (α + β)]
n=2
(a)n−1
(1 − µ)an,2
(c)n−1
≤ µ(1 − α) + (1 − µ)(1 − α)
≤ 1 − α,
which implies that g ∈ T S(λ, α, β). Hence T S(λ, α, β) is convex.
Theorem 2.4. (Extreme points) Let f1 (z) = z and
(2.4)
(1 − α)(c)n−1
fn (z) = z −
z n for n = 2, 3, 4, . . . .
(1 + nλ − λ)[n(1 + β) − (α + β)](a)n−1
Then f (z) ∈ T S(λ, α, β) if and only if f (z) can be expressed in the form
∞
∞
P
P
µn = 1.
µn fn (z), where µn ≥ 0 and
f (z) =
n=1
n=1
The proof of Theorem 2.4, follows on lines similar to the proof of the
theorem on extreme points given in Silverman [8].
Next we prove the following closure theorem.
Carlson - Shaffer operator...
137
Theorem 2.5. (Closure theorem ) Let the functions fj (z) (j = 1, 2, . . . m)
defined by (2.3) be in the classes T S(λ, αj , β) (j = 1, 2, . . . m) respectively.
Then the function h(z) defined by
à m
!
∞
1 X X
an,j z n
h(z) = z −
m n=2 j=1
is in the class T S(λ, α, β), where α = min {αj } where −1 ≤ αj < 1.
1≤j≤m
Proof. Since fj (z) ∈ T S(λ, αj , β) (j = 1, 2, 3, . . . m) by applying Theorem
2.2, to (2.3) we observe that
Ã
!
∞
m
X
(a)n−1 1 X
(1 + λ(n − 1))[n(1 + β) − (α + β)]
an,j
(c)
m
n−1
n=2
j=1
!
̰
m
1 X X
(a)n−1
=
(1 + λ(n − 1))[n(1 + β) − (α + β)]
an,j
m j=1 n=2
(c)n−1
m
1 X
(1 − αj ) ≤ 1 − α
≤
m j=1
which in view of Theorem 2.2, again implies that h(z) ∈ T S(λ, α, β) and so
the proof is complete.
Theorem 2.6. Let f ∈ T S(λ, α, β). Then
1. f isnstarlike
o of order δ(0 ≤ δ < 1) in the disc |z| < r1 ; that is,
zf ′ (z)
Re
> δ, (|z| < r1 ; 0 ≤ δ < 1), where
f (z)
r1 = inf
n≤2
½
(a)n−1
(c)n−1
µ
1−δ
n−δ
¶
(1 + nλ − λ)[n(1 + β) − (α + β)]
1−α
1
¾ n−1
.
2. f isnconvex ofoorder δ (0 ≤ δ < 1) in the disc |z| < r2 , that is
′′ (z)
Re 1 + zff ′ (z)
> δ, (|z| < r2 ; 0 ≤ δ < 1), where
r2 = inf
n≤2
½
(a)n−1 (1 − δ)(1 + nλ − λ)[n(1 + β) − (α + β)]
(c)n−1
n(n − δ)
1
¾ n−1
.
138
G.Murugusundaramoorthy, T. Rosy and K.Muthunagai
Each of these results are sharp for the extremal function f (z) given by (2.4).
Proof. Given f ∈ A, and f is starlike of order δ, we have
¯
¯ ′
¯
¯ zf (z)
¯ < 1 − δ.
¯
−
1
(2.5)
¯
¯ f (z)
For the left hand side of (2.5) we have
¯
¯ ′
¯ zf (z)
¯
¯
¯≤
−
1
¯ f (z)
¯
∞
P
(n − 1)an |z|n−1
n=2
1−
n=2
1−δ
.
an
|z|n−1
n=2
The last expression is less than 1 − δ if
∞
X
n−δ
∞
P
an |z|n−1 < 1.
Using the fact, that f ∈ T S(λ, α, β) if and only if
∞
X
(1 + λ(n − 1))[n(1 + β) − (α + β)] (a)n−1
1−α
n=2
(c)n−1
an < 1.
We can say (2.5) is true if
n − δ n−1 (1 + λ(n − 1))[n(1 + β) − (α + β)] (a)n−1
.
|z|
<
1−δ
1−α
(c)n−1
Or, equivalently,
|z|n−1 <
(1 − δ)(1 + λ(n − 1))[n(1 + β) − (α + β)] (a)n−1
(n − δ)(1 − α)
(c)n−1
which yields the starlikeness of the family.
(ii)
Using the fact that f is convex if and only if zf ′ is starlike, we can
prove (ii), on lines similar to the proof of (i).
Carlson - Shaffer operator...
3
139
Partial Sums
Following the earlier works by Silverman [8] and Silvia [9] on partial sums
of analytic functions. We consider in this section partial sums of functions
in the class T S(λ, α, β) and obtain sharp lower bounds for the ratios of real
part of f (z) to fk (z) and f ′ (z) to fk′ (z).
Theorem 3.1. Let f (z) ∈ T S(λ, α, β) be given by (1.1) and define the
partial sums f1 (z) and fk (z), by
(3.1)
f1 (z) = z; and fk (z) = z +
k
X
an z n , (k ∈ N/1)
n=2
Suppose also that
∞
X
dn |an | ≤ 1,
n=2
where
(3.2)
dn :=
(1 + λ(n − 1))[n(α + β) − (α + β)] (a)n−1
.
(1 − α)
(c)n−1
Then f ∈ T S(λ, α, β). Furthermore,
¾
½
1
f (z)
z ∈ U, k ∈ N
>1−
(3.3)
Re
fk (z)
dk+1
and
(3.4)
Re
½
fk (z)
f (z)
¾
>
dk+1
.
1 + dk+1
Proof. For the coefficients dn given by (3.2) it is not difficult to verify that
(3.5)
dn+1 > dn > 1.
Therefore we have
(3.6)
k
X
n=2
|an | + dk+1
∞
X
n=k+1
|an | ≤
∞
X
n=2
dn |an | ≤ 1
140
G.Murugusundaramoorthy, T. Rosy and K.Muthunagai
by using the hypothesis (3.2). By setting
½
µ
¶¾
f (z)
1
g1 (z) = dk+1
− 1−
fk (z)
dk+1
∞
P
dk+1
an z n−1
n=k+1
= 1+
(3.7)
k
P
1+
an z n−1
n=2
and applying (3.6), we find that
(3.8)
∞
P
dk+1
¯
¯
¯ g1 (z) − 1 ¯
¯
¯
¯ g1 (z) + 1 ¯ ≤
|an |
n=k+1
2−2
n
P
∞
P
|an | − dk+1
n=2
|an |
n=k+1
≤ 1, z ∈ U,
which readily yields the assertion (3.3) of Theorem 3.1. In order to see that
(3.9)
f (z) = z +
z k+1
dk+1
gives sharp result, we observe that for z = reiπ/k that
1
1 − dk+1
as z → 1− . Similarly, if we take
½
(3.10)
fk (z)
dk+1
g2 (z) = (1 + dk+1 )
−
f (z)
1 + dk+1
∞
P
(1 + dn+1 )
an z n−1
n=k+1
= 1−
∞
P
1+
an z n−1
f (z)
fk (z)
¾
n=2
and making use of (3.6), we can deduce that
(3.11)
¯
¯
¯ g2 (z) − 1 ¯
¯
¯
¯ g2 (z) + 1 ¯ ≤
(1 + dk+1 )
∞
P
|an |
n=k+1
2−2
k
P
n=2
|an | − (1 − dk+1 )
∞
P
n=k+1
|an |
= 1+
zk
dk+1
→
Carlson - Shaffer operator...
141
which leads us immediately to the assertion (3.4) of Theorem 3.1.
The bound in (3.4) is sharp for each k ∈ N with the extremal function
f (z) given by (3.9). The proof of the Theorem 3.1, is thus complete.
Theorem 3.2. If f (z) of the form (1.1) satisfies the condition (2.1). Then
½ ′ ¾
f (z)
k+1
(3.12)
Re
.
≥1−
′
fk (z)
dk+1
Proof. By setting
µ
¶¾
k+1
f ′ (z)
− 1−
g(z) = dk+1
fk′ (z)
dk+1
∞
∞
P
P
k+1
1 + dk+1
nan z n−1
nan z n−1 +
n=2
n=k+1
=
k
P
nan z n−1
1+
½
n=2
dk+1
k+1
= 1+
∞
P
nan z n−1
n=k+1
1+
k
P
.
nan
z n−1
n=2
(3.13)
Now
¯
¯
¯ g(z) − 1 ¯
¯
¯
¯ g(z) + 1 ¯ ≤
if
k
X
dk+1
k+1
2−2
k
P
∞
P
n|an |
n=k+1
n|an | −
n=2
dk+1
k+1
∞
P
.
n|an |
n=k+1
¯
¯
¯ g(z) − 1 ¯
¯
¯
¯ g(z) + 1 ¯ ≤ 1
∞
dk+1 X
n|an | +
n|an | ≤ 1
k + 1 n=k+1
n=2
(3.14)
since the left hand side of (3.14) is bounded above by
k
P
dn |an | if
n=2
(3.15)
k
X
n=2
(dn − n)|an | +
∞
X
n=k+1
dn −
dk+1
n|an | ≥ 0,
k+1
142
G.Murugusundaramoorthy, T. Rosy and K.Muthunagai
and the proof is complete.
The result is sharp for the extremal function f (z) = z +
z k+1
.
ck+1
Theorem 3.3. If f (z) of the form (1.1) satisfies the condition (2.1) then
½ ′ ¾
f (z)
dk+1
(3.16)
Re k′
.
≥
f (z)
k + 1 + dk+1
Proof. By setting
dk+1
f ′ (z)
−
g(z) = [(k + 1) + dk+1 ] k′
f (z) k + 1 + dk+1
³
´ P
∞
k+1
1 + dk+1
nan z n−1
n=k+1
= 1−
k
P
1+
nan z n−1
½
¾
n=2
and making use of (3.15), we deduce that
¯
¯
¯ g(z) − 1 ¯
¯
¯
¯ g(z) + 1 ¯ ≤
2−2
k
P
³
n=2
1+
dk+1
k+1
´ P
∞
n|an |
n=k+1
³
n|an | − 1 +
dk+1
k+1
´ P
∞
≤ 1,
n|an |
n=k+1
which leads us immediately to the assertion of Theorem 3.3.
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G.Murugusundaramoorthy,
School of Science and Humanities,
VIT University, Vellore - 632014,
India.
E-mail:gmsmoorthy@yahoo.com
K.Muthunagai
Department of Mathematics,
Hindustan College of Engineering,
Chennai,
India.
Thomas Rosy
Department of Mathematics
Madras Christian College,
Chennai - 600059,
India.