MATEMATIQKI VESNIK
originalni nauqni rad
research paper
65, 1 (2013), 14–28
March 2013
CERTAIN SUBCLASSES OF UNIFORMLY STARLIKE AND
CONVEX FUNCTIONS DEFINED BY CONVOLUTION WITH
NEGATIVE COEFFICIENTS
M.K. Aouf, A.A. Shamandy, A.O. Mostafa and A.K. Wagdy
Abstract. The aim of this paper is to obtain coefficient estimates, distortion theorems,
convex linear combinations and radii of close-to-convexity, starlikeness and convexity for functions
belonging to the class T S(g, λ; α, β). Furthermore partial sums fn (z) of functions f (z) in the class
T S(g, λ; α, β) are considered and sharp lower bounds for the ratios of real part of f (z) to fn (z)
and f 0 (z) to fn0 (z) are determined.
1. Introduction
Let A denote the class of functions of the form
∞
P
ak z k ,
f (z) = z +
(1.1)
k=2
which are analytic in the open unit disc U = {z ∈ C : |z| < 1}. Let g ∈ A be given
by
∞
P
g(z) = z +
bk z k ,
(1.2)
k=2
the Hadamard product (or convolution) f ∗ g of f and g is defined (as usual) by
∞
P
ak ck z k = (g ∗ f )(z).
(1.3)
(f ∗ g)(z) = z +
k=2
Following Goodman ([6] and [7]), Ronning ([11 and [12]) introduced and studied
the following subclasses:
(i) A function f (z) of the form (1.1) is said to be in the class Sp (α, β) of
β-uniformly starlike functions if it satisfies the condition:
¯ 0
¯
½ 0
¾
¯ zf (z)
¯
zf (z)
Re
− α > β ¯¯
− 1¯¯ (z ∈ U) ,
(1.4)
f (z)
f (z)
where −1 ≤ α < 1 and β ≥ 0.
2010 AMS Subject Classification: 30C45.
Keywords and phrases: Analytic function; Hadamard product; distortion theorems; partial
sums.
14
Subclasses of uniformly starlike and convex functions
15
(ii) A function f (z) of the form (1.1) is said to be in the class U CV (α, β) of
β-uniformly convex functions if it satisfies the condition:
¯ 00
¯
½
¾
¯ zf (z) ¯
zf 00 (z)
¯
¯ (z ∈ U) ,
Re 1 + 0
−α >β¯ 0
(1.5)
f (z)
f (z) ¯
where −1 ≤ α < 1 and β ≥ 0.
It follows from (1.4) and (1.5) that
f (z) ∈ U CV (α, β) ⇔ zf 0 (z) ∈ Sp (α, β).
(1.6)
For −1 ≤ α < 1, 0 ≤ λ ≤ 1 and β ≥ 0, we let S(g, λ; α, β) be the subclass of A
consisting of functions f (z) of the form (1.1) and functions g(z) of the form (1.2)
and satisfying the analytic criterion:
¯
¯
o
n
¯
¯
z(f ∗g)0 (z)
z(f ∗g)0 (z)
(1.7)
−
α
>
β
−
1
Re (1−λ)(f ∗g)(z)+λz
¯
¯.
0
0
(f ∗g) (z)
(1−λ)(f ∗g)(z)+λz(f ∗g) (z)
z
Remark 1. (i) Putting g(z) = (1−z)
in the class S(g, λ; α, β), we obtain the
class Sp (λ, α, β) defined by Murugusundaramoorthy and Magesh [10].
z
(ii) Putting g(z) = (1−z)
2 in the class S(g, λ; α, β), we obtain the class
U CV (λ, α, β) defined by Murugusundaramoorthy and Magesh [10].
Let T denote the subclass of A consisting of functions of the form:
f (z) = z −
∞
P
ak z k (ak ≥ 0) .
(1.8)
k=2
Further, we define the class T S(g, λ; α, β) by
T S(g, λ; α, β) = S(g, λ; α, β) ∩ T
(1.9)
We note that:
z
z
, 0; α, 1) = T Sp (α) and T S( (1−z)
(i) T S( (1−z)
2 , 0; α, 1) = U CT (α) (see Bharati
et al. [2]);
z
z
(ii) T S( (1−z)
, 0; α, β) = T Sp (α, β) and T S( (1−z)
2 , 0; α, β) = U CT (α, β) (see
Bharati et al. [2]);
z
z
(iii) T S( (1−z)
, 0; α, 0) = T ∗ (α) and T S( (1−z)
2 , 0; α, 0) = C (α) (see Silverman
[15]);
P∞
k−1 k
(iv) T S(z + k=2 (a)
(c)k−1 z , 0; α, β) = T S(α, β) (c 6= 0, −1, −2, . . . ) (see Murugusundaramoorthy and Magesh [8, 9]);
P∞
(v) T S(z + k=2 k n z k , 0; α, β) = T S(n, α, β) (n ∈ N0 = N ∪ {0}, where
N = {1, 2, . . . }) (see Rosy and Murugusundaramoorthy [13]);
z
z
(vi) T S( (1−z)
, λ; α, β) = T Sp (λ, α, β) and T S( (1−z)
2 , λ; α, β) = U CT (λ, α, β)
(see Murugusundaramoorthy and Magesh [10]);
¢
P∞ ¡
(vii) T S(z + k=2 k + δδ − 1 z k , 0; α, β) = D(β, α, δ) (δ > −1) (see Shams et
al. [14]);
16
M.K. Aouf, A.A. Shamandy, A.O. Mostafa, A.K. Wagdy
P∞
n
(viii) T S(z + k=2 [1 + δ (k − 1)] z k , 0; α, β) = T Sδ (n, α, β) (δ ≥ 0, n ∈ N0 )
(see Aouf and Mostafa [1]).
Also we note that:
∞
P
(i) T S(z +
Γk (α1 )z k , λ; α, β) = T Sq,s (α1 ; λ, α, β)
k=2
n
o
z(H (α1 ,β1 )f (z))0
= {f ∈ T : Re (1−λ)Hq,s (α1 ,β1q,s
0 − α
)f (z)+λz(Hq,s (α1 ,β1 )f (z))
¯
¯
¯
¯
z (Hq,s (α1 ,β1 )f (z))0
> β ¯ (1−λ)Hq,s (α1 ,β1 )f (z)+λz (Hq,s (α1 ,β1 )f (z))0 − 1¯},
where −1 ≤ α < 1, 0 ≤ λ ≤ 1, β ≥ 0, z ∈ U and Γk (α1 ) is defined by
Γk (α1 ) =
(α1 )k−1 · · · (αq )k−1
(β1 )k−1 · · · (βs )k−1 (1)k−1
(1.10)
(αi > 0, i = 1, . . . , q; βj > 0, j = 1, . . . , s; q ≤ s + 1, q, s ∈ N0 ), where the operator
Hq,s (α1 , β1 ) was introduced and studied by Dziok and Srivastava (see [4] and [5]),
which is a generalization of many other linear operators considered earlier;
im
P∞ h
(ii) T S(z + k=2 `+1+µ(k−1)
z k , λ; α, β) = T S(m, µ, `, λ; α, β)
`+1
o
n
n
z(I m (µ,`)f (z))0
= f ∈ T : Re (1−λ)I m (µ,`)f
0 − α
m
(z)+λz(I (µ,`)f (z))
¯
¯o
¯
¯
z(I m (µ,`)f (z))0
> β ¯ (1−λ)I m (µ,`)f (z)+λz(I m (µ,`)f (z))0 − 1¯ ,
where −1 ≤ α < 1, 0 ≤ λ ≤ 1, β ≥ 0, m ∈ N0 , µ, ` ≥ 0, z ∈ U and the operator
I m (µ, `) was defined by Cătaş et al. (see [3]), which is a generalization of many
other linear operators considered earlier;
P∞
(iii) T S(z + k=2 Ck (b, µ)z k , λ; α, β) = T S(b, µ, λ; α, β) =
¯
¯¾
¾
½
½
0
0
¯
¯
z (Jbµ f (z))
z (Jbµ f (z))
¯
¯ ,
>β¯
f ∈ T : Re
0 − α
0 − 1
¯
(1−λ)Jbµ f (z)+λz (Jbµ f (z))
(1−λ)Jbµ f (z)+λz (Jbµ f (z))
where −1 ≤ α < 1, 0 ≤ λ ≤ 1, β ≥ 0, z ∈ U and Ck (b, µ) is defined by
¶µ
µ
1+b
−
(b ∈ C \ Z−
(1.11)
Ck (b, µ) =
0 }, Z0 = Z \ N),
k+b
where the operator Jbµ was introduced by Srivastava and Attiya (see [18]), which
is a generalization of many other linear operators considered earlier.
2. Coefficient estimates
Unless otherwise mentioned, we shall assume in the reminder of this paper
that, −1 ≤ α < 1, 0 ≤ λ ≤ 1, β ≥ 0 and z ∈ U.
Theorem 1. A function f (z) of the form (1.1) is in the class S(g, λ; α, β) if
∞
P
{k (1 + β) − (α + β) [1 + λ (k − 1)]} bk |ak | ≤ 1 − α,
k=2
where bk+1 ≥ bk > 0 (k ≥ 2).
(2.1)
Subclasses of uniformly starlike and convex functions
17
Proof. Assume that the inequality (2.1) holds true. Then we have
¯
¯
¯
¯
z (f ∗ g)0 (z)
¯
β ¯¯
−
1
¯
0
(1 − λ) (f ∗ g)(z) + λz(f ∗ g) (z)
½
¾
z (f ∗ g)0 (z)
− Re
−
1
(1 − λ) (f ∗ g)(z) + λz(f ∗ g)0 (z)
¯
¯
¯
¯
z (f ∗ g)0 (z)
¯
≤ (1 + β) ¯¯
−
1
¯
0
(1 − λ) (f ∗ g)(z) + λz(f ∗ g) (z)
P∞
(1 + β) k=2 (1 − λ) (k − 1) bk |ak | z k−1
P∞
≤
≤ 1 − α.
1 − k=2 [1 + λ (k − 1)] bk |ak | z k−1
This completes the proof of Theorem 1.
Theorem 2. A necessary and sufficient condition for the function f (z) of the
form (1.8) to be in the class T S(g, λ; α, β) is that
∞
P
{k (1 + β) − (α + β) [1 + λ (k − 1)]} ak bk ≤ 1 − α.
(2.2)
k=2
Proof. In view of Theorem 1, we need only to prove the necessity. If f (z) ∈
T S(g, λ; α, β) and z is real, then
¯ ∞
¯
∞
P
¯ P
¯
¯
1−
k ak bk z k−1
(1 − λ) (k − 1) ak bk z k−1 ¯
¯
¯
k=2
¯.
− α ≥ β ¯¯ k=2P
∞
∞
¯
P
k−1
k−1 ¯
¯
1−
[1 + λ (k − 1)] ak bk z
[1 + λ (k − λ)] ak bk z
¯1 −
¯
k=2
k=2
Letting z −→ 1− along the real axis, we obtain
∞
P
{k (1 + β) − (α + β) [1 + λ (k − 1)]} ak bk ≤ 1 − α.
k=2
This completes the proof of Theorem 2.
Corollary 1.
T S(g, λ; α, β). Then
ak ≤
Let the function f (z) defined by (1.8) be in the class
1−α
(k ≥ 2).
{k (1 + β) − (α + β) [1 + λ (k − 1)]} bk
(2.3)
The result is sharp for the function
f (z) = z −
1−α
z k (k ≥ 2).
{k (1 + β) − (α + β) [1 + λ (k − 1)]} bk
(2.4)
By taking bk = Γk (α1 ), where Γk (α1 ) is defined by (1.10), in Theorem 2, we
have:
18
M.K. Aouf, A.A. Shamandy, A.O. Mostafa, A.K. Wagdy
Corollary 2. A necessary and sufficient condition for the function f (z) of
the form (1.8) to be in the class T Sq,s (α1 ; λ, α, β) is that
∞
P
{k (1 + β) − (α + β) [1 + λ (k − 1)]} Γk (α1 )ak ≤ 1 − α.
k=2
h
By taking bk =
`+1+µ(k−1)
`+1
im
(m ∈ N0 , µ, ` ≥ 0), in Theorem 2, we have:
Corollary 3. A necessary and sufficient condition for the function f (z) of
the form (1.8) to be in the class T S(m, µ, `, λ; α, β) is that
∞
P
h
{k (1 + β) − (α + β) [1 + λ (k − 1)]}
k=2
`+1+µ(k−1)
`+1
im
ak ≤ 1 − α.
By taking bk = Ck (b, µ), where Ck (b, µ) defined by (1.11), in Theorem 2, we
have:
Corollary 4. A necessary and sufficient condition for the function f (z) of
the form (1.8) to be in the class T S(b, µ, λ; α, β) is that
∞
P
{k (1 + β) − (α + β) [1 + λ (k − 1)]} |Ck (b, µ)| |ak | ≤ 1 − α.
k=2
3. Distortion theorem
Theorem 3. Let the function f (z) of the form (1.8) be in the class
T S(g, λ; α, β). Then for |z| = r < 1, we have
|f (z)| ≥ r −
1−α
r2
[2 + β − α − λ (α + β)] b2
(3.1)
|f (z)| ≤ r +
1−α
r2 ,
[2 + β − α − λ (α + β)] b2
(3.2)
and
provided that bk+1 ≥ bk > 0 (k ≥ 2). The equalities in (3.1) and (3.2) are attained
for the function f (z) given by
f (z) = z −
1−α
z 2,
[2 + β − α − λ (α + β)] b2
at z = r and z = rei(2k+1)π (k ∈ Z).
Proof. Since for k ≥ 2,
[2 + β − α − λ (α + β)] b2 ≤ {k (1 + β) − (α + β) [1 + λ (k − 1)]} bk ,
(3.3)
Subclasses of uniformly starlike and convex functions
19
using Theorem 2, we have
∞
P
[2 + β − α − λ (α + β)] b2
ak
k=2
≤
∞
P
{k (1 + β) − (α + β) [1 + λ (k − 1)]} ak bk ≤ 1 − α,
(3.4)
k=2
that is, that
∞
P
1−α
.
[2 + β − α − λ (α + β)] b2
ak ≤
k=2
From (1.8) and (3.5), we have
∞
P
|f (z)| ≥ r − r2
ak ≥ r −
k=2
and
∞
P
|f (z)| ≤ r + r2
ak ≤ r +
k=2
(3.5)
1−α
r2
[2 + β − α − λ (α + β)] b2
1−α
r2 .
[2 + β − α − λ (α + β)] b2
This completes the proof of Theorem 3.
Theorem 4. Let the function f (z) of the form (1.8) be in the class
T S(g, λ; α, β). Then for |z| = r < 1, we have
|f 0 (z)| ≥ r −
and
2 (1 − α)
r
[2 + β − α − λ (α + β)] b2
(3.6)
2 (1 − α)
r,
(3.7)
[2 + β − α − λ (α + β)] b2
≥ bk > 0 (k ≥ 2). The result is sharp for the function f (z)
|f 0 (z)| ≤ r +
provided that bk+1
given by (3.3).
Proof. From Theorem 2 and (3.5), we have
∞
P
kak ≤
k=2
2 (1 − α)
,
[2 + β − α − λ (α + β)] b2
and the remaining part of the proof is similar to the proof of Theorem 3.
4. Convex linear combinations
P`
Theorem 5. Let µυ ≥ 0 for υ = 1, 2, . . . , ` and υ=1 µυ ≤ 1. If the functions
Fυ (z) defined by
∞
P
Fυ (z) = z −
ak,υ z k (ak,υ ≥ 0; υ = 1, 2, . . . , `) ,
(4.1)
k=2
are in the class T S(g, λ; α, β) for every υ = 1, 2, . . . , `, then the function f (z)
defined by
µ
¶
∞
P
P̀
f (z) = z −
µυ ak,υ z k
k=2
is in the class T S(g, λ; α, β).
υ=1
20
M.K. Aouf, A.A. Shamandy, A.O. Mostafa, A.K. Wagdy
Proof. Since Fυ (z) ∈ T S(g, λ; α, β), it follows from Theorem 2 that
∞
P
{k (1 + β) − (α + β) [1 + λ (k − 1)]} ak,υ bk ≤ 1 − α,
(4.2)
k=2
for every υ = 1, 2, . . . , `. Hence
µ
∞
P
{k (1 + β) − (α + β) [1 + λ (k − 1)]}
k=2
=
P̀
υ=1
µ
µυ
P̀
υ=1
∞
P
¶
µυ ak,υ bk
¶
{k (1 + β) − (α + β) [1 + λ (k − 1)]} ak,υ bk
k=2
P̀
≤ (1 − α)
υ=1
µυ ≤ 1 − α.
By Theorem 2, it follows that f (z) ∈ T S(g, λ; α, β).
Corollary 5. The class T S(g, λ; α, β) is closed under convex linear combinations.
Theorem 6. Let f1 (z) = z and
fk (z) = z −
1−α
z k (k ≥ 2) .
{k (1 + β) − (α + β) [1 + λ (k − 1)]} bk
(4.3)
Then f (z) is in the class T S(g, λ; α, β) if and only if it can be expressed in the
form:
∞
P
f (z) =
µk fk (z),
(4.4)
where µk ≥ 0 and
P∞
k=1
k=1
µk = 1.
Proof. Assume that
f (z) =
∞
P
µk fk (z) = z −
k=1
∞
P
1−α
µk z k . (4.5)
{k
(1
+
β)
−
(α
+
β)
[1
+
λ
(k
−
1)]}
b
k
k=2
Then it follows that
∞
P
k=2
{k(1+β)−(α+β)[1+λ(k−1)]}bk
1−α
·
1−α
{k(1+β)−(α+β)[1+λ(k−1)]}bk µk
=
∞
P
µk = 1 − µ1 ≤ 1.
(4.6)
k=2
So, by Theorem 2, f (z) ∈ T S(g, λ; α, β).
Conversely, assume that the function f (z) defined by (1.8) belongs to the class
T S(g, λ; α, β). Then
ak ≤
1−α
{k (1 + β) − (α + β) [1 + λ (k − 1)]} bk
(k ≥ 2) .
(4.7)
21
Subclasses of uniformly starlike and convex functions
Setting
µk =
{k (1 + β) − (α + β) [1 + λ (k − 1)]} ak bk
1−α
and
µ1 = 1 −
∞
P
(k ≥ 2) ,
µk ,
(4.8)
(4.9)
k=2
we can see that f (z) can be expressed in the form (4.4). This completes the proof
of Theorem 6.
Corollary 6. The extreme points of the class T S(g, λ; α, β) are the functions
f1 (z) = z and
fk (z) = z −
1−α
z k (k ≥ 2) .
{k (1 + β) − (α + β) [1 + λ (k − 1)]} bk
(4.10)
5. Radii of close-to-convexity, starlikeness and convexity
Theorem 7.
Let the function f (z) defined by (1.8) be in the class
T S(g, λ; α, β). Then f (z) is close-to-convex of order ρ (0 ≤ ρ < 1) in |z| < r1 ,
where
½
¾ 1
(1 − ρ) {k (1 + β) − (α + β) [1 + λ (k − 1)]} bk k−1
r1 = inf
.
(5.1)
k≥2
k (1 − α)
The result is sharp, with the extremal function f (z) given by (2.4).
Proof. We must show that
|f 0 (z) − 1| ≤ 1 − ρ for |z| < r1 ,
where r1 is given by (5.1). Indeed we find from (1.8) that
|f 0 (z) − 1| ≤
∞
P
kak |z|k−1 .
k=2
Thus |f 0 (z) − 1| ≤ 1 − ρ, if
∞
P
k=2
µ
k
1−ρ
¶
ak |z|k−1 ≤ 1.
(5.2)
But, by Theorem 2, (5.2) will be true if
µ
¶
k
{k (1 + β) − (α + β) [1 + λ (k − 1)]} bk
|z|k−1 ≤
,
1−ρ
1−α
that is, if
½
|z| ≤
(1 − ρ) {k (1 + β) − (α + β) [1 + λ (k − 1)]} bk
k (1 − α)
Theorem 7 follows easily from (5.3).
1
¾ k−1
(k ≥ 2) .
(5.3)
22
M.K. Aouf, A.A. Shamandy, A.O. Mostafa, A.K. Wagdy
Theorem 8.
Let the function f (z) defined by (1.8) be in the class
T S(g, λ; α, β). Then f (z) is starlike of order ρ (0 ≤ ρ < 1) in |z| < r2 , where
½
r2 = inf
k≥2
(1 − ρ) {k (1 + β) − (α + β) [1 + λ (k − 1)]} bk
(k − ρ) (1 − α)
1
¾ k−1
.
(5.4)
The result is sharp, with the extremal function f (z) given by (2.4).
Proof. We must show that
¯
¯ 0
¯
¯ zf (z)
¯
¯
¯ f (z) − 1¯ ≤ 1 − ρ for |z| < r2 ,
where r2 is given by (5.4). Indeed we find from (1.8) that
¯
¯ 0
¯
¯ zf (z)
¯≤
¯
−
1
¯
¯ f (z)
∞
P
(k − 1) ak |z|k−1
k=2
1−
∞
P
.
ak |z|
k−1
k=2
¯ 0
¯
¯ (z)
¯
Thus ¯ zff (z)
− 1¯ ≤ 1 − ρ, if
∞ (k − ρ) a |z|k−1
P
k
≤ 1.
1
−
ρ
k=2
(5.5)
But, by Theorem 2, (5.5) will be true if
(k − ρ) |z|k−1
{k (1 + β) − (α + β) [1 + λ (k − 1)]} bk
≤
,
1−ρ
1−α
that is, if
½
|z| ≤
(1 − ρ) {k (1 + β) − (α + β) [1 + λ (k − 1)]} bk
(k − ρ) (1 − α)
1
¾ k−1
(k ≥ 2) .
(5.6)
Theorem 8 follows easily from (5.6).
Corollary 7. Let the function f (z) defined by (1.8) be in the class
T S(g, λ; α, β). Then f (z) is convex of order ρ (0 ≤ ρ < 1) in |z| < r3 , where
½
r3 = inf
k≥2
(1 − ρ) {k (1 + β) − (α + β) [1 + λ (k − 1)]} bk
k (k − ρ) (1 − α)
1
¾ k−1
The result is sharp, with the extremal function f (z) given by (2.4).
.
(5.7)
23
Subclasses of uniformly starlike and convex functions
6. A family of integral operators
P∞
In view of Theorem 2, we see that z − k=2 dk z k is in the class T S(g, λ; α, β)
as long as 0 ≤ dk ≤ ak for all k. In particular, we have:
Theorem 9.
Let the function f (z) defined by (1.8) be in the class
T S(g, λ; α, β) and c be a real number such that c > −1. Then the function F (z)
defined by
Z
c + 1 z c−1
t f (t) dt (c > −1) ,
(6.1)
F (z) =
zc
0
also belongs to the class T S(g, λ; α, β).
Proof. From the representation (6.1) of F (z), it follows that
F (z) = z −
∞
P
dk z k ,
k=2
where
µ
dk =
c+1
c+k
¶
ak ≤ ak (k ≥ 2) .
On the other hand, the converse is not true. This leads to a radius of univalence
result.
P∞
Theorem 10. Let the function F (z) = z − k=2 ak z k (ak ≥ 0) be in the class
T S(g, λ; α, β), and let c be a real number such that c > −1. Then the function f (z)
given by (6.1) is univalent in |z| < R∗ , where
½
∗
R = inf
k≥2
(c + 1) {k (1 + β) − (α + β) [1 + λ (k − 1)]} bk
k (c + k) (1 − α)
1
¾ k−1
.
(6.2)
The result is sharp.
Proof. From (6.1), we have
f (z) =
0
∞
P
z 1−c |z c F (z)|
=z−
c+1
k=2
µ
c+k
c+1
¶
ak z k .
In order to obtain the required result, it suffices to show that
|f 0 (z) − 1| < 1 wherever |z| < R∗ ,
where R∗ is given by (6.2). Now
|f 0 (z) − 1| ≤
Thus |f 0 (z) − 1| < 1 if
∞ k (c + k)
P
ak |z|k−1 .
k=2 (c + 1)
∞ k (c + k)
P
ak |z|k−1 < 1.
k=2 (c + 1)
(6.3)
24
M.K. Aouf, A.A. Shamandy, A.O. Mostafa, A.K. Wagdy
But Theorem 2 confirms that
∞ {k (1 + β) − (α + β) [1 + λ (k − 1)]} a b
P
k k
≤ 1.
1
−
α
k=2
(6.4)
Hence (6.3) will be satisfied if
k (c + k) k−1
{k (1 + β) − (α + β) [1 + λ (k − 1)]} bk
|z|
<
,
(c + 1)
1−α
that is, if
½
|z| <
(c + 1) {k (1 + β) − (α + β) [1 + λ (k − 1)]} bk
k (c + k) (1 − α)
1
¾ k−1
(k ≥ 2) .
(6.5)
Therefore, the function f (z) given by (6.1) is univalent in |z| < R∗ . Sharpness of
the result follows if we take
f (z) = z −
(c + k) (1 − α)
z k (k ≥ 2) .
(c + 1) {k (1 + β) − (α + β) [1 + λ (k − 1)]} bk
(6.6)
7. Partial sums
Following the earlier works by Silverman [16] and Siliva [17] on partial sums of
analytic functions, we consider in this section partial sums of functions in the class
T S(g, λ; α, β) and obtain sharp lower bounds for the ratios of real part of f (z) to
fn (z) and f 0 (z) to fn0 (z).
Theorem 11. Define the partial sums f1 (z) and fn (z) by
f1 (z) = z
and
n
P
fn (z) = z +
ak z k ,
(n ∈ N \ {1}) .
k=2
Let f (z) ∈ T S(g, λ; α, β) be given by (1.8) and satisfy condition (2.2) and
½
1,
k = 2, 3, . . . , n,
ck ≥
cn+1 , k = n + 1, n + 2, . . . ,
(7.1)
where, for convenience,
ck =
Then
{k (1 + β) − (α + β) [1 + λ (k − 1)]} bk
.
1−α
½
Re
f (z)
fn (z)
¾
and
>1−
½
Re
fn (z)
f (z)
1
cn+1
¾
>
(z ∈ U; n ∈ N) ,
cn+1
.
1 + cn+1
(7.2)
(7.3)
(7.4)
25
Subclasses of uniformly starlike and convex functions
Proof. For the coefficients ck given by (7.2) it is not difficult to verify that
ck+1 > ck > 1.
(7.5)
Therefore we have
n
P
∞
P
ak + cn+1
k=2
∞
P
ak ≤
k=n+1
ck ak ≤ 1.
(7.6)
k=2
By setting
½
g1 (z) = cn+1
∞
P
cn+1
ak z k−1
µ
¶¾
1
f (z)
k=n+1
− 1−
=1+
,
n
P
fn (z)
cn+1
1+
ak z k−1
(7.7)
k=2
and applying (7.6), we find that
¯
¯
¯ g1 (z) − 1 ¯
¯
¯
¯ g1 (z) + 1 ¯ ≤
∞
P
cn+1
ak
k=n+1
2−2
n
P
k=2
¯
¯
¯
¯
Now ¯ gg11 (z)−1
(z)+1 ¯ ≤ 1 if
n
P
k=2
(7.8)
ak
k=n+1
∞
P
ak + cn+1
.
∞
P
ak − cn+1
ak ≤ 1.
k=n+1
From condition (2.2), it is sufficient to show that
n
P
ak + cn+1
k=2
∞
P
ak ≤
k=n+1
∞
P
ck ak
k=2
which is equivalent to
n
P
k=2
(ck − 1) ak +
∞
P
(ck − cn+1 ) ak ≥ 0
(7.9)
k=n+1
which readily yields the assertion (7.3) of Theorem 11. In order to see that
f (z) = z +
z n+1
cn+1
iπ
(7.10)
gives sharp result, we observe that for z = r e n that
as z → 1− . Similarly, if we take
½
g2 (z) = (1 + cn+1 )
cn+1
fn (z)
−
f (z)
1 + cn+1
f (z)
fn (z)
=1+
∞
P
(1 + cn+1 )
¾
zn
cn+1
1+
∞
P
k=2
1
cn+1
ak z k−1
k=n+1
=1−
→1−
ak z k−1
, (7.11)
26
M.K. Aouf, A.A. Shamandy, A.O. Mostafa, A.K. Wagdy
and making use of (7.6), we can deduce that
¯
¯
¯ g2 (z) − 1 ¯
¯
¯
¯ g2 (z) + 1 ¯ ≤
∞
P
(1 + cn+1 )
ak
k=n+1
2−2
n
P
∞
P
ak − (1 − cn+1 )
k=2
(7.12)
ak
k=n+1
which leads us immediately to the assertion (7.4) of Theorem 11.
The bound in (7.4) is sharp for each n ∈ N with the extremal function f (z)
given by (7.10). The proof of Theorem 11 is thus completed.
Theorem 12. If f (z) of the form (1.8) satisfies condition (2.2), then
½ 0
¾
f (z)
n+1
Re
≥1−
,
(7.13)
0
fn (z)
cn+1
and
½
¾
fn0 (z)
cn+1
Re
≥
f 0 (z)
n + 1 + cn+1
where ck is defined by (7.2) and satisfies the condition
½
k,
k = 2, 3, . . . , n,
ck ≥ kcn+1
n+1 , k = n + 1, n + 2, . . .
(7.14)
(7.15)
The results are sharp with the function f (z) given by (7.10).
Proof. By setting
g(z) =
µ
¶¾
f 0 (z)
n+1
−
1
−
fn0 (z)
cn+1
∞
n
P
P
n+1
kak z k−1 +
kak z k−1
1 + cn+1
cn+1
n+1
½
k=n+1
=1+
1+
k=2
n
P
,
kak z k−1
k=2
cn+1
n+1
=1+
∞
P
kak z k−1
k=n+1
n
P
1+
,
kak z
(7.16)
k−1
k=2
we obtain
¯
¯
¯ g(z) − 1 ¯
¯
¯
¯ g(z) + 1 ¯ ≤
cn+1
n+1
2−2
n
P
k=2
¯
¯
¯
¯
Now ¯ g(z)−1
g(z)+1 ¯ ≤ 1 if
n
P
k=2
kak +
∞
P
kak
k=n+1
kak −
.
∞
cn+1 P
kak
n + 1 k=n+1
∞
cn+1 P
kak ≤ 1,
n + 1 k=n+1
(7.17)
(7.18)
27
Subclasses of uniformly starlike and convex functions
P∞
since the left-hand side of (7.18) is bounded above by k=2 ck ak if
´
³
n
∞
P
P
n+1
(ck − k) ak +
ck − cn+1
k ak ≥ 0
k=2
(7.19)
k=n+1
and the proof of (7.13) is completed.
To prove result (7.14), define the function g(z) by
³
µ
g(z) =
n + 1 + cn+1
n+1
¶½
cn+1
fn0 (z)
−
f 0 (z)
n + 1 + cn+1
1+
¾
cn+1
n+1
= 1−
1+
´ P
∞
kak z k−1
k=n+1
∞
P
kak z k−1
k=2
and making use of (7.19), we deduce that
³
´ P
∞
n+1
1 + cn+1
kak
¯
¯
¯ g(z) − 1 ¯
k=n+1
¯
¯≤
≤ 1,
³
´ P
n
∞
¯ g(z) + 1 ¯
P
n+1
2−2
kak − 1 − cn+1
kak
k=2
k=n+1
which leads us immediately to the assertion (7.14) of Theorem 12.
Acknowledgement. The authors would like to thank the referees of the
paper for their helpful suggestions.
REFERENCES
[1] M.K. Aouf, A.O. Mostafa, Some properties of a subclass of uniformly convex functions with
negative coefficients, Demonstratio Math. 2 (2008), 353–370.
[2] R. Bharati, R. Parvatham, A. Swaminathan, On subclasses of uniformly convex functions
and corresponding class of starlike functions, Tamakang J. Math. 28 (1997), 17–32.
[3] A. Cătaş, G.I. Oros, G. Oros, Differential subordinations associated with multiplier transformations, Abstract Appl. Anal. (2008), Article ID845724, 11 pages.
[4] J. Dziok, H.M. Srivastava, Classes of analytic functions with the generalized hypergeometric
function, Appl. Math. Comput. 103 (1999), 1–13.
[5] J. Dziok, H.M. Srivastava, Certain subclasses of analytic functions associated with the generalized hypergeometric function, Integral Transform. Spec. Funct. 14 (2003), 7–18.
[6] A.W. Goodman, On uniformly convex functions, Ann. Polon. Math. 56 (1991), 87–92.
[7] A.W. Goodman, On uniformly starlike functions, J. Math. Anal. Appl. 155 (1991), 364–370.
[8] G. Murugusundaramoorthy, N. Magesh, A new subclass of uniformly convex functions and
corresponding subclass of starlike functions with fixed second coefficient, J. Inequal. Pure
Appl. Math. 5:4 (2004), Art. 85, 1–10.
[9] G. Murugusundaramoorthy, N. Magesh, Linear operators associated with a subclass of uniformly convex functions, Internat. J. Pure Appl. Math. Sci. 3 (2006), 113–125.
[10] G. Murugusundaramoorthy, N. Magesh, On certain subclasses of analytic functions associated with hypergeometric functions, Appl. Math Lett. 24 (2011), 494–500.
[11] F. Ronning, On starlike functions associated with parabolic regions, Ann. Univ. MariaeCurie-Sklodowska, Sect. A 45 (1991), 117–122.
[12] F. Ronning, Uniformly convex functions and a corresponding class of starlike functions,
Proc. Amer. Math. Soc. 118 (1993), 189–196.
,
28
M.K. Aouf, A.A. Shamandy, A.O. Mostafa, A.K. Wagdy
[13] T. Rosy, G. Murugusundaramoorthy, Fractional calculus and their applications to certain
subclass of uniformly convex functions, Far East J. Math. Sci. 15 (2004), 231–242.
[14] S. Shams, S.R. Kulkarni, J.M. Jahangiri, Classes of uniformly starlike and convex functions,
Internat. J. Math. Math. Sci. 55 (2004), 2959–2961.
[15] H. Silverman, Univalent functions with negative coefficients, Proc. Amer. Math. Soc. 51
(1975), 109–116.
[16] H. Silverman, Partial sums of starlike and convex functions J. Math. Anal. Appl. 209 (1997),
221–227.
[17] E.M. Silvia, On partial sums of convex functions of order α, Houston J. Math. 11 (1985),
397–404.
[18] H.M. Srivastava, A.A. Attiya, An integral operator associated with the Hurwitz-Lerch Zeta
function and differential subordination, Integral Transform. Spec. Funct. 18 (2007), 207–216.
(received 13.03.2011; in revised form 01.04.2012; available online 01.05.2012)
Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt
E-mail: mkaouf127@yahoo.com, aashamandy@hotmail.com, adelaeg254@yahoo.com,
awagdyfos@yahoo.com