General Mathematics Vol. 17, No. 4 (2009), 05–12
Notes on radius problems of certain
univalent functions
Hiro Kobashi, Kazuo Kuroki, Shigeyoshi Owa
Abstract
For analytic functions f (z) normalized by f (0) = f ′ (0) − 1 = 0
in the open unit disk U, a class P(β1 , β2 ; λ) of f (z) defined by some
conditions with some complex numbers β1 and β2 is introduced. The
object of the present paper is to consider some radius problems of
1
f (αz) for f (z) ∈ S.
α
2000 Mathematics Subject Classification: Primary 30C45.
Key words and phrases: Analytic, univalent, Cauchy-Schwarz
inequality.
1
Introduction
Let A be the class of functions f (z) of the form
(1.1)
f (z) = z +
∞
X
n=2
5
an z n
6
Hiro Kobashi, Kazuo Kuroki, Shigeyoshi Owa
which are analytic in the open unit disk U = {z∈C : |z| < 1}. Let S be the
subclass of A consisting of all univalent functions f (z) in U. For f (z)∈A,
f (z)
6= 0 (z∈U) and
we say that f (z)∈P(β1 , β2 ; λ) if f (z) satisfies
z
(1.2)
′′
′′′ z
z
β1
≦λ
+
β
2
f (z)
f (z) (z∈U)
for some complex numbers β1 and β2 , and for some real λ > 0. Obradović
and Ponnusamy [2] have studied the subclass P2 (λ) of A consisting of f (z)
f (z)
6= 0 (z∈U) and
satisfying
z
(1.3)
for some real λ > 0.
′′ z
≦λ
f (z) (z∈U)
Let us consider a function f (z) given by
(1.4)
f (z) =
z
(1 − z)δ
(δ ≧ 0).
Then, we see that
(1.5)
(1.6)
and
f (z)
1
=
6= 0 (z∈U),
z
(1 − z)δ
′′ z
δ−2 = δ(δ − 1)(1 − z) < δ(δ − 1)2δ−2
f (z) (1.7)
′′′ z
δ−3 = δ(δ − 1)(δ − 2)(1 − z) < δ(δ − 1)(δ − 2)2δ−3
f (z) (δ ≧ 2)
(δ ≧ 3) .
Notes on radius problems of certain univalent functions
Therefore, Koebe function f (z) =
and P(0, 1; λ) for any λ > 0.
7
z
belongs to the class P(1, 0; 2)
(1 − z)2
If we consider the function f (z) defined by
z
f (z) = P
n
,
zk
k=0
then
′′
′′′ n
n
X
X
z
z
β1
+ β2
< |β1 |
k(k − 1) + |β2 |
k(k − 1)(k − 2)
f (z)
f (z) k=2
k=3
=
n(n + 1)(n − 1)(4|β1 | + 3(n − 2)|β2 |)
.
12
This means that f (z) ∈ P(β1 , β2 ; λ) with
λ=
2
n(n + 1)(n − 1)(4|β1 | + 3(n − 2)|β2 |)
.
12
Main results
To consider our problem for the class P(β1 , β2 ; λ), we need the following
lemma due to Goodman [1].
Lemma 1 If f (z) ∈ S and
(2.1)
∞
X
z
=1+
bn z n ,
f (z)
n=2
then we have
(2.2)
∞
X
n=2
2
(n − 1)bn ≦ 1.
8
Hiro Kobashi, Kazuo Kuroki, Shigeyoshi Owa
Further, we need the following lemma.
Lemma 2 Let f (z) ∈ A and
f (z) satisfies
(2.3)
2|β1 ||b2 | +
∞
X
n=3
∞
P
z
= 1+
bn z n 6= 0
f (z)
n=1
(z∈U). If
n(n − 1)(|β1 | + (n − 2)|β2 |)|bn | ≦ λ,
for some complex numbers β1 and β2 , then f (z)∈P(β1 , β2 ; λ).
Proof. We note that
(2.4)
′′
′′′ ∞
X
z
z
β1
< 2|β1 ||b2 | +
+ β2
n(n − 1)(|β1 | + (n − 2)|β2 |)|bn |.
f (z)
f (z) n=3
Thus, if f (z) satisfies the inequality (2.3), then f (z)∈P(β1 , β2 ; λ).
Now, we derive
Theorem 1 Let f (z) ∈ S and α ∈ C (|α| < 1). Then the func1
tion f (αz) belongs to the class P(β1 , β2 ; λ) for 0 < |α| ≦ |α0 (λ)|, where
α
|α0 | = |α0 (λ)| is the smallest root of the equation
Notes on radius problems of certain univalent functions
9
(2.5)
q
q
12
3
|α| 2
|α| 6 3|α|4 + 14|α|2 + 3 +2
|β1 |
=λ
+ |β2 |
1 − |b2 |2
2
3
1 − |α|2
1 − |α|2
2
|α|2
in 0 < |α| < 1.
Proof. Since
z
6= 0 (z∈U) for f (z) ∈ S, if we write
f (z)
∞
X
z
=1+
bn z n ,
f (z)
n=1
then
z
(2.6)
1
f (αz)
α
=1+
∞
X
α n bn z n
n=1
for 0 < |α| < 1.
1
To show that f (αz)∈P(β1 , β2 ; λ), we have to prove that
α
(2.7)
|β1 |
∞
X
n=2
∞
X
n! n n! n α bn + |β2 |
α bn ≦ λ
(n − 2)!
(n
−
3)!
n=3
which is equivalent to (2.3) by means of Lemma 2. Indeed, applying the
Cauchy-Schwarz inequality for the left hand of (2.7), we obtain that
∞
∞
P
P
n! n n! n α bn + |β2 |
α bn = |β1 |
n2 (n −
n=3 (n − 3)!
n=2
n=2 (n − 2)!
∞
1
1
1
P
(n − 1)|bn |2 2 + |β2 |
n2 (n − 1)(n − 2)2 |α|2n 2 (n − 1)|bn |2 2
(2.8) |β1 |
1)|α|2n
12
∞
P
n=3
10
Hiro Kobashi, Kazuo Kuroki, Shigeyoshi Owa
∞
21 ∞
12
P 2
P
2n
2
≦ |β1 |
n (n − 1)|α|
(n − 1)|bn |
n=2
n=2
X
21 X
12
∞
∞
2
2
2n
2
+|β2 |
n (n − 1)(n − 2) |α|
(n − 1)|bn |
n=3
n=3
21
∞
12 ∞
21
P
P 2
1−|b2 |2
n2 (n−1)(n−2)2 |α|2n
≦ |β1 |
n (n−1)|α|2n +|β2 |
n=3
n=2
q
q
21
|α|2 2 |α|2 + 2
|α|3 6 3|α|4 + 14|α|2 + 3 2
= |β1 |
.
+ |β2 |
1 − |b2 |
2
3
1 − |α|2
1 − |α|2
Now, we consider the complex number α 0 < |α| < 1 such that
(2.9)
|β1 |
|α|2
q
q
21
2 |α|2 + 2
|α|3 6 3|α|4 + 14|α|2 + 3 2
= λ.
+ |β2 |
1 − |b2 |
2
3
1 − |α|2
1 − |α|2
This give that
p
h(|α|) = −λ|α|6 + 3λ + |β1 | 2(|α|2 + 2) |α|4
p
p
3
4
2
2
2
−|β2 | 6(3|α| + 14|α| + 3)(1 − |b2 | )|α| − 3λ + |β1 | 2(|α| + 2) |α|2 +λ = 0.
p
Noting that h(0) = λ > 0 and h(1) = −2 30(1 − |b2 |2 )|β2 | < 0, h(|α|) = 0
has a root |α0 | = |α0 (λ)| in 0 < |α| < 1. This completes the proof of the
theorem.
Remark 1
In the proof of Theorem 1, we calculate
X
∞
n=2
n2 (n − 1)|α|2n
21
=
|α|2
q
2 2 + |α|2
2
1 − |α|2
11
Notes on radius problems of certain univalent functions
as follows. Note that
∞
X
n=2
n
2
2
n (n − 1)t = t
2
=t
Letting t = |α|2 , we have
X
∞
n=2
X
∞
nt
n
n=2
t3 − 3t2 + 4t
(1 − t)3
n2 (n − 1)|α|2n
12
′
′′
=t
=
2t2 (t + 2)
.
(1 − t)4
2
=
|α|2
=
|α|3
−t3 + 2t2
(1 − t)2
′′
q
2 2 + |α|2
.
2
1 − |α|2
Further, we prove
X
∞
n=2
n2 (n − 1)(n − 2)2 |α|2n
21
as follows. Note that
∞
X
n=2
2 n
2
3
n (n − 1)(n − 2) t = t
= t3
X
∞
n=3
q
6 3|α|4 + 14|α|2 + 3
3
1 − |α|2
n
n(n − 2)t
′′′
3
=t
−t4 + 3t3
(1 − t)3
′′′
′′
2
′
2
t4 − 4t3 + 9t2
3 6t + 18t
3 3t + 14t + 3
=
t
=
6t
.
(1 − t)4
(1 − t)5
(1 − t)6
Letting t = |α|2 , we have
X
∞
n=2
n2 (n − 1)(n − 2)2 |α|2n
Remark 2
12
=
|α|3
q
6 3|α|4 + 14|α|2 + 3
.
3
1 − |α|2
1
If we take α = eiθ in (2.5), then we have
2
√
√
p
2 642
2 2
|β1 | +
|β2 | 1 − |b2 |2 .
λ=
3
27
12
Hiro Kobashi, Kazuo Kuroki, Shigeyoshi Owa
1
in (2.5), then we have
2
√
p
p
3
3
3
2
2
(1−|α|) |α| 2(|α|2 + 2)+|α| 6(3|α|4 + 14|α|2 + 3)
− 1 − |α|2 = 0.
2
If we put λ = |β1 | = |β2 | = 1 and |b2 | =
It is easy to see that the above equation has a root |α0 | such that 0.3999 <
|α0 | < 0.4002.
References
[1] A. W. Goodman, Univalent Functions, Vol.I and II, Mariner, Tampa,
Florida, 1983.
[2] M. Obradovć and S. Ponnusamy, Radius properties for subclasses of
univalent functions, Analysis, 25(2005), 183-188.
Hiro Kobashi, Kazuo Kuroki, Shigeyoshi Owa
Kinki University
Department of Mathematics
Higashi-Osaka, Osaka 577-8502, Japan
E-mails: hero of earth oo1@hotmail.com
freedom@sakai.zaq.ne.jp
owa@math.kindai.ac.jp