Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2011, Article ID 865496, 14 pages
doi:10.1155/2011/865496
Research Article
A New Roper-Suffridge Extension Operator on
a Reinhardt Domain
Jianfei Wang and Cailing Gao
Department of Mathematics and Physics, Information Engineering, Zhejiang Normal University,
Zhejiang, Jinhua 321004, China
Correspondence should be addressed to Jianfei Wang, wjfustc@zjnu.cn
Received 5 July 2011; Revised 28 September 2011; Accepted 5 October 2011
Academic Editor: Sung Guen Kim
Copyright q 2011 J. Wang and C. Gao. 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.
We introduce a new Roper-Suffridge extension operator on the following Reinhardt domain
pj
Ωn, p2 ,..., pn {z ∈ Cn : |z1 |2 nj2 |zj |pj < 1}given by Fz fz1 f z1 nj2 aj zj , f z1 1/p2 z2 ,
. . . , f z1 1/pn zn , where f is a normalized locally biholomorphic function on the unit disc D, pj
are positive integer, aj are complex constants, and j 2, . . . , n. Some conditions for aj are found
under which the operator preserves almost starlike mappings of order α and starlike mappings of
order α, respectively. In particular, our results reduce to many well-known results when all αj 0.
1. Introduction
In 1995, Roper and Suffridge 1 introduced an extension operator. This operator is defined
as follows:
Φn f z fz1 , f z1 z0 ,
1.1
where f is a normalized locally biholomorphic function on the unit disk D in C, z z1 , z0 n
n
n−1
belonging to the unit
ball B in C , z0 z2 , . . . , zn ∈ C and the branch of the square root
is chosen such that f 0 1.
It is well known that the Roper-Suffridge extension operator has the following remarkable properties:
i if f is a normalized convex function on D, then Φn f is a normalized convex
mapping on Bn ;
2
Abstract and Applied Analysis
ii if f is a normalized starlike function on D, then Φn f is a normalized starlike
mapping on Bn ;
iii if f is a normalized Bloch function on D, then Φn f is a normalized Bloch mapping
on Bn .
The above result i was proved by Roper and Suffridge 1 and the result ii and iii
was proved by Graham and Kohr 2, 3. Until now, it is difficult to construct the concrete
convex mappings, starlike mappings, and Bloch mappings on Bn . By making use of the
Roper-Suffridge extension operator, we may easily give many concrete examples about these
mappings. This is one important reason why people are interested in this extension operator.
In 2005, Muir 4 modified the Roper-Suffridge extension operator as follows:
Fz fz1 f z1 P z0 ,
f z1 z0
,
1.2
where P z0 is a homogeneous polynomial of degree 2 with respect to z0 , and f, z1 , and z0
are defined as above. They proved that this operator preserves starlikeness and convexity if
and only if P 1/4 and P 1/2, respectively. The modified operator plays a key role to
study the extreme points of convex mappings on Bn see 5, 6. Later, Kohr 7 and Muir 8
used the Loewner chain to study the modified Roper-Suffridge extension operator. Recently,
the modified Roper-Suffridge extension operator on the unit ball is also studied by Wang and
Liu 9 and Feng and Yu 10.
On the other hand, people also considered the generalized Roper-Suffridge extension
operator on the general Reinhardt domains. For example, Gong and Liu 11, 12 induced the
definition of ε starlike mappings and obtained that the operator
1/p z0
Φn, 1/p f z fz1 , f z1 1.3
maps the ε starlike functions on D to the ε starlike mappings on the Reinhardt domain Ωn,p {z ∈ Cn : |z1 |2 nj2 |zj |p < 1}, where p 1, f, z1 , and z0 are defined as above. When
ε 0 and ε 1, Φn, 1/p maps the starlike function and convex function on D to the starlike
mapping and the convex mapping on Ωn, p , respectively.
Furthermore, Gong and Liu 13 proved that the operator
1/p2
1/pn z2 , . . . , f z1 zn
Φn, 1/p2 , ..., 1/pn f z fz1 , f z1 1.4
maps the ε starlike functions on D to the ε starlike mappings on the domain Ωn,p2 ,...,pn {z ∈
Cn : |z1 |2 nj2 |zj |pj < 1}, where pj 1, j 2, . . . , n, f, z1 , and z0 are defined as above.
Liu and Liu 14 proved that this operator preserves starlikeness of order α on the domain
Ωn, p2 , ..., pn . On the other hand, Feng and Liu 15 proved that this operator preserves almost
starlikeness of order α on the domain Ωn, p2 , ..., pn .
Abstract and Applied Analysis
3
In contrast to the modified Roper-Suffridge extension operator in the unit ball, it is
natural to ask if we can modify the Roper-Suffridge extension operator on the Reinhardt
domains. In this paper, we will introduce the following modified operator:
⎞
n
1/p2
1/pn
pj z2 , . . . , f z1 zn ⎠
Fz ⎝fz1 f z1 aj zj , f z1 ⎛
1.5
j2
on the Reinhardt domain Ωn,p2 ,...,pn . We will give some sufficient conditions for aj under which
the above Roper-Suffridge operator preserves an almost starlike mappings of order α and
starlike mappings of order α, respectively.
In the following, we give some notation and definitions. Let Cn be the space of n
complex variables z z1 , . . . , zn with the Euclidean inner product z, w ni1 zi wi and
the Euclidean norm z z, z1/2 , where z, w ∈ Cn and the symbol “ ” means transpose.
The unit ball of Cn is the set Bn {z ∈ Cn : z < 1}, and the unit sphere is denoted by
∂Bn {z ∈ Cn : z 1}. In the case of one complex variable, B1 is the unit disk, usually
denoted by D. Let Ω be a domain in Cn . Denote HΩ by the space of all holomorphic
mappings from Ω into Cn . A mapping f ∈ HBn is called normalized if f0 0 and
Jf 0 In , where Jf 0 is the complex Jacobian matrix of f at the origin and In is the identity
operator on Cn . A mapping f ∈ HBn is said to be locally biholomorphic if det Jf z / 0 for
n
n
every z ∈ B . A normalized mapping f ∈ HB is said to be convex if λω1 1−λω2 ∈ fBn for arbitrary ω1 , ω2 ∈ fBn and 0 λ 1. A normalized mapping f ∈ HBn is said to be
starlike with respect to the origin if λfBn ⊂ fBn , 0 λ 1. A normalized mapping
f ∈ HBn is said to be ε starlike if there exists a positive number ε, 0 ε 1, such that
fBn is starlike with respect to every point in εfBn .
A domain Ω is called a Reinhardt domain if eiθ1 z1 , eiθ2 z2 , . . . , eiθn zn ∈ Ω holds for
any z z1 , z2 , . . . , zn ∈ Ω and θ1 , θ2 , . . . , θn ∈ R. A domain Ω is called a circular domain
if eiθ z ∈ Ω holds for any z ∈ Ω and θ ∈ R. The Minkowski functional ρz of the Reinhardt
domain
Ωn,p2 ,...,pn
⎧
⎨
⎫
n ⎬
p
z ∈ Cn : |z1 |2 zj j < 1 ,
⎩
⎭
j2
pj 1, j 2, . . . , n
1.6
z ∈ Cn .
1.7
is defined as
z
ρz inf t > 0, ∈ Ωn, p2 , ..., pn ,
t
Then, the Minkowski functional ρz is a norm of Cn and Ωn,p2 ,··· ,pn is the unit ball in the
Banach space Cn with respect to this norm. The Minkowski functional ρz is C1 on Ωn,p2 ,...,pn
4
Abstract and Applied Analysis
except for a lower-dimensional manifold Ω0 . Moreover, we give the following properties of
the Minkowski functional ρz see 16:
2
2
∂ρ
zz ρz,
∂z
∂ρ
zz 1,
∂z
∂ρ
∂ρ
λz z,
∂z
∂z
∀z ∈ Cn \ Ω0 ,
∀z ∈ ∂Ωn, p2 ,..., pn \ Ω0 ,
1.8
n
∀λ ∈ 0, ∞, z ∈ C \ Ω0 ,
∂ρ iθ ∂ρ
e z e−iθ z,
∂z
∂z
∀z ∈ Cn \ Ω0 , θ ∈ R.
Definition 1.1 see 17. Suppose that Ω is a bounded starlike circular domain in Cn . Its
Minkowski functional ρz is C1 except for a lower-dimensional manifold. Let 0 α < 1.
We say that a normalized locally biholomorphic mapping f ∈ HΩ is an almost starlike
mapping of order α if the following condition holds:
R
2 ∂ρ
zJf−1 zfz α,
ρz ∂z
z ∈ Ω \ {0}.
1.9
When Ω Bn , its Minkowski functional ρz z, the above inequality becomes
Rz Jf−1 zfz αz2 ,
z ∈ Bn .
1.10
In particular, when α 0, f reduces to a starlike mapping on Ω.
Definition 1.2 see 18. Suppose that Ω ∈ Cn is a bounded starlike circular domain. Its
Minkowski functional ρz is C1 except for a lower-dimensional manifold. Let 0 < α < 1.
We say that a normalized locally biholomorphic mapping f ∈ HΩ is a starlike mapping of
order α if the following condition holds:
2 ∂ρ
1
1 −1
<
,
−
zfz
zJ
ρz ∂z
f
2α 2α
z ∈ Ω \ {0}.
1.11
When Ω Bn , the above inequality reduces to
1
1
1 −1
,
z J zfz −
<
z2 f
2α
2α
z ∈ Bn \ {0}.
1.12
Abstract and Applied Analysis
5
2. Some Lemmas
In order to prove the main results, we need the following three lemmas.
Lemma 2.1 see 19. Let p be a holomorphic function on D. If Rpz > 0 and p0 > 0, then
2Rpz
p z .
1 − |z|2
2.1
Lemma 2.2 see 19. Let f be a normalized biholomorphic function on D. Then,
1 − |z|2 f z − 2z 4
f z
2.2
holds for all z ∈ D.
Lemma 2.3 see 20. If ρz is a Minkowski function of the domain Ωn, p2 ,..., pn , z /
0, then
∂ρ
z1
z p ,
2
∂z1
ρz 2z1 /ρz nj2 pj zj /ρz j
p −2
pj zj zj /ρz j
∂ρ
z 2 p ,
∂zj
2ρz 2z1 /ρz nj2 pj zj /ρz j
2.3
j 2, . . . , n.
3. Main Results
Theorem 3.1. Let 0 α < 1 and let f be an almost starlike function of order α on the unit disc D. If
complex numbers aj satisfy the condition |aj | 1 − α/4, j 2, . . . , n, then
⎛
Fz ⎝fz1 f z1 n
j2
⎞
1/p2
1/pn
pj aj zj , f z1 z2 , . . . , f z1 zn ⎠
3.1
is an almost starlike mapping of order α on the domain Ωn, p2 ,..., pn , where pj are positive integer and
pj 2; the branches are chosen such that f z1 1/pj |z1 0 1.
Proof. By the definition of almost starlike mapping of order α, we need only to prove that the
following inequality:
R
2 ∂ρ
zJF−1 zFz α
ρz ∂z
holds for all z ∈ Ωn, p2 ,..., pn and z /
0.
3.2
6
Abstract and Applied Analysis
The case of z0 0 is trivial. So, we need only consider that z z1 , z0 ∈ Ωn,p2 ,...,pn ,
z0 /
0. Let z ζu |ζ|eiθ u, u ∈ ∂Ωn, p2 ,..., pn , and ζ ∈ D \ {0}, then we have
R
2 ∂ρ
zJF−1 zFz α
ρz ∂z
∂ρ iθ −1 iθ iθ 2
⇐⇒ R iθ |ζ|e u JF |ζ|e u F |ζ|e u α
ρ |ζ|e u ∂z
3.3
2 e−iθ ∂ρ
⇐⇒ R
uJF−1 |ζ|eiθ u F |ζ|eiθ u α
|ζ| ∂z
⇐⇒ R
J −1 ζuFζu
2∂ρ
α.
u F
∂z
ζ
For a fixed u, the expression R2∂ρ/∂zuJF−1 ζuFζu/ζ − α is the real part of a
holomorphic function with respect to ζ, so it is a harmonic function. By the minimum of
harmonic function principle, we know that it attains its minimum on |ζ| 1, so we need only
to prove for all z ∈ ∂Ωn, p2 ,..., pn and z0 / 0. Hence, ρz 1 and inequality 3.2 becomes
R
2∂ρ
zJF−1 zFz α,
∂z
3.4
z ∈ ∂Ωn, p2 ,..., pn , z0 / 0.
In the following, we will prove inequality 3.4.
Since
⎛
Fz ⎝fz1 f z1 n
j2
⎞
1/p2
1/pn
pj aj zj , f z1 z2 , . . . , f z1 zn ⎠
,
3.5
we have
⎛
n
pj
p −1
f
aj zj
a2 p2 f z1 z22
f
z
z
1
1
⎜
⎜
j2
⎜
⎜ 1
1/p2
⎜
1/p2 −1 f z1 z2
f z1 f z1 ⎜
JF z ⎜
⎜ p2
⎜
⎜
···
···
⎜
⎜
⎝1
1/pn −1 f z1 zn
0
f z1 pn
⎞
p −1
· · · an pn f z1 znn ⎟
⎟
⎟
⎟
⎟
···
0
⎟
⎟.
⎟
⎟
⎟
···
···
⎟
⎟
1/pn ⎠
···
f z1 3.6
Abstract and Applied Analysis
7
Suppose that JF−1 zFz A x1 , x2 , . . . , xn , then Fz JF zA; that is,
⎡
⎤
n
n
n
p
pj −1
pj
j
x1 ⎣f z1 f z1 aj zj ⎦ f z1 aj pj xj zj fz1 f z1 aj zj ,
j2
j2
x1
j2
f z1 z2 x2 z2 ,
p2 f z1 3.7
..
.
x1
f z1 zn xn zn .
pn f zn Some computation shows that
x1 n
pj
fz1 aj pj − 1 zj ,
−
f z1 j2
⎤
n
pj
f z1 fz1 f z1 x2 ⎣1 − 2 p f z aj pj − 1 zj ⎦z2 ,
2
1 j2
p2 f z1 ⎡
3.8
..
.
⎤
n
pj
f z1 fz1 f z1 xn ⎣1 − 2 p f z aj pj − 1 zj ⎦zn .
n
1 j2
pn f z1 ⎡
From Lemma 2.3, we obtain
∂ρ
z1
z1
z p ,
n
2 n
pj 2
∂z1
2|z1 | j2 pj zj j
ρz 2 z1 /ρz j2 pj zj /ρz
p −2
p −2
pj zj zj j
pj zj zj j
∂ρ
z 2 p p .
∂zj
2ρz 2z1 /ρz nj2 pj zj j
2 2|z1 |2 nj2 pj zj j
3.9
In terms of 3.8 and 3.9, we obtain
2∂ρ
Gz
zJF−1 zFz p ,
2
∂z
2|z1 | nj2 pj zj j
3.10
8
Abstract and Applied Analysis
where
⎡
⎤
n
pj
fz1 − aj pj − 1 zj ⎦
Gz 2z1 ⎣ f z1 j2
!
"
n
n
pj
pk
fz
f
f
z
z
1
1
1
pj zj 1 − 2 p f z ak pk − 1 zk
j
1 k2
pj f z1 j2
!
"
n
pj
fz
f
z
1
1
2 fz1 2|z1 |
pj zj 1 − 2
z1 f z1 j2
pj f z1 ⎡
⎤
n
n pk f z1 p
j
zj − 2z1 ⎦.
ak pk − 1 zk ⎣ f z1 j2
k2
By making use of the equality |z1 |2 n
j2 |zj |
3.11
pj
1, we then get
!
"
n
pj
fz
f
z
1
1
2 fz1 Gz 2|z1 |
pj zj 1 − 2
z1 f z1 j2
pj f z1 #
$
n
pj f z1 2
aj pj − 1 zj
1 − |z1 | − 2z1 .
f z1 j2
3.12
Let hz1 fz1 /z1 f z1 − α. Notice that f is an almost starlike function of order α on the
unit disc; hence, Rhz1 > 0 and h0 1 − α > 0. By Lemma 2.1, we can obtain that
2Rhz
h z .
1 − |z|2
3.13
fz1 f z1 &2 1 − α − hz1 − z1 h z1 .
% f z1 3.14
Furthermore, we get
Substituting 3.14 into 3.12, we have
n
p
Gz 2|z1 | hz1 α pj zj j
2
j2
'
1
α hz1 z1 1−
h z1 pj pj
pj
pj
(
$
pj f z1 2
−
2z
1
−
aj pj − 1 zj
|z
|
1
1
f z1 j2
⎛
⎞
n n
p
p
2
j
hz1 ⎝2|z1 | zj ⎠ 2α|z1 |2 pj − 1 α zj j
n
#
j2
j2
#
$
n
n
pj pj f z1 2
z1 zj h z1 aj pj − 1 zj
1 − |z1 | − 2z1 .
f z1 j2
j2
3.15
Abstract and Applied Analysis
9
Hence,
⎞
n n
p
p
pj − 1 α zj j
RGz ⎝2|z1 |2 zj j ⎠Rhz1 2α|z1 |2 ⎛
j2
−
j2
n zj pj z1 h z1 −
j2
aj pj − 1 zj pj f z1 1 − |z1 |2 − 2z1 .
f z 1
j2
n 3.16
By Lemma 2.2 and 3.13, we can get that
n
2|z |Rhz p 1
1
RGz 1 |z1 |2 Rhz1 2α|z1 |2 pj − 1 α zj j − 1 − |z1 |2
2
1
−
|z
|
1
j2
n p
− 4 aj pj − 1 zj j
j2
n
p
pj − 1 α zj j − 2|z1 |Rhz1 1 |z1 |2 Rhz1 2α|z1 |2 j2
n p
− 4 aj pj − 1 zj j
j2
1 − |z1 |2 Rhz1 2α|z1 |2 n &
%
zj pj α 1 − 4aj pj − 1 .
j2
3.17
Hence, when |aj | 1 − α/4, we have
n
n
p
p
RGz 1 − |z1 |2 Rhz1 2α|z1 |2 α pj zj j 2α|z1 |2 α pj zj j .
j2
j2
3.18
In terms of 3.10 and 3.18, we obtain
R
2∂ρ
zJF−1 zFz α,
∂z
3.19
which completes the proof of Theorem 3.1.
Remark 3.2. When a2 a3 · · · an 0, the result of Theorem 3.1 has been obtained by Liu
and Liu 14.
Corollary 3.3. Let f be a normalized biholomorphic starlike function on the unit disc D. If |aj | 1/4, j 2, . . . , n, then
⎛
⎞
n
pj
1/p2
1/pn
z2 , . . . , f z1 zn ⎠
Fz ⎝fz1 f z1 aj zj , f z1 j2
3.20
10
Abstract and Applied Analysis
is a normalized biholomorphic starlike mapping on the domain Ωn, p2 ,..., pn , where pj are positive integer
and pj 2; the branches are chosen such that f z1 1/pj |z1 0 1.
Theorem 3.4. Let 0 < α < 1 and let f be a starlike function of order α on the unit disc D. If complex
numbers aj satisfy the condition |aj | 1 − |2α − 1|/8α, j 2, . . . , n, then
⎞
n
pj
1/p2
1/pn
z2 , . . . , f z1 zn ⎠
Fz ⎝fz1 f z1 aj zj , f z1 ⎛
3.21
j2
is a starlike mapping of order α on the domain Ωn, p2 ,..., pn , where pj are positive integer and pj 1; the
branches are chosen such that f z1 1/pj |z1 0 1.
Proof. By the definition of starlike mapping of order α, we need only to prove that the
following inequality:
2 ∂ρ
1
1 −1
ρz ∂z zJF zFz − 2α < 2α
3.22
0.
holds for all z ∈ Ωn, p2 ,..., pn and z0 /
Similar to the proof of Theorem 3.1, we need only to prove that 3.22 holds for ρz 1
0 according to the maximum modulus theorem for analytic functions. So, it is suffice
and z0 /
to show that
2∂ρ
1
1 −1
∂z zJF zFz − 2α < 2α .
3.23
From the proof of Theorem 3.1, we can get
∂ρ
zJF−1 zFz
∂z
2 2|z1 |2 ⎧
⎨
!
"
n
pj
fz
fz1 f
z
1
1
2
pj ⎩2|z1 | z f z pj zj 1 − zj 1
1
pj f z1 p
j2
j
j2
1
n
2
3.24
⎫
#
$⎬
n
pj f z1 .
aj pj − 1 zj
1 − |z1 |2 − 2z1
⎭
f z1 j2
Hence,
2∂ρ
Hz
1
,
zJF−1 zFz −
∂z
2α 2α 2|z |2 n p z pj
1
j2 j j
3.25
Abstract and Applied Analysis
11
where
!
"
$
#
n
pj
fz1 f z1 fz1 1
1−
−
− 1 2α pj zj
Hz 2|z1 | 2α
z1 f z1 2α pj f z1 2
j2
#
$
n
pj f z1 2
1 − |z1 | − 2z1 .
2α aj pj − 1 zj
f z1 j2
2
3.26
Let hz1 2αfz1 /z1 f z1 − 1. Then, |hz1 | < 1 because f is a starlike function of order
α on the unit disc D. By the Schwarz-Pick lemma, we obtain that
2
h z1 1 − |hz1 | .
1 − |z1 |2
3.27
fz1 f z1 hz1 z1 h z1 1
.
&2 1 − 2α − 2α −
% 2α
f z1 3.28
On the other hand, we can get
Substituting 3.28 into 3.26, we have
("
'
!
$
#
n
pj
fz1 1
1
h
1
hz
z
z
1
1
1
−
Hz 2|z1 | 2α
− 1 2α pj zj 1 −
−
−
−
z1 f z1 2α
pj 2αpj
2αpj
2αpj
j2
2
#
$
n
pj f z1 2
1 − |z1 | − 2z1
2α aj pj − 1 zj
f z1 j2
2|z1 |2 hz1 hz1 n n
n
p
zj pj z1 h z1 zj pj 2α − 1
pj − 1 zj j
j2
j2
#
j2
$
n
pj f z1 2
− 2z1 .
1
−
2α aj pj − 1 zj
|z
|
1
f z1 j2
3.29
Hence,
n n
p
p
pj − 1 zj j
|Hz| 1 |z1 |2 |hz1 | z1 h z1 zj j |2α − 1|
j2
j2
n pj f z1 2
aj pj − 1 zj 2α
1 − |z1 | − 2z1 .
f z1 j2
3.30
12
Abstract and Applied Analysis
By Lemma 2.2 and 3.27, we have
n
p
1 − |hz1 |2 2
1
−
−
1|
pj − 1 zj j
|z
|2α
|Hz| 1 |z1 |2 |hz1 | |z1 |
|
1
2
1 − |z1 |
j2
n p
8α aj pj − 1 zj j
j2
1 |z1 |2 |hz1 | − 1 1 |z1 |2 2|z1 |1 − |hz1 |
|2α − 1|
3.31
n
n p
p
pj − 1 zj j 8α aj pj − 1 zj j
j2
j2
n
p
1 |z1 |2 |hz1 | − 11 − |z1 |2 |2α − 1| 8αaj pj − 1 zj j .
j2
If |aj | 1 − |2α − 1|/8α, then we obtain
n
p
1 − |2α − 1| pj − 1 zj j
|Hz| < 1 |z1 |2 |2α − 1| 8α
8α
j2
≤ 1 |z1 |2 n
p
pj − 1 zj j
j2
2|z1 |2 3.32
n
p
pj zj j .
j2
The equality 3.25 and 3.32 show that
2∂ρ
1
1 −1
∂z zJF zFz − 2α < 2α ,
3.33
which completes the proof of Theorem 3.4.
4. Problem
In 2003, Gong and Liu 13 proved that the Roper-Suffridge extension operator
1/p2
1/pn Φn,1/p2 ,...,1/pn f z fz1 , f z1 z2 , . . . , f z1 zn
4.1
does preserve convexity on Ωn , p2 , . . . , pn , which solved the open problem posed by Graham
and Kohr 2. Naturally, we will propose the following problem on the new Roper-Suffridge
extension operator.
Abstract and Applied Analysis
13
Problem 1. Let pj be positive integer. Under what conditions for aj such that if f is a convex
function in the disc D, then the mapping defined by the new Roper-Suffridge extension
operator
⎛
⎞
n
p
1/p
1/p
j
2
n
z2 , . . . , f z1 zn ⎠
Fz ⎝fz1 f z1 aj zj , f z1 4.2
j2
is a convex mapping in the Reinhardt domain Ωn , p2 , . . . , pn ?
Acknowledgments
The authors cordially thank the referees thorough reviewing with useful suggestions and
comments made to the paper. The project was supported by the National Natural Science
Foundation of China Grant no. 11001246 and no. 11101139, the Natural Science Foundation
of Zhejiang province Grant no. Y6090694, and the Zhejiang Innovation Project Grant no.
T200905.
References
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