# A Note on Strongly Starlike Mappings in Several Complex Variables

Authors: Hidetaka Hamada; Tatsuhiro Honda; Gabriela Kohr; Kwang Ho Shon
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

Letf be a normalized biholomorphic mapping on the Euclidean unit ball 𝔹
n in ℂ
n and let α
∈
0,1. In this paper, we will show that if f is strongly starlike of order α in the sense of Liczberski and Starkov, then it is also strongly starlike of order α in the sense of Kohr and Liczberski. We also give an example which shows that the converse of the above result does not hold in dimension n
≥
2.


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## Body

## 1. Introduction and Preliminaries

Letℂ
n denote the space of n complex variables z
=
(
z
1
,
…
,
z
n
) with the Euclidean inner product 〈
z
,
w
〉
=
∑
j
=
1
n
‍
z
j
w
¯
jand the norm ∥
z
∥
=
〈
z
,
z
〉
1
/
2. The open unit ball {
z
∈
ℂ
n
:
∥
z
∥
<
1
} is denoted by 𝔹
n. In the case of one complex variable, 𝔹
1 is denoted by U.IfΩ is a domain in ℂ
n, let H
(
Ω
) be the set of holomorphic mappings from Ω to ℂ
n. If Ω is a domain in ℂ
n which contains the origin and f
∈
H
(
Ω
), we say that f is normalized if f
(
0
)
=
0 and D
f
(
0
)
=
I
n, where I
n is the identity matrix.A normalized mappingf
∈
H
(
𝔹
n
) is said to bestarlike if f is biholomorphic on 𝔹
n and t
f
(
𝔹
n
)
⊂
f
(
𝔹
n
) for t
∈
[
0,1
], where the last condition says that the image f
(
𝔹
n
) is a starlike domain with respect to the origin. For a normalized locally biholomorphic mapping f on 𝔹
n, f is starlike if and only if
(1)
ℜ
〈
[
D
f
(
z
)
]
-
1
f
(
z
)
,
z
〉
>
0
,
z
∈
𝔹
n
∖
{
0
}
(see [1–4] and the references therein, cf. [5]).Letα
∈
(
0,1
]. A function f
∈
H
(
U
), normalized by f
(
0
)
=
0 and f
′
(
0
)
=
1, is said to be strongly starlike
of order
α if
(2)
|
arg
z
f
′
(
z
)
f
(
z
)
|
<
α
π
2
,
z
∈
U
.Iff is strongly starlike of order α, then f is also starlike and thus univalent on U. Stankiewicz [6] proved that if α
∈
(
0,1
), then a domain Ω
≠
ℂ which contains the origin is α-accessible if and only if Ω
=
f
(
U
), where U is the unit disc in ℂ and f is a strongly starlike function of order 1
-
α on U. For strongly starlike functions on U, see also Brannan and Kirwan [7], Ma and Minda [8], and Sugawa [9].Kohr and Liczberski [10] introduced the following definition of strongly starlike mappings of order α on 𝔹
n.Definition 1.
Let0
<
α
≤
1. A normalized locally biholomorphic mapping f
∈
H
(
𝔹
n
) is said to be strongly starlike of order α if
(3)
|
arg
⁡
〈
[
D
f
(
z
)
]
-
1
f
(
z
)
,
z
〉
|
<
α
π
2
,
z
∈
𝔹
n
∖
{
0
}
.
Obviously, iff is strongly starlike of order α, then f is also starlike, and if α
=
1 in (3), one obtains the usual notion of starlikeness on the unit ball 𝔹
n.
Using this definition, Hamada and Honda [11], Hamada and Kohr [12], Liczberski [13], and Liu and Li [14] obtained various results for strongly starlike mappings of order α in several complex variables.
Recently, Liczberski and Starkov [15] gave another definition of strongly starlike mappings of order α on the Euclidean unit ball 𝔹
n in ℂ
n, where α
∈
(
0,1
], and proved that a normalized biholomorphic mapping f on 𝔹
n is strongly starlike of order 1
-
α if and only if f
(
𝔹
n
) is an α-accessible domain in ℂ
n for α
∈
(
0,1
). Their definition is as follows.Definition 2.
Let0
<
α
≤
1. A normalized locally biholomorphic mapping f
∈
H
(
𝔹
n
) is said to be strongly starlike of order α
(in the sense of Liczberski and Starkov) if
(4)
ℜ
〈
[
D
f
(
z
)
]
-
1
f
(
z
)
,
z
〉
≥
∥
(
[
D
f
(
z
)
]
-
1
)
*
z
∥
·
∥
f
(
z
)
∥
sin
(
(
1
-
α
)
π
2
)
,
z
∈
𝔹
n
∖
{
0
}
.In the casen
=
1, it is obvious that both notions of strong starlikeness of order α are equivalent. Thus, the following natural question arises in dimension n
≥
2.Question 1.
Letα
∈
(
0,1
). Is there any relation between the above two definitions of strong starlikeness of order α?Letf be a normalized biholomorphic mapping on the Euclidean unit ball 𝔹
n in ℂ
n and let α
∈
(
0,1
). In this paper, we will show that if f is strongly starlike of order α in the sense of Definition 2, then it is also strongly starlike of order α in the sense of Definition 1. As a corollary, the results obtained in [11–14] for strongly starlike mappings of order α in the sense of Definition 1 also hold for strongly starlike mappings of order α in the sense of Definition 2. We also give an example which shows that the converse of the above result does not hold in dimension n
≥
2.


## 2. Main Results

Let∠
(
a
,
b
) denote the angle between a
,
b
∈
ℂ
n
∖
{
0
} regarding a, b as real vectors in ℝ
2
n.Lemma 3.
Leta
,
b
∈
ℂ
n
∖
{
0
} be such that 〈
a
,
b
〉
≠
0. If |
arg
〈
a
,
b
〉
|
≤
π and 0
≤
∠
(
a
,
b
)
<
π
/
2, then
(5)
|
arg
〈
a
,
b
〉
|
≤
∠
(
a
,
b
)
.Proof.
Letθ
=
arg
〈
a
,
b
〉, φ
=
∠
(
a
,
b
). Then we have 〈
a
,
b
〉
=
r
e
i
θ for some r
≥
0 and
(6)
ℜ
〈
a
,
b
〉
=
∥
a
∥
·
∥
b
∥
cos
⁡
φ
=
r
cos
⁡
θ
.
Since cos
⁡
φ
>
0 and r
=
|
〈
a
,
b
〉
|
≤
∥
a
∥
·
∥
b
∥, we have
(7)
cos
⁡
φ
≤
cos
⁡
θ
.
Therefore, we have |
θ
|
≤
φ, as desired.Theorem 4.
Letf be a normalized biholomorphic mapping on the Euclidean unit ball 𝔹
n in ℂ
n and let α
∈
(
0,1
). If f is strongly starlike of order α in the sense of Definition 2, then it is also strongly starlike of order α in the sense of Definition 1.Proof.
Assume thatf is strongly starlike of order α in the sense of Definition 2. Then by (4), we have 〈
[
D
f
(
z
)
]
-
1
f
(
z
)
,
z
〉
≠
0 and
(8)
∠
(
(
[
D
f
(
z
)
]
-
1
)
*
z
,
f
(
z
)
)
≤
α
π
2
,
z
∈
𝔹
n
∖
{
0
}
.
Using Lemma 3, we have
(9)
|
arg
〈
[
D
f
(
z
)
]
-
1
f
(
z
)
,
z
〉
|
=
|
arg
〈
f
(
z
)
,
(
[
D
f
(
z
)
]
-
1
)
*
z
〉
|
≤
∠
(
(
[
D
f
(
z
)
]
-
1
)
*
z
,
f
(
z
)
)
≤
α
π
2
,
z
∈
𝔹
n
∖
{
0
}
.
For fixed z
∈
𝔹
n
∖
{
0
}, let w
=
z
/
∥
z
∥ and
(10)
p
(
ζ
)
=
{
1
ζ
〈
[
D
f
(
ζ
w
)
]
-
1
f
(
ζ
w
)
,
w
〉
,
for
ζ
∈
U
∖
{
0
}
,
1
,
for
ζ
=
0
.
Then p is a holomorphic function on U with |
arg
p
(
ζ
)
|
≤
π
α
/
2 for ζ
∈
U. Since arg
p is a harmonic function on U and arg
p
(
0
)
=
0, by applying the maximum and minimum principles for harmonic functions, we obtain |
arg
p
(
ζ
)
|
<
π
α
/
2 for ζ
∈
U. Thus, we have
(11)
|
arg
〈
[
D
f
(
z
)
]
-
1
f
(
z
)
,
z
〉
|
<
α
π
2
,
z
∈
𝔹
n
∖
{
0
}
.
Hence f is strongly starlike of order α in the sense of Definition 1, as desired.The following example shows that the converse of the above theorem does not hold in dimensionn
≥
2.Example 5.
Forα
∈
(
0,1
), let
(12)
f
(
z
)
=
f
α
(
z
)
=
(
z
1
+
b
z
2
2
,
z
2
)
,
z
=
(
z
1
,
z
2
)
∈
𝔹
2
,
where
(13)
b
=
3
3
2
sin
(
α
π
2
)
.
Then
(14)
D
f
(
z
)
=
[
1
2
b
z
2
0
1
]
,
[
D
f
(
z
)
]
-
1
=
[
1
-
2
b
z
2
0
1
]
.
Therefore,
(15)
〈
[
D
f
(
z
)
]
-
1
f
(
z
)
,
z
〉
=
(
z
1
+
b
z
2
2
-
2
b
z
2
2
)
z
1
¯
+
|
z
2
|
2
=
|
z
1
|
2
+
|
z
2
|
2
-
b
z
1
¯
z
2
2
.
Since |
z
1
z
2
2
|
≤
2
/
(
3
3
), for z
∈
∂
𝔹
2, we obtain that |
b
z
1
z
2
2
|
≤
sin
(
α
π
/
2
)
∥
z
∥
3 for z
∈
𝔹
2. This implies that 〈
[
D
f
(
z
)
]
-
1
f
(
z
)
,
z
〉 lies in the disc of center ∥
z
∥
2 and radius sin
(
α
π
/
2
)
∥
z
∥
2 for each z
∈
𝔹
2
∖
{
0
} and thus
(16)
|
arg
〈
[
D
f
(
z
)
]
-
1
f
(
z
)
,
z
〉
|
<
α
π
2
,
z
∈
𝔹
2
∖
{
0
}
.
Therefore, f
=
f
α is strongly starlike of order α in the sense of Definition 1.
On the other hand,(17)
(
[
D
f
(
z
)
]
-
1
)
*
z
=
(
z
1
,
z
2
-
2
b
z
¯
2
z
1
)
.
So, for z
0
=
(
1
/
3
,
2
/
3
), we have
(18)
〈
[
D
f
(
z
0
)
]
-
1
f
(
z
0
)
,
z
0
〉
=
1
-
m
,
∥
(
[
D
f
(
z
0
)
]
-
1
)
*
z
0
∥
2
=
1
3
+
2
3
(
1
-
3
m
)
2
,
∥
f
(
z
0
)
∥
2
=
1
3
(
1
+
3
m
)
2
+
2
3
,
sin
(
(
1
-
α
)
π
2
)
=
1
-
m
2
,
where
(19)
m
=
sin
(
α
π
2
)
.
Then, we obtain
(20)
∥
(
[
D
f
(
z
0
)
]
-
1
)
*
z
0
∥
2
∥
f
(
z
0
)
∥
2
sin
2
(
(
1
-
α
)
π
2
)
-
(
ℜ
〈
[
D
f
(
z
0
)
]
-
1
f
(
z
0
)
,
z
0
〉
)
2
=
(
1
-
m
)
{
[
1
3
+
2
3
(
1
-
3
m
)
2
]
[
1
3
(
1
+
3
m
)
2
+
2
3
]
×
(
1
+
m
)
-
(
1
-
m
)
1
3
}
.
Since
(21)
[
1
3
+
2
3
(
1
-
3
m
)
2
]
[
1
3
(
1
+
3
m
)
2
+
2
3
]
(
1
+
m
)
-
(
1
-
m
)
is increasing on [
1
/
3,1
] and positive for m
=
1
/
3, we have
(22)
ℜ
〈
[
D
f
(
z
0
)
]
-
1
f
(
z
0
)
,
z
0
〉
<
∥
(
[
D
f
(
z
0
)
]
-
1
)
*
z
0
∥
×
∥
f
(
z
0
)
∥
sin
(
(
1
-
α
)
π
2
)
for m
∈
[
1
/
3,1
).
On the other hand, forz
~
0
=
(
i
/
3
,
2
/
3
), we have
(23)
〈
[
D
f
(
z
~
0
)
]
-
1
f
(
z
~
0
)
,
z
~
0
〉
=
1
+
m
i
,
∥
(
[
D
f
(
z
~
0
)
]
-
1
)
*
z
~
0
∥
2
=
1
3
+
2
3
|
1
-
3
m
i
|
2
=
6
m
2
+
1
,
∥
f
(
z
~
0
)
∥
2
=
1
3
|
i
+
3
m
|
2
+
2
3
=
3
m
2
+
1
.
Then, we obtain
(24)
∥
(
[
D
f
(
z
~
0
)
]
-
1
)
*
z
~
0
∥
2
∥
f
(
z
~
0
)
∥
2
sin
2
(
(
1
-
α
)
π
2
)
-
(
ℜ
〈
[
D
f
(
z
~
0
)
]
-
1
f
(
z
~
0
)
,
z
~
0
〉
)
2
=
(
6
m
2
+
1
)
(
3
m
2
+
1
)
(
1
-
m
2
)
-
1
=
m
2
(
-
18
m
4
+
9
m
2
+
8
)
.
Since -
18
m
4
+
9
m
2
+
8 is positive for m
∈
[
0,1
/
3
], we have
(25)
ℜ
〈
[
D
f
(
z
~
0
)
]
-
1
f
(
z
~
0
)
,
z
~
0
〉
<
∥
(
[
D
f
(
z
~
0
)
]
-
1
)
*
z
~
0
∥
×
∥
f
(
z
~
0
)
∥
sin
(
(
1
-
α
)
π
2
)
for m
∈
(
0,1
/
3
].
Thus,f
=
f
α is not strongly starlike of order α in the sense of Definition 2 for α
∈
(
0,1
).



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*Source: 265718-2014-03-03.xml*