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Abstract and Applied Analysis
Volume 2012, Article ID 647972, 9 pages
doi:10.1155/2012/647972
Research Article
Schwarz-Pick Estimates for Holomorphic Mappings
with Values in Homogeneous Ball
Jianfei Wang
Department of Mathematics, Zhejiang Normal University, Zhejiang, Jinhua 321004, China
Correspondence should be addressed to Jianfei Wang, wjfustc@zjnu.cn
Received 11 July 2012; Accepted 21 October 2012
Academic Editor: Abdelghani Bellouquid
Copyright q 2012 Jianfei Wang. 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.
Let BX be the unit ball in a complex Banach space X. Assume BX is homogeneous. The
generalization of the Schwarz-Pick estimates of partial derivatives of arbitrary order is established
for holomorphic mappings from the unit ball B n to BX associated with the Carathéodory metric,
which extend the corresponding Chen and Liu, Dai et al. results.
1. Introduction
By the classical Pick’s invariant form of Schwarz’s lemma, a holomorphic function fz which
is bounded by one in the unit disk D ⊂ C satisfies the following inequlity
1 − fz2
f z ≤
1 − |z|2
1.1
at each point z of D. Ruscheweyh in 1 firstly obtained best-possible estimates of higher
order derivatives of bounded holomorphic functions on the unit disk in 1985. Recently, a lot
of attention see Ghatage et al. 2, MacCluer et al. 3, Avkhadiev and Wirths 4, Ghatage
and Zheng 5, Dai and Pan 6 has been paid to the Schwarz-Pick estimates of high-order
derivative estimates in one complex variable. The best result is given as follows:
k! 1 − fz2
k k−1
f z ≤
k 1 |z| ,
2
1 − |z|
z ∈ D, k ≥ 1.
1.2
2
Abstract and Applied Analysis
It is natural to consider an extension of the above Schwarz-Pick estimates to higher
dimensions. Anderson et al. 7 gave Schwarz-Pick estimates of derivatives of arbitrary order
of functions in the Schur-Agler class on the unit polydisk and the unit ball of Cn , respectively.
Recently, Chen and Liu in 8 obtained estimates of high-order derivatives for all the bounded
holomorphic functions on the unit ball of Cn . Later, Dai et al. in 9, 10 generalized the
high order Schwarz-Pick estimates for holomorphic mappings between unit balls in complex
Hilbert space. Their main result is expressed as follows.
Theorem A. Suppose fz is holomorphic mapping from Bn to Bm . Then for any multiindex k ≥ 1
and β ∈ Cn \ {0},
k
k
Hfz D f, z, β , D f, z, β
where Dk f, z, β metric on Bn .
≤ k! 1 |α|k k!/α!∂f
k
|β, z|
1 −
|z|2 |β|2
k−1
|β, z|2 1/2
k
Hz β, β ,
1.3
z/∂zα1 1 ∂zα2 2 · · · ∂zαNN βα and Hz β, β is the Bergman
In this paper, we will extend Theorem A to holomorphic mappings from the unit ball
Bn to BX associated with the Carathéodory metric. In particular, when BX Bm , our result
coincides with Theorem A. Furthermore, our result shows that the high-order Schwarz-Pick
estimates on the unit ball do depend on the geometric property of the image domain BX .
Throughout this paper, the symbol X is used to denote a complex Banach space with
norm || · ||, and BX {z ∈ X : ||z|| < 1} is the unit ball in X. Let Cn be the space of n complex
variables z z1 , . . . , zn with the Euclidean inner product z, w nj1 zj wj , where the
symbol ’ stands for the transpose of vector or matrix. The unit ball of Cn is always written by
Bn . It is well known that if f is a holomorphic mapping from BX into X, then the following
well-known expansion
∞
n 1 n
D fx y − x
f y n!
n0
1.4
holds for all y in some neighborhood of x ∈ BX , where Dn fx means the nth Fréchet
derivative of f at the point x, and
n Dn fx y − x, y − x, . . . , y − x .
Dn fx y − x
1.5
Furthermore, Dn fx is a bounded symmetric n-linear mapping from nj1 X into X. For a
domain Ω ∈ X, a mapping f : Ω → X is called to be biholomorphic if fΩ is a domain; the
inverse f −1 exists and is holomorphic on fΩ. Let AutΩ denote the set of biholomorphic
mappings of Ω onto itself. Ω is said to be homogeneous, if for each pair of points x, y ∈ Ω,
there is an f ∈ AutΩ such that fx y.
In multiindex notation, α α1 , α2 , . . . , αn ∈ Nn is an n-tuple of nonnegative integers,
|α| α1 α2 · · · αn , α! α1 ! · · · αn !, zα zα1 1 · · · zαnn .
Abstract and Applied Analysis
3
Let Kz, z be the Bergman kernel function. Then the Bergman metric Hz β, β can be
defined as
n
∂2 log Kz, z
uj uk ,
Hz β, β ∂zj ∂zk
j,k1
1.6
where z ∈ Ω, u u1 , u2 , . . . , un ∈ Cn . It is well known that Hz β, β 1−|z|2 |β, z|2 /1−
|z|2 2 in 9.
Let FcBX z, ξ be the infinitesimal form of Carathéodory metric of domain BX . By the
definition of the Carathéodory metric 11, we have for any ξ ∈ X,
FcBX z, ξ sup Dfzξ : f ∈ HBX , BX , fz 0 ,
1.7
where HBX , BX denotes the family of holomorphic mappings which map BX into BX .
2. Some Lemmas
In order to prove the main results, we need the following lemmas. Let BX be the unit ball in a
complex Banach space X, and BX is homogeneous.
Lemma 2.1 see 11. If f ∈ HBX , BX , then
FcBX fz, Dfzξ ≤ FcBX z, ξ,
z ∈ BX , ξ ∈ X.
2.1
In particular, when f is biholomorphic mapping, then FcBX fz, Dfzξ FcBX z, ξ.
Lemma 2.2 see 12. Consider the following:
FcBX 0, ξ ξ,
ξ ∈ X.
2.2
Lemma 2.3. Let f ∈ HD, BX . Then f can be written with the following n-variable power series
given by
fz ∞
aj zj ,
z ∈ D.
2.3
j0
Then the following holds
FcBX a0 , ak ≤ 1
for any integer k ≥ 0.
2.4
4
Abstract and Applied Analysis
Proof. For the fixed k, we define
k f ei2πj/k z
.
fk z k
j1
2.5
Then fk ∈ HD, BX . It is clear that
1,
0,
k
1
ei2πjl/k k j1
if l ≡ 0 modk,
otherwise.
2.6
From the power series expansion of the holomorphic function f, we get
⎞⎞
⎛ ⎛
k
∞
1 ⎝
i2πjl/k
α
⎝a0 e
fk z aα z ⎠⎠
k j1
l1
|α|l
a0 ∞
2.7
alk zlk .
l1
In terms of the homogeneity of BX , we can take Ψ ∈ AutBX and Ψa0 0, then Ψ ◦ fk ∈
HD, BX . This implies that
Ψ ◦ fk z Ψ a0 ∞
lk
alk z
l1
∞
∞
lk
2
lk
alk z
alk z
Ψa0 DΨa0 D Ψa0 ···
l1
2.8
l1
DΨa0 ak zk DΨa0 a2k z2k DΨa0 a3k z3k · · · .
By making use of the orthogonality, we obtain
DΨa0 aα zα 1
2π
2π
Ψ ◦ fk
zeiθ e−iαθ dθ.
2.9
0
Hence,
1
DΨa0 aα z ≤
2π
α
2π Ψ ◦ fk zeiθ e−iαθ dθ ≤ 1.
2.10
0
This implies the following inequality
DΨa0 aα |z|α ≤ 1
2.11
Abstract and Applied Analysis
5
holds for any z ∈ D. Thus,
DΨa0 aα zα ≤ 1
2.12
holds for any z ∈ D. It means that ||DΨa0 aα || ≤ 1.
By Lemmas 2.1 and 2.2, we obtain
FcBX a0 , aα FcBX 0, DΨa0 aα DΨa0 aα ≤ 1,
2.13
which is the desired result.
3. Main Results
Theorem 3.1. Let f : D → BX be a holomorphic mapping. Then the following inequality
1 |z|k−1
FcBX fz, f k z ≤ k! k
1 − |z|2
3.1
holds for k ≥ 1 and z ∈ D.
Proof. Let gξ be a holomorphic function on D defined by
gξ f
zξ
,
1 zξ
ξ ∈ D.
3.2
Then g can be written as a power series as follows:
gξ ∞
aj ξj .
3.3
j0
In order to obtain Theorem 3.1, we need to prove the following equality:
f k z k k−1
k!
1 − |z|
2
j0
j
ak−j z|j| .
3.4
Let 0 < r < 1 such that Dz, r ⊂ D, the Cauchy integral formula shows that
fz 1
2πi
|w|r
fw
dw.
w−z
3.5
6
Abstract and Applied Analysis
Thus,
f k z k!
2πi
fw
|w|r
w − zk1
3.6
dw.
Let w z ξ/1 zξ. Then
1 − |z|2
dw
,
dξ
1 zξ2
w−zξ
1 − |z|2
1 zξ2
3.7
.
Substituting 3.7 into 3.6, we get
f
k
z k!
2πi 1 − |z|2
k!
1 − |z|
k
|zξ/1zξ̄|r
gξ1 zξk−1
dξ
ξk1
3.8
k−1 k−1
ak−j zj ,
k
j
2
j0
which prove the equality 3.4.
From Lemma 2.3, we have for any integer k ≥ 1,
3.9
FcBX a0 , ak ≤ 1.
This implies that
⎛
⎜
FcBX fz, f k z ≤ FcBX ⎝a0 , k!
1 − |z|2
≤
k!
1 − |z|
2
k
k 1 |z|
k−1 k−1
j0
j
⎞
⎟
ak−j |z|j ⎠
3.10
k−1
which completes the desired result.
Remark 3.2. If BX D, then the inequality 3.1 reduces to
2
1 − fz
k k−1
f z ≤ k! k 1 |z|
1 − |z|2
which coincides with the Theorem 1.1 of Dai and Pan 6 in one complex variable.
3.11
Abstract and Applied Analysis
7
Theorem 3.3. Let f : Bn → BX be a holomorphic mapping. Then the following inequality
FcBX fz, Dk f, z, β
⎛
⎞k−1
n
β, z
k
⎜
⎟
B
≤ k!⎝1 F
z,
β
⎠
c
2 1/2
2
1 − |z|2 β β, z 3.12
holds for k ≥ 1, β ∈ Cn \ {0} and z ∈ Bn .
Proof. For any fixed k ≥ 1, β ∈ ∂Bn , and ξ ∈ Bn . Define the following disk:
2
Δ λ ∈ C : ξ λβ < 1 .
3.13
Notice that β, ξ − ξ, ββ 0. Hence,
ξ λβ2 λ ξ, β β ξ − ξ, β β2
2 2
λ ξ, β ξ − ξ, β β < 1.
3.14
That is,
λ ξ, β < 1 − ξ − ξ, β β2 1 − |ξ|2 ξ, β 2 .
Set σ 3.15
1 − |ξ|2 |ξ, β|2 . For the fixed ξ and β, we define
gω f ξ ωσ − ξ, β β ,
ω ∈ D.
3.16
Then gω is holomorphic mapping from the unit disk D to the homogeneous domain Ω.
According to Theorem 3.1 to the functions g and ω ξ, β/σ, we have
1 |ω |k−1
FcBX g ω , g k ω ≤ k! k ,
2
1 − |ω |
3.17
which holds for k ≥ 1. Since gω fξ, and
ω β, ξ 2 ,
1 − |ξ|2 ξ, β 2
1 − ω 1 − |ξ|2
2 .
1 − |ξ|2 ξ, β 3.18
In terms of the chain rule, we have
k!
k!
α
∂f k ξ
∂f k ξ
k
α
k k
g k ω σ
σβ
αN
αN β σ D f, ξ, β .
α1
α2
α1
α2
α! ∂z1 ∂z2 · · · ∂zN
α! ∂z1 ∂z2 · · · ∂zN
|α|k
|α|k
3.19
8
Abstract and Applied Analysis
Hence,
FcBX fξ, σ k Dk f, ξ, β
⎡
⎤
k−1
2 k
2
β, ξ ⎢ 1 − |ξ| β, ξ ⎥ k
≤ k! 1 ⎣
⎦ σ .
2 1/2
2
2
2
1 − |ξ| β, ξ 1 − |ξ|
3.20
n
Note the definition of Carathéodory metric and FcB z, β 1 − |z|2 |β, z|2 /1 − |z|2 2 in
11, we can get
FcBX
k
fz, D f, z, β
k−1
n
β, z k
B
≤ k! 1 F
.
z,
β
c
2
1 − |z|2 β, z 1/2
3.21
This gives the proof of the case z ξ and β ∈ ∂Bn . For general vector β ∈ Cn \ {0}, we may
substitute β/β for β. By the homogeneous of β from the above inequality, we can obtain the
same result, which completes the proof of the Theorem 3.3.
m
Remark 3.4. If BX Bm , then Hfz Dk f, z, β, Dk f, z, β FcB fz, Dk f, z, β and
m
Hz β, β FcB z, β. Thus, the Theorem 3.3 reduces to Theorem A established by Dai et
al. 9.
Acknowledgments
The author cordially thanks the referees’ thorough reviewing with useful suggestions and
comments made to the paper. The author would also like to express this appreciation to Dr.
Liu Yang for giving him some useful discussions. This work was supported by the National
Natural Science Foundation of China nos. 11001246, 11101139, NSF of Zhejiang province
nos. Y6110260, Y6110053, and LQ12A01004, and Zhejiang Innovation Project no. T200905.
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Abstract and Applied Analysis
9
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