Journal of Inequalities in Pure and
Applied Mathematics
A NEW SUBCLASS OF UNIFORMLY CONVEX FUNCTIONS AND
A CORRESPONDING SUBCLASS OF STARLIKE FUNCTIONS
WITH FIXED SECOND COEFFICIENT
volume 5, issue 4, article 85,
2004.
G. MURUGUSUNDARAMOORTHY AND N. MAGESH
Department of Mathematics
Vellore Institute of Technology,
Deemed University
Vellore - 632014, India.
EMail: gmsmoorthy@yahoo.com
Department of Mathematics
Adhiyamaan College of Engineering
Hosur - 635109, India.
EMail: nmagi_2000@yahoo.co.in
Received 22 June, 2004;
accepted 25 August, 2004.
Communicated by: A. Sofo
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
125-04
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Abstract
Making use of Linear operator theory, we define a new subclass of uniformly
convex functions and a corresponding subclass of starlike functions with negative coefficients. The main object of this paper is to obtain coefficient estimates
distortion bounds, closure theorems and extreme points for functions belonging
to this new class. The results are generalized to families with fixed finitely many
coefficients.
2000 Mathematics Subject Classification: 30C45.
Key words: Univalent, Convex, Starlike, Uniformly convex, Uniformly starlike, Linear
operator.
The authors would like to thank the referee for his insightful suggestions.
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
The Class S(α, β) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3
Distortion Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4
The Class T Sbn ,k (α, β) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
References
A New Subclass of Uniformly
Convex Functions and a
Corresponding Subclass of
Starlike Functions with Fixed
Second Coefficient
G. Murugusundaramoorthy and
N. Magesh
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1.
Introduction
Denoted by S the class of functions of the form
(1.1)
f (z) = z +
∞
X
an z n
n=2
that are analytic and univalent in the unit disc 4 = {z : |z| < 1} and by ST
and CV the subclasses of S that are respectively, starlike and convex. Goodman
[2, 3] introduced and defined the following subclasses of CV and ST.
A function f (z) is uniformly convex (uniformly starlike) in 4 if f (z) is in
CV (ST ) and has the property that for every circular arc γ contained in 4,
with center ξ also in 4, the arc f (γ) is convex (starlike) with respect to f (ξ).
The class of uniformly convex functions is denoted by U CV and the class of
uniformly starlike functions by U ST (for details see [2]). It is well known from
[4, 5] that
00 00
zf (z) zf
(z)
≤ Re 1 + 0
f ∈ U CV ⇔ 0
.
f (z) f (z)
In [5], Rønning introduced a new class of starlike functions related to U CV
defined as
0
0 zf (z)
zf (z)
f ∈ Sp ⇔ − 1 ≤ Re
.
f (z)
f (z)
Note that f (z) ∈ U CV ⇔ zf 0 (z) ∈ Sp . Further, Rønning generalized the class
Sp by introducing a parameter α, − 1 ≤ α < 1,
0
0
zf (z)
zf
(z)
f ∈ Sp (α) ⇔ − 1 ≤ Re
−α .
f (z)
f (z)
A New Subclass of Uniformly
Convex Functions and a
Corresponding Subclass of
Starlike Functions with Fixed
Second Coefficient
G. Murugusundaramoorthy and
N. Magesh
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Now we define the function φ(a, c; z) by
φ(a, c; z) = z +
(1.2)
∞
X
(a)n−1
n=2
(c)n−1
zn,
for c 6= 0, −1, −2, . . . , a 6= −1; z ∈ ∆ where (λ)n is the Pochhammer symbol
defined by
(1.3) (λ)n =
=
Γ(n + λ)
Γ(λ)
(
1;
n=0
λ(λ + 1)(λ + 2) . . . (λ + n − 1), n ∈ N = {1, 2, . . . }
)
.
Carlson and Shaffer [1] introduced a linear operator L(a, c), by
(1.4)
L(a, c)f (z) = φ(a, c; z) ∗ f (z)
∞
X
(a)n−1
=z+
an z n , z ∈ 4,
(c)
n−1
n=2
where ∗ stands for the Hadamard product or convolution product of two power
series
∞
∞
X
X
n
ϕ(z) =
ϕn z and ψ(z) =
ψn z n
n=1
defined by
(ϕ ∗ ψ)(z) = ϕ(z) ∗ ψ(z) =
n=1
∞
X
n=1
ϕn ψn z n .
A New Subclass of Uniformly
Convex Functions and a
Corresponding Subclass of
Starlike Functions with Fixed
Second Coefficient
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N. Magesh
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We note that L(a, a)f (z) = f (z), L(2, 1)f (z) = zf 0 (z), L(m + 1, 1)f (z) =
Dm f (z), where Dm f (z) is the Ruscheweyh derivative of f (z) defined by Ruscheweyh [6] as
z
(1.5)
Dm f (z) =
∗ f (z), m > −1.
(1 − z)m+1
Which is equivalently,
Dm f (z) =
z dm m−1
{z
f (z)}.
m! dz m
For β ≥ 0 and −1 ≤ α < 1, we let S(α, β) denote the subclass of S consisting of functions f (z) of the form (1.1) and satisfying the analytic criterion
z(L(a, c)f (z))0
z(L(a, c)f (z))
(1.6)
Re
− α > β − 1 , z ∈ 4.
L(a, c)f (z)
L(a, c)f (z)
T
We also let T S(α, β) = S(α, β) T where T, the subclass of S consisting
of functions of the form
(1.7)
0
f (z) = z −
∞
X
an z n , an ≥ 0,
∀ n ≥ 2,
n=2
was introduced and studied by Silverman [7].
The main object of this paper is to obtain necessary and sufficient conditions
for the functions f (z) ∈ T S(α, β). Furthermore we obtain extreme points, distortion bounds and closure properties for f (z) ∈ T S(α, β) by fixing the second
coefficient.
A New Subclass of Uniformly
Convex Functions and a
Corresponding Subclass of
Starlike Functions with Fixed
Second Coefficient
G. Murugusundaramoorthy and
N. Magesh
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2.
The Class S(α, β)
In this section we obtain necessary and sufficient conditions for functions f (z)
in the classes T S(α, β).
Theorem 2.1. A function f (z) of the form (1.1) is in S(α, β) if
(2.1)
∞
X
[n(1 + β) − (α + β)]
n=2
(a)n−1
|an | ≤ 1 − α,
(c)n−1
−1 ≤ α < 1, β ≥ 0.
Proof. It suffices to show that
z(L(a, c)f (z))0
z(L(a, c)f (z))0
β
− 1 − Re
− 1 ≤ 1 − α.
L(a, c)f (z)
L(a, c)f (z)
A New Subclass of Uniformly
Convex Functions and a
Corresponding Subclass of
Starlike Functions with Fixed
Second Coefficient
G. Murugusundaramoorthy and
N. Magesh
We have
z(L(a, c)f (z))0
z(L(a, c)f (z))0
β
− 1 − Re
−1
L(a, c)f (z)
L(a, c)f (z)
z(L(a, c)f (z))0
≤ (1 + β) − 1
L(a, c)f (z)
P∞
n−1
(1 + β) n=2 (n − 1) (a)
|a |
(c)n−1 n
≤
.
P∞ (a)n−1
1 − n=2 (c)n−1 |an |
This last expression is bounded above by (1 − α) if
∞
X
n=2
[n(1 + β) − (α + β)]
(a)n−1
|an | ≤ 1 − α,
(c)n−1
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and hence the proof is complete.
Theorem 2.2. A necessary and sufficient condition for f (z) of the form (1.7) to
be in the class T S(α, β), −1 ≤ α < 1, β ≥ 0 is that
∞
X
(2.2)
[n(1 + β) − (α + β)]
n=2
(a)n−1
an ≤ 1 − α.
(c)n−1
Proof. In view of Theorem 2.1, we need only to prove the necessity. If f (z) ∈
T S(α, β) and z is real then
P∞
P
(a)n−1
(a)n−1
n−1 n−1
(n
−
1)
a
z
a
z
1− ∞
n
n
n
n=2 (c)n−1
(c)n−1
n=2
−α≥β
.
P∞ (a)n−1
P∞ (a)n−1
n−1
n−1
1−
an z
an z
1−
n=2 (c)n−1
n=2 (c)n−1
A New Subclass of Uniformly
Convex Functions and a
Corresponding Subclass of
Starlike Functions with Fixed
Second Coefficient
G. Murugusundaramoorthy and
N. Magesh
Letting z → 1 along the real axis, we obtain the desired inequality
∞
X
(a)n−1
an ≤ 1 − α, −1 ≤ α < 1, β ≥ 0.
[n(1 + β) − (α + β)]
(c)n−1
n=2
Corollary 2.3. Let the function f (z) defined by (1.7) be in the class T S(α, β).
Then
an ≤
(1 − α)(c)n−1
, n ≥ 2, −1 ≤ α ≤ 1, β ≥ 0.
[n(1 + β) − (α + β)](a)n−1
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Remark 1. In view of Theorem 2.2, we can see that if f (z) is of the form (1.7)
and is in the class T S(α, β) then
a2 =
(2.3)
(1 − α)(c)
.
(2 + β − α)(a)
By fixing the second coefficient, we introduce a new subclass T Sb (α, β) of
T S(α, β) and obtain the following theorems.
Let T Sb (α, β) denote the class of functions f (z) in T S(α, β) and be of the
form
(2.4)
∞
b(1 − α)(c) 2 X
f (z) = z −
z −
an z n (an ≥ 0), 0 ≤ b ≤ 1.
(2 + β − α)(a)
n=3
Theorem 2.4. Let function f (z) be defined by (2.4). Then f (z) ∈ T Sb (α, β) if
and only if
(2.5)
A New Subclass of Uniformly
Convex Functions and a
Corresponding Subclass of
Starlike Functions with Fixed
Second Coefficient
G. Murugusundaramoorthy and
N. Magesh
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∞
X
(a)n−1
an ≤ (1 − b)(1 − α),
[n(1 + β) − (α + β)]
(c)
n−1
n=3
−1 ≤ α < 1, β ≥ 0.
Proof. Substituting
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a2 =
b(1 − α) (c)
, 0 ≤ b ≤ 1.
(2 + β − α) (a)
in (2.2) and simple computation leads to the desired result.
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Corollary 2.5. Let the function f (z) defined by (2.4) be in the class T Sb (α, β).
Then
(2.6)
an ≤
(1 − b)(1 − α)(c)n−1
, n ≥ 3, −1 ≤ α ≤ 1, β ≥ 0.
[n(1 + β) − (α + β)](a)n−1
Theorem 2.6. The class T Sb (α, β) is closed under convex linear combination.
Proof. Let the function f (z) be defined by (2.4) and g(z) defined by
∞
b(1 − α) (c) 2 X
g(z) = z −
z −
dn z n ,
(2 + β − α) (a)
n=3
(2.7)
A New Subclass of Uniformly
Convex Functions and a
Corresponding Subclass of
Starlike Functions with Fixed
Second Coefficient
where dn ≥ 0 and 0 ≤ b ≤ 1.
Assuming that f (z) and g(z) are in the class T Sb (α, β), it is sufficient to
prove that the function H(z) defined by
G. Murugusundaramoorthy and
N. Magesh
H(z) = λf (z) + (1 − λ)g(z), (0 ≤ λ ≤ 1)
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(2.8)
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is also in the class T Sb (α, β).
Since
H(z) = z −
(2.9)
∞
X
b(1 − α)(c) 2
z −
{λan + (1 − λ)dn }z n ,
(2 + β − α)(a)
n=3
an ≥ 0, dn ≥ 0, 0 ≤ b ≤ 1, we observe that
(2.10)
∞
X
n=3
[n(1 + β) − (α + β)]
(a)n−1
(λan + (1 − λ)dn ) ≤ (1 − b)(1 − α)
(c)n−1
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which is, in view of Theorem 2.4, again, implies that H(z) ∈ T Sb (α, β) which
completes the proof of the theorem.
Theorem 2.7. Let the functions
(2.11)
∞
b(1 − α)(c) 2 X
z −
fj (z) = z −
an, j z n , an,j ≥ 0
(2 + β − α)(a)
n=3
be in the class T Sb (α, β) for every j (j = 1, 2, . . . , m). Then the function F (z)
defined by
F (z) =
(2.12)
m
X
µj fj (z),
A New Subclass of Uniformly
Convex Functions and a
Corresponding Subclass of
Starlike Functions with Fixed
Second Coefficient
j=1
G. Murugusundaramoorthy and
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is also in the class T Sb (α, β), where
m
X
(2.13)
µj = 1.
j=1
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Proof. Combining the definitions (2.11) and (2.12), further by (2.13) we have
!
∞
m
b(1 − α)(c) 2 X X
(2.14)
F (z) = z −
z −
µj an,j z n .
(2 + β − α)(a)
n=3
j=1
Since fj (z) ∈ T Sb (α, β) for every j = 1, 2, . . . , m, Theorem 2.4 yields
(2.15)
∞
X
n=3
[n(1 + β) − (α + β)]
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(a)n−1
an,j ≤ (1 − b)(1 − α),
(c)n−1
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for j = 1, 2, . . . , m. Thus we obtain
∞
X
[n(1 + β) − (α + β)]
n=3
=
m
X
(a)n−1
(c)n−1
m
X
!
µj an,j
j=1
∞
X
(a)n−1
[n(1 + β) − (α + β)]
an,j
(c)n−1
n=3
µj
j=1
!
≤ (1 − b)(1 − α)
in view of Theorem 2.4. So, F (z) ∈ T Sb (α, β).
Theorem 2.8. Let
f2 (z) = z −
(2.16)
b(1 − α)(c) 2
z
(2 + β − α)(a)
and
(2.17)
A New Subclass of Uniformly
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Second Coefficient
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b(1 − α)(c) 2
(1 − b)(1 − α)(c)n−1
fn (z) = z −
z −
zn
(2 + β − α)(a)
[n(1 + β) − (α + β)](a)n−1
for n = 3, 4, . . . . Then f (z) is in the class T Sb (α, β) if and only if it can be
expressed in the form
f (z) =
(2.18)
∞
X
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n=2
where λn ≥ 0 and
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λn = 1.
Page 11 of 20
Proof. We suppose that f (z) can be expressed in the form (2.18). Then we have
∞
(1 − b)(1 − α)(c)n−1
b(1 − α)(c) 2 X
zn
f (z) = z −
z −
λn
(2 + β − α)(a)
[n(1 + β) − (α + β)](a)n−1
n=3
(2.19)
=z−
∞
X
An z n ,
n=2
where
A2 =
(2.20)
b(1 − α)(c)
(2 + β − α)(a)
and
(2.21)
An =
λn (1 − b)(1 − α)(c)n−1
, n = 3, 4, . . . .
[n(1 + β) − (α + β)](a)n−1
Therefore
∞
∞
X
X
(a)n−1
An = b(1 − α) +
λn (1 − b)(1 − α)
[n(1 + β) − (α + β)]
(c)
n−1
n=3
n=2
(2.22)
= (1 − α)[b + (1 − λ2 )(1 − b)]
≤ (1 − α),
it follows from Theorem 2.2 and Theorem 2.4 that f (z) is in the class T Sb (α, β).
Conversely, we suppose that f (z) defined by (2.4) is in the class T Sb (α, β).
Then by using (2.6), we get
(2.23)
(1 − b)(1 − α)(c)n−1
, (n ≥ 3).
an ≤
[n(1 + β) − (α + β)](a)n−1
A New Subclass of Uniformly
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Corresponding Subclass of
Starlike Functions with Fixed
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Setting
(2.24)
λn =
[n(1 + β) − (α + β)](a)n−1
an , (n ≥ 3)
(1 − b)(1 − α)(c)n−1
and
(2.25)
λ2 = 1 −
∞
X
λn ,
n=3
we have (2.18). This completes the proof of Theorem 2.8.
Corollary 2.9. The extreme points of the class T Sb (α, β) are functions fn (z),
n ≥ 2 given by Theorem 2.8.
A New Subclass of Uniformly
Convex Functions and a
Corresponding Subclass of
Starlike Functions with Fixed
Second Coefficient
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3.
Distortion Theorems
In order to obtain distortion bounds for the function f ∈ T Sb (α, β), we first
prove the following lemmas.
Lemma 3.1. Let the function f3 (z) be defined by
(3.1)
f3 (z) = z −
b(1 − α)(c) 2 (1 − b)(1 − α)(c)2 3
z −
z .
(2 + β − α)(a)
(3 + 2β − α)(a)2
Then, for 0 ≤ r < 1 and 0 ≤ b ≤ 1,
(3.2)
|f3 (reiθ )| ≥ r −
b(1 − α)(c) 2 (1 − b)(1 − α)(c)2 3
r −
r
(2 + β − α)(a)
(3 + 2β − α)(a)2
with equality for θ = 0. For either 0 ≤ b < b0 and 0 ≤ r ≤ r0 or b0 ≤ b ≤ 1,
(3.3)
|f3 (reiθ )| ≤ r +
b(1 − α)(c) 2 (1 − b)(1 − α)(c)2 3
r −
r
(2 + β − α)(a)
(3 + 2β − α)(a)2
with equality for θ = π, where
1
2(1 − α)(c)(c)2
× {−[(3 + 2β − α)(a)2 (c) + 4(2 + β − α)(a)(c)2 − (1 − α)(c)(c)2 ]
+ [((3 + 2β − α)(a)2 (c) + 4(2 + β − α)(a)(c)2 − (1 − α)(c)(c)22
(3.4) b0 =
+ 16(2 + β − α)(1 − α)(a)(c)(c)22 ]1/2 }
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and
(3.5) r0 =
1
{−2(1 − b)(2 + β − α)(a)(c + 1)
b(1 − b)(1 − α)(c)2
+ [4(1 − b)2 (2 + β − α)2 (a)2 (c + 1)2
+ b2 (1 − b)(3 + 2β − α)(1 − α)(a)2 (c)2 ]1/2 }.
Proof. We employ the technique as used by Silverman and Silvia [8]. Since
(3.6)
∂|f3 (reiθ )|2
∂θ
3
= 2(1 − α)r sin θ
b(c)
4(1 − b)(c)2
+
r cos θ
(2 + β − α)(a) (3 + 2β − α)(a)2
b(1 − b)(1 − α)(c)(c)2
2
−
r
(2 + β − α)(3 + 2β − α)(a)(a)2
we can see that
(3.7)
A New Subclass of Uniformly
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Second Coefficient
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∂|f3 (reiθ )|2
=0
∂θ
for θ1 = 0, θ2 = π, and
1 b[(1 − b)(1 − α)(c)2 r2 − (3 + 2β − α)(a)2 ]
−1
(3.8)
θ3 = cos
r
4(1 − b)(2 + β − α)(a)(c + 1)
since θ3 is a valid root only when −1 ≤ cos θ3 ≤ 1. Hence we have a third root
if and only if r0 ≤ r < 1 and 0 ≤ b ≤ b0 . Thus the results of the theorem follow
from comparing the extremal values |f3 (reiθk )|, k = 1, 2, 3 on the appropriate
intervals.
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Lemma 3.2. Let the functions fn (z) be defined by (2.17) and n ≥ 4. Then
|fn (reiθ )| ≤ |f4 (−r)|.
(3.9)
Proof. Since
fn (z) = z −
and
rn
n
b(1 − α)(c) 2
(1 − b)(1 − α)(c)n−1
zn
z −
(2 + β − α)(a)
[n(1 + β) − (α + β)](a)n−1
is a decreasing function of n, we have
b(1 − α)(c) 2 (1 − b)(1 − α)(c)3 4
r −
r
(2 + β − α)(a)
[4 + 3β − α](a)3
= −f4 (−r),
|fn (reiθ | ≤ r +
A New Subclass of Uniformly
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G. Murugusundaramoorthy and
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which shows (3.9).
Theorem 3.3. Let the function f (z) defined by (2.4) belong to the class T Sb (α, β),
then for 0 ≤ r < 1,
b(1 − α)(c) 2 (1 − b)(1 − α)(c)2 3
|f (reiθ )| ≥ r −
r −
r
(2 + β − α)(a)
[3 + 2β − α](a)2
(3.10)
with equality for f3 (z) at z = r, and
n
o
(3.11)
|f (reiθ )| ≤ max max |f3 (reiθ )|, −f4 (−r) ,
θ
where max |f3 (reiθ )| is given by Lemma 3.1.
θ
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Proof. The proof of Theorem 3.3 is obtained by comparing the bounds of Lemma
3.1 and Lemma 3.2.
Remark 2. Taking b = 1 in Theorem 3.3 we obtain the following result.
Corollary 3.4. Let the function f (z) defined by (1.7) be in the class T S(α, β).
Then for |z| = r < 1, we have
r−
(3.12)
(1 − α)(c)
(1 − α)(c)
r2 ≤ |f (z)| ≤ r +
r2 .
(2 + β − α)(a)
(2 + β − α)(a)
Lemma 3.5. Let the function f3 (z) be defined by (3.1). Then, for 0 ≤ r < 1,
and 0 ≤ b ≤ 1,
(3.13)
|f30 (reiθ )| ≥ 1 −
2b(1 − α)(c)
3(1 − b)(1 − α)(c)2 2
r−
r
(2 + β − α)(a)
(3 + 2β − α)(a)2
A New Subclass of Uniformly
Convex Functions and a
Corresponding Subclass of
Starlike Functions with Fixed
Second Coefficient
G. Murugusundaramoorthy and
N. Magesh
with equality for θ = 0. For either 0 ≤ b < b1 and 0 ≤ r ≤ r1 or b1 ≤ b ≤ 1,
(3.14)
|f30 (reiθ )|
2b(1 − α)(c)
3(1 − b)(1 − α)(c)2 2
≤1+
r−
r
(2 + β − α)(a)
(3 + 2β − α)(a)2
with equality for θ = π, where
1
6(1 − α)(c)(c2 )
× {−[(3 + 2β − α)(a)2 (c) + 6(2 + β − α)(a)(c)2 − 3(1 − α)(c)(c)2 ]
+ {((3 + 2β − α)(a)2 (c) + 6(2 + β − α)(a)(c)2 − 3(1 − α)(c)(c)2 )2
(3.15) b1 =
+ 72(2 + β − α)(1 −
α)(a)(c)(c22 )}1/2 }
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and
(3.16) r1 =
1
{−3(1 − b)(2 + β − α)(a)(c + 1)
3b(1 − b)(1 − α)(c2 )
+ [8(1 − b)2 (2 + β − α)2 (a)2 (c + 1)2
+ 3b2 (1 − b)(3 + 2β − α)(1 − α)(a)2 (c)2 ]1/2 }.
Proof. The proof of Lemma 3.5 is much akin to the proof of Lemma 3.1.
Theorem 3.6. Let the function f (z) defined by (2.4) belong to the class T Sb (α, β),
then for 0 ≤ r < 1,
|f 0 (reiθ )| ≥ 1 −
(3.17)
2b(1 − α)(c)
3(1 − b)(1 − α)(c)2 2
r−
r
(2 + β − α)(a)
[3 + 2β − α](a)2
with equality for f30 (z) at z = r, and
n
o
(3.18)
|f 0 (reiθ )| ≤ max max |f30 (reiθ )|, −f40 (−r) ,
θ
where max |f30 (reiθ )| is given by Lemma 3.5.
θ
Remark 3. Putting b = 1 in Theorem 3.6 we obtain the following result.
Corollary 3.7. Let the function f (z) defined by (1.2) be in the class T S(α, β).
Then for |z| = r < 1, we have
(3.19)
2(1 − α)(c)
2(1 − α)(c)
1−
r ≤ |f 0 (z)| ≤ 1 +
r.
(2 + β − α)(a)
(2 + β − α)(a)
A New Subclass of Uniformly
Convex Functions and a
Corresponding Subclass of
Starlike Functions with Fixed
Second Coefficient
G. Murugusundaramoorthy and
N. Magesh
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4.
The Class T Sbn,k (α, β)
Instead of fixing just the second coefficient, we can fix finitely many coefficients. Let T Sbn ,k (α, β) denote the class of functions in T Sb (α, β) of the form
(4.1)
f (z) = z −
k
X
n=2
where 0 ≤
∞
X
bn (1 − α)(c)n−1
n
z −
an z n ,
[n(1 + β) − (α + β)](a)n−1
n=k+1
Pk
n=2 bn = b ≤ 1. Note that T Sb2 ,2 (α, β) = T Sb (α, β).
Theorem 4.1. The extreme points of the class T Sbn ,k (α, β) are
fk (z) = z −
k
X
n=2
bn (1 − α)(c)n−1
zn
[n(1 + β) − (α + β)](a)n−1
G. Murugusundaramoorthy and
N. Magesh
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and
fn (z) = z −
A New Subclass of Uniformly
Convex Functions and a
Corresponding Subclass of
Starlike Functions with Fixed
Second Coefficient
k
X
n=2
bn (1 − α)(c)n−1
zn
[n(1 + β) − (α + β)](a)n−1
−
∞
X
(1 − b)(1 − α)(c)n−1
zn
[n(1
+
β)
−
(α
+
β)](a)
n−1
n=k+1
The details of the proof are omitted, since the characterization of the extreme
points enables us to solve the standard extremal problems in the same manner
as was done for T Sb (α, β).
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References
[1] B.C. CARLSON AND S.B. SHAFFER, Starlike and prestarlike hypergrometric functions, SIAM J.Math. Anal., 15 (2002), 737–745.
[2] A.W. GOODMAN, On uniformly convex functions, Ann. Polon. Math., 56
(1991), 87–92.
[3] A.W. GOODMAN, On uniformly starlike functions, J. Math. Anal. & Appl.,
155 (1991), 364–370.
[4] W. MA AND D. MINDA, Uniformly convex functions, Ann. Polon. Math.,
57 (1992), 165–175.
A New Subclass of Uniformly
Convex Functions and a
Corresponding Subclass of
Starlike Functions with Fixed
Second Coefficient
[5] F. RØNNING, Uniformly convex functions and a corresponding class of
starlike functions, Proc. Amer. Math. Soc., 118 (1993), 189–196.
G. Murugusundaramoorthy and
N. Magesh
[6] S. RUSCHEWEYH, New criteria for Univalent Functions, Proc. Amer.
Math. Soc., 49 (1975), 109–115.
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[7] H. SILVERMAN, Univalent functions with negative coefficients, Proc.
Amer. Math. Soc., 51 (1975), 109–116.
[8] H. SILVERMAN AND E.M. SILVIA, Fixed coefficients for subclasses of
starlike functions, Houston. J. Math., 7 (1997), 129–136.
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