1. Introduction and Preliminaries
Let
ℂ
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
¯
j
and 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
.
If
Ω
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 mapping
f
∈
H
(
𝔹
n
)
is said to be starlike 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]).
Let
α
∈
(
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
.
If
f
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.
Let
0
<
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, if
f
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.
Let
0
<
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 case
n
=
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.
Let
α
∈
(
0,1
)
. Is there any relation between the above two definitions of strong starlikeness of order
α
?
Let
f
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.
Let
a
,
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.
Let
θ
=
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.
Let
f
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 that
f
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 dimension
n
≥
2
.
Example 5.
For
α
∈
(
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, for
z
~
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
)
.