In this paper, we will show that if? is strongly starlike of order ? in the sense of Liczberski and Starkov, then it is also strongly starlike of order ? in the sense of Kohr and Liczber
Trang 1Research Article
A Note on Strongly Starlike Mappings in Several
Complex Variables
Hidetaka Hamada,1Tatsuhiro Honda,2Gabriela Kohr,3and Kwang Ho Shon4
1 Faculty of Engineering, Kyushu Sangyo University, Fukuoka 813-8503, Japan
2 Hiroshima Institute of Technology, Hiroshima 731-5193, Japan
3 Faculty of Mathematics and Computer Science, Babes¸-Bolyai University, 1 M Kog˘alniceanu Street, 400084 Cluj-Napoca, Romania
4 Department of Mathematics, College of Natural Sciences, Pusan National University, Busan 609-735, Republic of Korea
Correspondence should be addressed to Kwang Ho Shon; khshon@pusan.ac.kr
Received 3 December 2013; Accepted 27 January 2014; Published 3 March 2014
Academic Editor: Junesang Choi
Copyright © 2014 Hidetaka Hamada et al This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited
Let𝑓 be a normalized biholomorphic mapping on the Euclidean unit ball B𝑛inC𝑛and let𝛼 ∈ (0, 1) In this paper, we will show that if𝑓 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 dimension𝑛 ≥ 2
1 Introduction and Preliminaries
Let C𝑛 denote the space of 𝑛 complex variables 𝑧 =
(𝑧1, , 𝑧𝑛) with the Euclidean inner product ⟨𝑧, 𝑤⟩ =
∑𝑛𝑗=1𝑧𝑗𝑤𝑗 and the norm‖ 𝑧 ‖= ⟨𝑧, 𝑧⟩1/2 The open unit ball
{𝑧 ∈ C𝑛 : ‖𝑧‖ < 1} is denoted by B𝑛 In the case of one
complex variable,B1is denoted by𝑈
IfΩ is a domain in C𝑛, let𝐻(Ω) be the set of holomorphic
mappings fromΩ to C𝑛 IfΩ is a domain in C𝑛which contains
the origin and𝑓 ∈ 𝐻(Ω), we say that 𝑓 is normalized if
𝑓(0) = 0 and 𝐷𝑓(0) = 𝐼𝑛, where𝐼𝑛is the identity matrix
A normalized mapping𝑓 ∈ 𝐻(B𝑛) is said to be starlike if
𝑓 is biholomorphic on B𝑛and𝑡𝑓(B𝑛) ⊂ 𝑓(B𝑛) for 𝑡 ∈ [0, 1],
where the last condition says that the image𝑓(B𝑛) is a starlike
domain with respect to the origin For a normalized locally
biholomorphic mapping𝑓 on B𝑛,𝑓 is starlike if and only if
R ⟨[𝐷𝑓 (𝑧)]−1𝑓 (𝑧) , 𝑧⟩ > 0, 𝑧 ∈ B𝑛\ {0} (1)
(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
arg
𝑧𝑓(𝑧)
𝑓 (𝑧)
< 𝛼
𝜋
If𝑓 is strongly starlike of order 𝛼, then 𝑓 is also starlike and thus univalent on𝑈 Stankiewicz [6] proved that if𝛼 ∈ (0, 1), then a domain Ω ̸= C which contains the origin is 𝛼-accessible if and only ifΩ = 𝑓(𝑈), where 𝑈 is the unit disc
inC 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 defini-tion of strongly starlike mappings of order𝛼 on B𝑛
Definition 1 Let 0 < 𝛼 ≤ 1 A normalized locally biholomorphic mapping𝑓 ∈ 𝐻(B𝑛) is said to be strongly starlike of order𝛼 if
arg ⟨[𝐷𝑓 (𝑧)]−1𝑓 (𝑧) , 𝑧⟩ < 𝛼𝜋
2, 𝑧 ∈ B𝑛\ {0} (3) Obviously, if𝑓 is strongly starlike of order 𝛼, then 𝑓 is also starlike, and if𝛼 = 1 in (3), one obtains the usual notion
of starlikeness on the unit ballB𝑛 Using this definition, Hamada and Honda [11], Hamada and Kohr [12], Liczberski [13], and Liu and Li [14] obtained
http://dx.doi.org/10.1155/2014/265718
Trang 22 Abstract and Applied Analysis
various results for strongly starlike mappings of order𝛼 in
several complex variables
Recently, Liczberski and Starkov [15] gave another
defini-tion of strongly starlike mappings of order𝛼 on the Euclidean
unit ball B𝑛 in C𝑛, where 𝛼 ∈ (0, 1], and proved that
a normalized biholomorphic mapping𝑓 on B𝑛 is strongly
starlike of order1 − 𝛼 if and only if 𝑓(B𝑛) is an 𝛼-accessible
domain inC𝑛for𝛼 ∈ (0, 1) Their definition is as follows
Definition 2 Let 0 < 𝛼 ≤ 1 A normalized locally
biholomorphic mapping𝑓 ∈ 𝐻(B𝑛) is said to be strongly
starlike of order𝛼 (in the sense of Liczberski and Starkov)
if
R ⟨[𝐷𝑓 (𝑧)]−1𝑓 (𝑧) , 𝑧⟩
≥([𝐷𝑓 (𝑧)]−1)∗𝑧 ⋅𝑓(𝑧)sin((1 − 𝛼) 𝜋2) ,
𝑧 ∈ B𝑛\ {0}
(4)
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 dimension𝑛 ≥ 2
Question 1 Let𝛼 ∈ (0, 1) Is there any relation between the
above two definitions of strong starlikeness of order𝛼?
Let 𝑓 be a normalized biholomorphic mapping on the
Euclidean unit ballB𝑛inC𝑛and let𝛼 ∈ (0, 1) In this paper,
we will show that if𝑓 is strongly starlike of order 𝛼 in the sense
ofDefinition 2, then it is also strongly starlike of order𝛼 in the
sense ofDefinition 1 As a corollary, the results obtained in
[11–14] for strongly starlike mappings of order𝛼 in the sense
of Definition 1 also hold for strongly starlike mappings of
order𝛼 in the sense ofDefinition 2 We also give an example
which shows that the converse of the above result does not
hold in dimension𝑛 ≥ 2
2 Main Results
Let∠(𝑎, 𝑏) denote the angle between 𝑎, 𝑏 ∈ C𝑛\ {0} regarding
𝑎, 𝑏 as real vectors in R2𝑛
Lemma 3 Let 𝑎, 𝑏 ∈ C𝑛 \ {0} be such that ⟨𝑎, 𝑏⟩ ̸= 0 If
| arg⟨𝑎, 𝑏⟩| ≤ 𝜋 and 0 ≤ ∠(𝑎, 𝑏) < 𝜋/2, then
arg⟨𝑎,𝑏⟩ ≤ ∠(𝑎,𝑏) (5)
Proof Let𝜃 = arg⟨𝑎, 𝑏⟩, 𝜑 = ∠(𝑎, 𝑏) Then we have ⟨𝑎, 𝑏⟩ =
𝑟𝑒𝑖𝜃for some𝑟 ≥ 0 and
R ⟨𝑎, 𝑏⟩ = ‖𝑎‖ ⋅ ‖𝑏‖ cos 𝜑 = 𝑟 cos 𝜃 (6)
Since cos𝜑 > 0 and 𝑟 = |⟨𝑎, 𝑏⟩| ≤ ‖𝑎‖ ⋅ ‖𝑏‖, we have
Therefore, we have|𝜃| ≤ 𝜑, as desired
Theorem 4 Let 𝑓 be a normalized biholomorphic mapping
on the Euclidean unit ballB𝑛 inC𝑛and 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
⟨[𝐷𝑓(𝑧)]−1𝑓(𝑧), 𝑧⟩ ̸= 0 and
∠ (([𝐷𝑓 (𝑧)]−1)∗𝑧, 𝑓 (𝑧)) ≤ 𝛼𝜋2, 𝑧 ∈ B𝑛\ {0} (8) UsingLemma 3, we have
arg ⟨[𝐷𝑓 (𝑧)]−1𝑓 (𝑧) , 𝑧⟩ =arg ⟨𝑓 (𝑧) , ([𝐷𝑓 (𝑧)]−1)∗𝑧⟩
≤ ∠ (([𝐷𝑓 (𝑧)]−1)∗𝑧, 𝑓 (𝑧))
≤ 𝛼𝜋2, 𝑧 ∈ B𝑛\ {0}
(9) For fixed𝑧 ∈ B𝑛\ {0}, let 𝑤 = 𝑧/‖𝑧‖ and
𝑝 (𝜁) ={{ {
1
𝜁⟨[𝐷𝑓 (𝜁𝑤)]−1𝑓 (𝜁𝑤) , 𝑤⟩ , for 𝜁 ∈ 𝑈 \ {0} ,
(10) Then 𝑝 is a holomorphic function on 𝑈 with | arg 𝑝(𝜁)| ≤ 𝜋𝛼/2 for 𝜁 ∈ 𝑈 Since arg 𝑝 is a harmonic function on 𝑈 and arg𝑝(0) = 0, by applying the maximum and minimum principles for harmonic functions, we obtain | arg 𝑝(𝜁)| < 𝜋𝛼/2 for 𝜁 ∈ 𝑈 Thus, we have
arg ⟨[𝐷𝑓 (𝑧)]−1𝑓 (𝑧) , 𝑧⟩ < 𝛼𝜋
2, 𝑧 ∈ B𝑛\ {0} (11) 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
𝑓 (𝑧) = 𝑓𝛼(𝑧) = (𝑧1+ 𝑏𝑧22, 𝑧2) , 𝑧 = (𝑧1, 𝑧2) ∈ B2, (12) where
𝑏 = 3√32 sin(𝛼𝜋2) (13) Then
𝐷𝑓 (𝑧) = [1 2𝑏𝑧2
0 1 ] , [𝐷𝑓 (𝑧)]−1= [1 −2𝑏𝑧2
0 1 ] (14) Therefore,
⟨[𝐷𝑓 (𝑧)]−1𝑓 (𝑧) , 𝑧⟩ = (𝑧1+ 𝑏𝑧22− 2𝑏𝑧22) 𝑧1
+ 𝑧22
= 𝑧12 + 𝑧22− 𝑏𝑧1𝑧2
2 (15)
Trang 3Since |𝑧1𝑧2
2| ≤ 2/(3√3), for 𝑧 ∈ 𝜕B2, we obtain that
|𝑏𝑧1𝑧22| ≤ sin(𝛼𝜋/2)‖𝑧‖3 for 𝑧 ∈ B2 This implies that
⟨[𝐷𝑓(𝑧)]−1𝑓(𝑧), 𝑧⟩ lies in the disc of center ‖𝑧‖2and radius
sin(𝛼𝜋/2)‖𝑧‖2for each𝑧 ∈ B2\ {0} and thus
arg ⟨[𝐷𝑓 (𝑧)]−1𝑓 (𝑧) , 𝑧⟩ < 𝛼𝜋
2, 𝑧 ∈ B2\ {0} (16) Therefore,𝑓 = 𝑓𝛼is strongly starlike of order𝛼 in the sense
ofDefinition 1
On the other hand,
([𝐷𝑓 (𝑧)]−1)∗𝑧 = (𝑧1, 𝑧2− 2𝑏𝑧2𝑧1) (17)
So, for𝑧0= (1/√3, √2/√3), we have
⟨[𝐷𝑓 (𝑧0)]−1𝑓 (𝑧0) , 𝑧0⟩ = 1 − 𝑚,
([𝐷𝑓(𝑧0)]−1)∗𝑧02
= 13+23(1 − 3𝑚)2,
𝑓 (𝑧0)2= 1
3(1 + 3𝑚)2+
2
3, sin((1 − 𝛼)𝜋2) = √1 − 𝑚2,
(18)
where
𝑚 = sin (𝛼𝜋
Then, we obtain
([𝐷𝑓(𝑧0)]−1)∗𝑧02
𝑓 (𝑧0)2sin2((1 − 𝛼)𝜋2)
− (R ⟨[𝐷𝑓 (𝑧0)]−1𝑓 (𝑧0) , 𝑧0⟩)2
= (1 − 𝑚) {[13+23(1 − 3𝑚)2] [13(1 + 3𝑚)2+23]
× (1 + 𝑚) − (1 − 𝑚) }
(20)
Since
[13+23(1 − 3𝑚)2] [13(1 + 3𝑚)2+23] (1 + 𝑚) − (1 − 𝑚)
(21)
is increasing on[1/3, 1] and positive for 𝑚 = 1/3, we have
R ⟨[𝐷𝑓 (𝑧0)]−1𝑓 (𝑧0) , 𝑧0⟩ <([𝐷𝑓(𝑧0)]−1)∗𝑧0
× 𝑓(𝑧0)sin((1 − 𝛼)𝜋2)
(22) for𝑚 ∈ [1/3, 1)
On the other hand, for̃𝑧0= (𝑖/√3, √2/√3), we have
⟨[𝐷𝑓 (̃𝑧0)]−1𝑓 (̃𝑧0) , ̃𝑧0⟩ = 1 + 𝑚𝑖,
([𝐷𝑓(̃𝑧0)]−1)∗̃𝑧02
=1
3 +
2
3|1 − 3𝑚𝑖|2= 6𝑚2+ 1,
𝑓 (̃𝑧0)2=13|𝑖 + 3𝑚|2+23 = 3𝑚2+ 1
(23)
Then, we obtain
([𝐷𝑓(̃𝑧0)]−1)∗̃𝑧02
𝑓 (̃𝑧0)2sin2((1 − 𝛼)𝜋2)
− (R ⟨[𝐷𝑓 (̃𝑧0)]−1𝑓 (̃𝑧0) , ̃𝑧0⟩)2
= (6𝑚2+ 1) (3𝑚2+ 1) (1 − 𝑚2) − 1
= 𝑚2(−18𝑚4+ 9𝑚2+ 8)
(24)
Since−18𝑚4+ 9𝑚2+ 8 is positive for 𝑚 ∈ [0, 1/3], we have
R ⟨[𝐷𝑓 (̃𝑧0)]−1𝑓 (̃𝑧0) , ̃𝑧0⟩ <([𝐷𝑓(̃𝑧0)]−1)∗̃𝑧0
× 𝑓(̃𝑧0)sin((1 − 𝛼)𝜋2)
(25) for𝑚 ∈ (0, 1/3]
Thus,𝑓 = 𝑓𝛼is not strongly starlike of order𝛼 in the sense
ofDefinition 2for𝛼 ∈ (0, 1)
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper
Acknowledgments
Hidetaka Hamada is supported by JSPS KAKENHI Grant
no 25400151 Tatsuhiro Honda is partially supported by Brain Korea Project, 2013 The work of Gabriela Kohr was supported by a grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, Project no PN-II-ID-PCE-2011-3-0899 Kwang Ho Shon was supported by a 2-year research grant of Pusan National University
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