General Mathematics Vol. 15, No. 4 (2007), 107-119
Some Subclasses of Close-to-Convex and
Quasi-Convex Functions with Respect to
k-Symmetric Points1
Zhi-Gang Wang, Chun-Yi Gao, Halit Orhan
and Sezgin Akbulut
Abstract
In the present paper, the authors introduce two new subclasses
C (λ, α) of close-to-convex functions and QC(k) (λ, α) of quasi-convex
functions with respect to k-symmetric points. The integral representations and convolution conditions for these classes are provided.
Some coefficient inequalities for functions belonging to these classes
and their subclasses with negative coefficients are also provided.
(k)
2000 Mathematical Subject Classification: 30C45.
Key words: Close-to-convex functions, quasi-convex functions,
k-symmetric points, Hadamard product.
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 17 July, 2007
Accepted for publication (in revised form) 20 December, 2007
107
108
Zhi-Gang Wang and Chun-Yi Gao
which are analytic in the open unit disk U = {z ∈ C : |z| < 1}. Let S
denotes the subclass of A consisting of all functions which are univalent in
U. Also let T(n, p) denote the class of functions of the form
p
f (z) = z −
∞
X
al+p z l+p (al+p ≥ 0; p, n ∈ N = {1, 2, 3, . . .}),
l=n
which are analytic in U. Write T(1, 1) simple as T.
We denote by S∗ , K, C and QC the familiar subclasses of A consisting
of functions which are, respectively, starlike, convex, close-to-convex and
quasi-convex in U. Thus, by definition, we have (see, for details, [4, 6, 7, 9])
½
½ ′ ¾
¾
zf (z)
∗
S = f : f ∈ A and ℜ
> 0 (z ∈ U) ,
f (z)
¾
¾
½
½
zf ′′ (z)
> 0 (z ∈ U) ,
K = f : f ∈ A and ℜ 1 + ′
f (z)
½
½ ′ ¾
¾
zf (z)
∗
C = f : f ∈ A, ∃g ∈ S , such that ℜ
> 0 (z ∈ U) ,
g(z)
and
¾
¾
½
½
(zf ′ (z))′
> 0 (z ∈ U) .
QC = f : f ∈ A, ∃g ∈ K, such that ℜ
g ′ (z)
Let T(n, p, λ, α) be the subclass of T(n, p) consisting of functions f (z)
which satisfy the inequality
½
¾
zf ′ (z) + λz 2 f ′′ (z)
ℜ
> α (z ∈ U)
(1 − λ)f (z) + λzf ′ (z)
for some α (0 ≤ α < 1) and λ (0 ≤ λ ≤ 1). Altintas [1] once introduced
and investigated the class T(n, 1, λ, α). In a later paper, Altintas, Irmak
and Srivastava [2] derived some other interesting properties of the class
T(n, p, λ, α). Write T(1, 1, λ, α) simple as T(λ, α).
Let C(n, λ, α) be the subclass of T(n, 1) consisting of functions f (z)
which satisfy the inequality
¾
½
λz 2 f ′′′ (z) + (2λ + 1)zf ′′ (z) + f ′ (z)
> α (z ∈ U)
ℜ z
λz 2 f ′′ (z) + zf ′ (z)
109
Some Subclasses of Close-to-Convex and...
for some α (0 ≤ α < 1) and λ (0 ≤ λ ≤ 1). The class C(n, λ, α) was
introduced and investigated recently by Kamali and Akbulut [5]. Write
C(1, λ, α) simple as C(λ, α).
Sakaguchi [8] once introduced a class S∗s of functions starlike with respect
to symmetric points, it consists of functions f (z) ∈ S satisfying
¾
½
zf ′ (z)
> 0 (z ∈ U).
ℜ
f (z) − f (−z)
Following him, many authors discussed this class and its subclasses. And a
function f (z) ∈ A is in the class Cs if and only if zf ′ (z) ∈ S∗s .
(k)
Let Ss (α) denote the class of functions in S satisfying the following
inequality
½ ′ ¾
zf (z)
ℜ
> α (z ∈ U),
fk (z)
where 0 ≤ α < 1, k ≥ 2 is a fixed positive integer and fk (z) is defined by
the following equality
k−1
(1.2)
1 X −ν
ε f (εν z) (ε = exp(2πi/k); z ∈ U).
fk (z) =
k ν=0
(k)
(k)
And a function f (z) ∈ A is in the class Cs (α) if and only if zf ′ (z) ∈ Ss (α).
(k)
The class Ss (α) of functions starlike with respect to k-symmetric points
(k)
of order α was studied by Chand and Singh [3], and the class Cs (α) of
functions convex with respect to k-symmetric points of order α is a corresponding special class defined in [10].
(k)
(k)
Motivated by the classes T(λ, α), C(λ, α), Ss (α) and Cs (α), we now
introduce and investigate the following subclasses of A with respect to ksymmetric points, and obtain some interesting results.
Definition 1. Let C(k) (λ, α) denote the class of functions in A satisfying
the following inequality
½
¾
zf ′ (z) + λz 2 f ′′ (z)
(1.3)
ℜ
> α (z ∈ U),
(1 − λ)fk (z) + λzfk′ (z)
110
Zhi-Gang Wang and Chun-Yi Gao
where 0 ≤ α < 1, 0 ≤ λ ≤ 1, k ≥ 2 is a fixed positive integer and fk (z) is
defined by equality (1.2).
Definition 2. Let QC(k) (λ, α) denote the class of functions in A satisfying
the following inequality
¾
½
λz 2 f ′′′ (z) + (2λ + 1)zf ′′ (z) + f ′ (z)
> α (z ∈ U),
ℜ z
λz 2 fk′′ (z) + zfk′ (z)
where 0 ≤ α < 1, 0 ≤ λ ≤ 1, k ≥ 2 is a fixed positive integer and fk (z) is
defined by equality (1.2).
(k)
For convenience, we write C(k) (λ, α) ∩ T simple as CT (λ, α) , and
(k)
QC(k) (λ, α) ∩ T simple as QCT (λ, α) .
In our proposed investigation of functions in the classes C(k) (λ, α) and
QC(k) (λ, α), we shall also make use of the following lemmas.
Lemma 1. Let γ ≥ 0 and f ∈ C, then
Z
1+γ z
F (z) =
f (t)tγ−1 dt ∈ C.
zγ
0
This lemma is a special case of Theorem 4 in [11].
Lemma 2 [6]. Let 0 < λ ≤ 1 and f ∈ QC, then
Z
1
1 1− 1 z
F (z) = z λ
f (t)t λ −2 dt ∈ QC ⊂ C.
λ
0
Lemma 3. C(k) (λ, α) ⊂ C ⊂ S.
Proof. Let F (z) = (1 − λ)f (z) + λzf ′ (z), Fk (z) = (1 − λ)fk (z) + λzfk′ (z)
with f (z) ∈ C(k) (λ, α), substituting z by εµ z in (1.1) (µ = 0, 1, 2, . . . , k − 1),
we get
¾
½ µ ′ µ
ε zf (ε z) + λ(εµ z)2 f ′′ (εµ z)
> α (z ∈ U).
(1.4)
ℜ
(1 − λ)fk (εµ z) + λεµ zfk′ (εµ z)
Note that fk (εµ z) = εµ fk (z) and fk′ (εµ z) = fk′ (z), thus, inequality (1.4) can
be written as
¾
½ ′ µ
zf (ε z) + λz 2 εµ f ′′ (εµ z)
> α (z ∈ U).
(1.5)
ℜ
(1 − λ)fk (z) + λzfk′ (z)
Some Subclasses of Close-to-Convex and...
111
Letting µ = 0, 1, 2, . . . , k − 1 in (1.5), respectively, and summing them we
can obtain
( k−1
)
1 X zf ′ (εµ z) + λz 2 εµ f ′′ (εµ z)
ℜ
> α (z ∈ U),
k µ=0 (1 − λ)fk (z) + λzfk′ (z)
or equivalently,
¾
½ ′ ¾
½
zFk (z)
zfk′ (z) + λz 2 fk′′ (z)
=ℜ
> α (z ∈ U),
ℜ
′
(1 − λ)fk (z) + λzfk (z)
Fk (z)
that is Fk (z) ∈ S∗ (α), which is the usual class of starlike functions of order α
in U. Note that S∗ (0) = S∗ , this implies that F (z) = (1−λ)f (z)+λzf ′ (z) ∈
C. We now split it into two cases to prove.
Case 1. When λ = 0. It is obvious that f (z) = F (z) ∈ C.
Case 2. When 0 < λ ≤ 1. From F (z) = (1 − λ)f (z) + λzf ′ (z) and
0 < λ ≤ 1, we have
Z
1
1 1− 1 z
F (t)t λ −2 dt.
f (z) = z λ
λ
0
Since γ = λ1 − 1 ≥ 0, by Lemma 1, we obtain that f (z) ∈ C. Hence
C(k) (λ, α) ⊂ C ⊂ S, and the proof of Lemma 3 is complete.
By means of Lemma 2, using the similar method as in Lemma 3, we
may prove the following Lemma.
Lemma 4. QC(k) (λ, α) ⊂ QC ⊂ C.
In the present paper, we shall provide the integral representations and
convolution conditions for the classes C(k) (λ, α) and QC(k) (λ, α), we shall
also provide some coefficient inequalities for functions belonging to these
classes and their subclasses with negative coefficients.
2
Integral Representations
At first, we give the integral representations of functions belonging to the
classes C(k) (λ, α) and QC(k) (λ, α).
112
Zhi-Gang Wang and Chun-Yi Gao
Theorem 1. Let f (z) ∈ C(k) (λ, α) with 0 < λ ≤ 1, then we have
)
( k−1 Z µ
Z
1
1 1− 1 z
1 X ε u 2(1 − α)ω(t)
dt u λ −1 du,
(2.1)
fk (z) = z λ
exp
λ
k µ=0 0
t(1 − ω(t))
0
where fk (z) is defined by equality (1.2), ω(z) is analytic in U and ω(0) = 0,
|ω(z)| < 1.
Proof. Suppose that f (z) ∈ C(k) (λ, α), it is easy to know that the condition
(1.3) can be written as
1 + (1 − 2α)z
zf ′ (z) + λz 2 f ′′ (z)
≺
,
′
(1 − λ)fk (z) + λzfk (z)
1−z
where ” ≺ ” stands for the usual subordination, it follows that
zf ′ (z) + λz 2 f ′′ (z)
1 + (1 − 2α)ω(z)
=
,
′
(1 − λ)fk (z) + λzfk (z)
1 − ω(z)
(2.2)
where ω(z) is analytic in U and ω(0) = 0, |ω(z)| < 1. By applying the
similar method as in Lemma 3 to equality (2.2), we can obtain
k−1
(1 − λ)zfk′ (z) + λz(zfk′ (z))′
1 X 1 + (1 − 2α)ω(εµ z)
=
,
(1 − λ)fk (z) + λzfk′ (z)
k µ=0
1 − ω(εµ z)
(2.3)
from equality (2.3), we get
k−1
(1 − λ)fk′ (z) + λ(zfk′ (z))′ 1
1 X 2(1 − α)ω(εµ z)
−
=
.
(1 − λ)fk (z) + λzfk′ (z)
z
k µ=0 z(1 − ω(εµ z))
(2.4)
Integrating equality (2.4), we have
log
½
(1 − λ)fk (z) + λzfk′ (z)
z
¾
k−1
1X
=
k µ=0
Z
z
1X
=
k µ=0
Z
εµ z
k−1
0
0
2(1 − α)ω(εµ ζ)
dζ
ζ(1 − ω(εµ ζ))
2(1 − α)ω(t)
dt,
t(1 − ω(t))
113
Some Subclasses of Close-to-Convex and...
that is,
(2.5)
(1 − λ)fk (z) + λzfk′ (z) = z · exp
(
k−1
1X
k µ=0
Z
εµ z
0
)
2(1 − α)ω(t)
dt .
t(1 − ω(t))
From equality (2.5), we can get equality (2.1) easily. Hence the proof of
Theorem 1 is complete.
Theorem 2. Let f (z) ∈ C(k) (λ, α) with 0 < λ ≤ 1, then we have
1
1
f (z) = z 1− λ
λ
(2.6)
Z
0
·
z
Z
u
exp
0
(
k−1
1X
k µ=0
Z
0
εµ ζ
)
2(1 − α)ω(t)
dt
t(1 − ω(t))
1
1 + (1 − 2α)ω(ζ)
dζu λ −2 du,
1 − ω(ζ)
where ω(z) is analytic in U and ω(0) = 0, |ω(z)| < 1.
Proof. Suppose that f (z) ∈ C(k) (λ, α), from equalities (2.2) and (2.5), we
can get
(1 − λ)fk (z) + λzfk′ (z) 1 + (1 − 2α)ω(z)
·
(1 − λ)f (z) + λ(zf (z)) =
z
1 − ω(z)
)
( k−1 Z µ
ε
z
2(1 − α)ω(t)
1 + (1 − 2α)ω(z)
1X
dt ·
.
= exp
k µ=0 0
t(1 − ω(t))
1 − ω(z)
′
′
′
Integrating the above equality, we can get equality (2.6) easily.
Similarly, for the class QC(k) (λ, α), we have
Corollary 1. Let f (z) ∈ QC(k) (λ, α) with 0 < λ ≤ 1, then we have
1
1
fk (z) = z 1− λ
λ
Z
0
z
Z
0
u
exp
(
k−1
1X
k µ=0
Z
0
εµ ζ
)
1
2(1 − α)ω(t)
dt dζu λ −2 du,
t(1 − ω(t))
where fk (z) is defined by equality (1.2), ω(z) is analytic in U and ω(0) = 0,
|ω(z)| < 1.
114
Zhi-Gang Wang and Chun-Yi Gao
Corollary 2. Let f (z) ∈ QC(k) (λ, α) with 0 < λ ≤ 1, then we have
)
( k−1 Z µ
Z Z
Z
1 1− 1 z u 1 ξ
1 X ε ζ 2(1 − α)ω(t)
f (z) = z λ
dt
exp
λ
k µ=0 0
t(1 − ω(t))
0
0 ξ 0
·
1
1 + (1 − 2α)ω(ζ)
dζdξu λ −2 du,
1 − ω(ζ)
where ω(z) is analytic in U and ω(0) = 0, |ω(z)| < 1.
3
Convolution Conditions
In this section, we give the convolution conditions for the classes C(k) (λ, α)
and QC(k) (λ, α). Let f, g ∈ A, where f (z) is given by (1.1) and g(z) is
defined by
∞
X
g(z) = z +
bn z n ,
n=2
then the Hadamard product (or convolution) f ∗ g is defined (as usual) by
(f ∗ g)(z) = z +
∞
X
an bn z n = (g ∗ f )(z).
n=2
Theorem 3. A function f (z) ∈ C(k) (λ, α) if and only if
(
(
½
¾
1
z
iθ
iθ
f ∗ (1 − λ)
(1 − e ) − [1 + (1 − 2α)e ]h
z
(1 − z)2
(3.1)
+λz
½
¾′ ) )
z
(1 − eiθ ) − [1 + (1 − 2α)eiθ ]h
(z) 6= 0
(1 − z)2
for all z ∈ U and 0 ≤ θ < 2π, where h(z) is given by (3.6).
Proof. Suppose that f (z) ∈ C(k) (λ, α), since (1.3) is equivalent to
(3.2)
1 + (1 − 2α)eiθ
zf ′ (z) + λz 2 f ′′ (z)
=
6
(1 − λ)fk (z) + λzfk′ (z)
1 − eiθ
Some Subclasses of Close-to-Convex and...
115
for all z ∈ U and 0 ≤ θ < 2π. And the condition (3.2) can be written as
1
{[(1 − λ)zf ′ (z) + λz(zf ′ (z))′ ](1 − eiθ ) − [(1 − λ)fk (z) + λzfk′ (z)]
z
[1 + (1 − 2α)eiθ ]} 6= 0.
(3.3)
On the other hand, it is well known that
zf ′ (z) = f (z) ∗
(3.4)
z
.
(1 − z)2
And from the definition of fk (z), we know
(3.5)
fk (z) = (f ∗ h)(z),
where
k−1
z
1X
.
h(z) =
k υ=0 1 − ευ z
(3.6)
Substituting (3.4) and (3.5) into (3.3), we can get (3.1) easily. This completes the proof of Theorem 3.
Similarly, for the class QC(k) (λ, α), we have
Corollary 3. A function f (z) ∈ QC(k) (λ, α) if and only if
(
( (
½
¾
1
z
iθ
iθ
f ∗ z (1 − λ)
(1 − e ) − [1 + (1 − 2α)e ]h
z
(1 − z)2
+λz
½
¾′ )′ ) )
z
(z) 6= 0
(1 − eiθ ) − [1 + (1 − 2α)eiθ ]h
2
(1 − z)
for all z ∈ U and 0 ≤ θ < 2π, where h(z) is given by (3.6).
4
Coefficient Inequalities
In this section, we first provide the sufficient conditions for functions belonging to the classes C(k) (λ, α) and QC(k) (λ, α).
116
Zhi-Gang Wang and Chun-Yi Gao
Theorem 4. Let 0 ≤ α < 1 and 0 ≤ λ < 1. If
(4.1)
∞
X
(1 + λnk)(nk + 1 − α)|ank+1 | +
n=1
∞
X
[1 + λ(n − 1)]n|an | ≤ 1 − α,
n=2
n6=lk+1
then f (z) ∈ C(k) (λ, α).
Proof. It suffices to show that
|
zf ′ (z) + λz 2 f ′′ (z)
− 1| < 1 − α.
(1 − λ)fk (z) + λzfk′ (z)
Note that for |z| = r < 1, we have
P∞
[1 + λ(n − 1)](n − bn )an z n−1
zf ′ (z) + λz 2 f ′′ (z)
n=2P
−
1|
=
|
|
n−1
(1 − λ)fk (z) + λzfk′ (z)
1− ∞
n=2 [λn + (1 − λ)]bn an z
P∞
[1 + λ(n − 1)](n − bn )|an |
n=2P
.
≤
1− ∞
n=2 [λn + (1 − λ)]bn |an |
where
(4.2)
bn =
1
k
k−1
X
ν=0
ε(n−1)ν =


 1, n = lk + 1,


0, n 6= lk + 1.
This last expression is bounded above by 1 − α if
(4.3)
∞
X
[1 + λ(n − 1)](n − αbn )|an | ≤ 1 − α.
n=2
Since inequality (4.3) can be written as inequality (4.1), hence f (z) satisfies
the condition (1.3). This completes the proof of Theorem 4.
Similarly, for the class QC(k) (λ, α), we have
Corollary 4. Let 0 ≤ α < 1 and 0 ≤ λ < 1. If
∞
X
n=1
(nk + 1)(1 + λnk)(nk + 1 − α)|ank+1 | +
∞
X
n=2
[1 + λ(n − 1)]n2 |an | ≤ 1 − α,
n6=lk+1
117
Some Subclasses of Close-to-Convex and...
then f (z) ∈ QC(k) (λ, α).
We now provide the necessary and sufficient coefficient conditions for
(k)
(k)
functions belonging to the classes CT (λ, α) and QCT (λ, α).
Theorem 5. Let 0 ≤ α < 1, 0 ≤ λ < 1 and f (z) ∈ T, then f (z) ∈
(k)
CT (λ, α) if and only if
(4.4)
∞
X
(1 + λnk)(nk + 1 − α)ank+1 +
n=1
∞
X
[1 + λ(n − 1)]nan ≤ 1 − α.
n=2
n6=lk+1
Proof. In view of Theorem 4, we need only to prove the necessity. Suppose
(k)
that f (z) ∈ CT (λ, α), then from (1.3), we can get


∞
∞
X
X


1 −


n(n − 1)an z n−1 
nan z n−1 − λ




n=2
n=2
(4.5)
ℜ
> α,
∞
X




n−1


[λn + (1 − λ)]bn an z
1−




n=2
where bn is given by (4.2). By letting |z| = r → 1− through real values in
(4.5), we can get
1−
∞
X
nan − λ
n=2
∞
X
n(n − 1)an
n=2
∞
X
1−
[λn + (1 − λ)]bn an
≥ α,
n=2
or equivalently,
(4.6)
∞
X
[1 + λ(n − 1)](n − αbn )an ≤ 1 − α.
n=2
Substituting (4.2) into inequality (4.6), we can get inequality (4.4) easily.
This completes the proof of Theorem 5.
(k)
Similarly, for the class QCT (λ, α), we have
118
Zhi-Gang Wang and Chun-Yi Gao
Corollary 5. Let 0 ≤ α < 1, 0 ≤ λ < 1 and f (z) ∈ T, then f (z) ∈
(k)
QCT (λ, α) if and only if
∞
X
(nk + 1)(1 + λnk)(nk + 1 − α)|ank+1 | +
n=1
∞
X
[1 + λ(n − 1)]n2 |an | ≤ 1 − α.
n=2
n6=lk+1
Acknowledgements. This work was supported by the Scientific
Research Fund of Hunan Provincial Education Department and the Hunan
Provincial Natural Science Foundation (No. 05JJ30013) of People’s Republic of China. The authors would like to thank Professor Ming-Sheng Liu for
his helpful suggestions in the preparation of this paper.
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Zhi-Gang Wang and Chun-Yi Gao
School of Mathematics and Computing Science
Changsha University of Science and Technology,
Changsha,
410076 Hunan,
People’s Republic of China
E-Mail: zhigwang@163.com
Halit Orhan and Sezgin Akbulut
Department of Mathematics,
Faculty of Science and Arts,
Atatürk University,
25240 Erzurum,
Turkey