# Schwarz-Pick Estimates for Holomorphic Mappings with Values in Homogeneous Ball

Authors: Jianfei Wang
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

LetBX 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 Bn to BX associated with the Carathéodory metric, which extend the corresponding Chen and Liu, Dai et al. results.


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## Body

## 1. Introduction

By the classical Pick’s invariant form of Schwarz’s lemma, a holomorphic functionf(z) which is bounded by one in the unit disk D⊂ℂ satisfies the following inequlity
(1.1)|f′(z)|≤1-|f(z)|21-|z|2
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:
(1.2)|f(k)(z)|≤k!(1-|f(z)|2)(1-|z|2)k(1+|z|)k-1,z∈D,k≥1.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.
Supposef(z) is holomorphic mapping from Bn to Bm. Then for any multiindex k≥1 and β∈ℂn∖{0},
(1.3)Hf(z)(Dk(f,z,β),Dk(f,z,β))≤k!(1+(|〈β,z〉|)((1-|z|2)|β|2+|〈β,z〉|2)1/2)k-1(Hz(β,β))k,
where Dk(f,z,β)=∑|α|=k(k!/α!)(∂fk(z)/∂z1α1∂z2α2⋯∂zNαN)βα and Hz(β,β) is the Bergman metric on Bn.In this paper, we will extend Theorem A to holomorphic mappings from the unit ballBn 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 symbolX 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〉=∑j=1nzjwj¯, 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
(1.4)f(y)=∑n=0∞‍1n!Dnf(x)((y-x)n)
holds for all y in some neighborhood of x∈BX, where Dnf(x) means the nth Fréchet derivative of f at the point x, and
(1.5)Dnf(x)((y-x)n)=Dnf(x)(y-x,y-x,…,y-x).
Furthermore, Dnf(x) is a bounded symmetric n-linear mapping from ∏j=1n‍X into X. For a domain Ω∈X, a mapping f:Ω→X is called to be biholomorphic if f(Ω) is a domain; the inverse f-1 exists and is holomorphic on f(Ω). Let Aut(Ω) 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∈Aut(Ω) such that f(x)=y.In multiindex notation,α=(α1,α2,…,αn)∈ℕn is an n-tuple of nonnegative integers, |α|=α1+α2+⋯+αn,  α!=α1!⋯αn!,  zα=z1α1⋯znαn.LetK(z,z) be the Bergman kernel function. Then the Bergman metric Hz(β,β) can be defined as
(1.6)Hz(β,β)=∑j,k=1n‍∂2logK(z,z)∂zj∂z-kuju-k,
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].LetFcBX(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,
(1.7)FcBX(z,ξ)=sup{|Df(z)ξ|:f∈H(BX,BX),f(z)=0},
where H(BX,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. LetBX be the unit ball in a complex Banach space X, and BX is homogeneous.Lemma 2.1 (see [11]).
Iff∈H(BX,BX), then
(2.1)FcBX(f(z),Df(z)ξ)≤FcBX(z,ξ),z∈BX,ξ∈X.
In particular, when f is biholomorphic mapping, then FcBX(f(z),Df(z)ξ)=FcBX(z,ξ).Lemma 2.2 (see [12]).
Consider the following:(2.2)FcBX(0,ξ)=∥ξ∥,ξ∈X.Lemma 2.3.
Letf∈H(D,BX). Then f can be written with the following n-variable power series given by
(2.3)f(z)=∑j=0∞‍ajzj,z∈D.
Then the following holds
(2.4)FcBX(a0,ak)≤1
for any integer k≥0.Proof.
For the fixedk, we define
(2.5)fk(z)=∑j=1k‍f(ei(2πj/k)z)k.
Then fk∈H(D,BX). It is clear that
(2.6)1k∑j=1k‍ei(2πjl/k)={1,ifl≡0(modk),0,otherwise.
From the power series expansion of the holomorphic function f, we get
(2.7)fk(z)=1k(∑j=1k‍(a0+∑l=1∞‍ei(2πjl/k)∑|α|=l‍aαzα))=a0+∑l=1∞‍alkzlk.
In terms of the homogeneity of BX, we can take Ψ∈Aut(BX) and Ψ(a0)=0, then Ψ∘fk∈H(D,BX). This implies that
(2.8)Ψ∘fk(z)=Ψ(a0+∑l=1∞‍alkzlk)=Ψ(a0)+DΨ(a0)(∑l=1∞‍alkzlk)+D2Ψ(a0)(∑l=1∞‍alkzlk)+⋯=DΨ(a0)(ak)zk+DΨ(a0)(a2k)z2k+DΨ(a0)(a3k)z3k+⋯.
By making use of the orthogonality, we obtain
(2.9)DΨ(a0)(aα)zα=12π∫02π‍(Ψ∘fk)(zeiθ)e-iαθdθ.
Hence,
(2.10)∥DΨ(a0)(aα)zα∥≤12π∫02π‍∥(Ψ∘fk)(zeiθ)e-iαθ∥dθ≤1.
This implies the following inequality
(2.11)∥DΨ(a0)(aα)∥|z|α≤1
holds for any z∈D. Thus,
(2.12)∥DΨ(a0)(aαzα)∥≤1
holds for any z∈D¯. It means that ||DΨ(a0)(aα)||≤1.
By Lemmas2.1 and 2.2, we obtain
(2.13)FcBX(a0,aα)=FcBX(0,DΨ(a0)(aα))=∥DΨ(a0)(aα)∥≤1,
which is the desired result.


## 3. Main Results

Theorem 3.1.
Letf:D→BX be a holomorphic mapping. Then the following inequality
(3.1)FcBX(f(z),f(k)(z))≤k!(1+|z|)k-1(1-|z|2)k
holds for k≥1 and z∈D.Proof.
Letg(ξ) be a holomorphic function on D defined by
(3.2)g(ξ)=f(z+ξ1+z-ξ),ξ∈D.
Then g can be written as a power series as follows:
(3.3)g(ξ)=∑j=0∞‍ajξj.
In order to obtain Theorem 3.1, we need to prove the following equality:
(3.4)f(k)(z)=k!1-|z|2∑j=0k‍(k-1j)ak-jz-|j|.
Let 0<r<1 such that D(z,r)⊂D, the Cauchy integral formula shows that
(3.5)f(z)=12πi∫|w|=r‍f(w)w-zdw.
Thus,
(3.6)f(k)(z)=k!2πi∫|w|=r‍f(w)(w-z)k+1dw.
Let w=(z+ξ)/(1+z-ξ). Then
(3.7)dwdξ=1-|z|2(1+z-ξ)2,w-z=ξ1-|z|2(1+z-ξ)2.
Substituting (3.7) into (3.6), we get
(3.8)f(k)(z)=k!2πi(1-|z|2)k∫|(z+ξ)/(1+z̅ξ)|=r‍g(ξ)(1+z-ξ)k-1ξk+1dξ=k!(1-|z|2)k∑j=0k-1‍(k-1j)ak-jz-j,
which prove the equality (3.4).
From Lemma2.3, we have for any integer k≥1,
(3.9)FcBX(a0,ak)≤1.
This implies that
(3.10)FcBX(f(z),f(k)(z))≤FcBX(a0,k!(1-|z|2)k∑j=0k-1‍(k-1j)ak-j|z|j)≤k!(1-|z|2)k(1+|z|)k-1
which completes the desired result.Remark 3.2.
IfBX=D, then the inequality (3.1) reduces to
(3.11)|f(k)(z)|≤k!1-|f(z)|2(1-|z|2)k(1+|z|)k-1
which coincides with the Theorem 1.1 of Dai and Pan [6] in one complex variable.Theorem 3.3.
Letf:Bn→BX be a holomorphic mapping. Then the following inequality
(3.12)FcBX(f(z),Dk(f,z,β))≤k!(1+|〈β,z〉|((1-|z|2)|β|2+|〈β,z〉|2)1/2)k-1[FcBn(z,β)]k
holds for k≥1, β∈ℂn∖{0} and z∈Bn.Proof.
For any fixedk≥1,β∈∂Bn, and ξ∈Bn. Define the following disk:
(3.13)Δ={λ∈ℂ:|ξ+λβ|2<1}.
Notice that 〈β,ξ-〈ξ,β〉β〉=0. Hence,
(3.14)|ξ+λβ|2=|(λ+〈ξ,β〉)β+ξ-〈ξ,β〉β|2=|λ+〈ξ,β〉|2+|ξ-〈ξ,β〉β|2<1.
That is,
(3.15)|λ+〈ξ,β〉|<1-|ξ-〈ξ,β〉β|2=1-|ξ|2+|〈ξ,β〉|2.
Set σ=1-|ξ|2+|〈ξ,β〉|2. For the fixed ξ and β, we define
(3.16)g(ω)=f(ξ+(ωσ-〈ξ,β〉)β),ω∈D.
Then g(ω) is holomorphic mapping from the unit disk D to the homogeneous domain Ω.
According to Theorem3.1 to the functions g and ω'=(〈ξ,β〉)/σ, we have
(3.17)FcBX(g(ω′),g(k)(ω′))≤k!(1+|ω′|)k-1(1-|ω′|2)k,
which holds for k≥1. Since g(ω')=f(ξ), and
(3.18)|ω′|=|〈β,ξ〉|1-|ξ|2+|〈ξ,β〉|2,1-|ω′|2=1-|ξ|21-|ξ|2+|〈ξ,β〉|2.
In terms of the chain rule, we have
(3.19)g(k)(ω′)=∑|α|=k‍k!α!∂fk(ξ)∂z1α1∂z2α2⋯∂zNαN(σβ)α=σk∑|α|=k‍k!α!∂fk(ξ)∂z1α1∂z2α2⋯∂zNαNβα=σkDk(f,ξ,β).
Hence,
(3.20)FcBX(f(ξ),σkDk(f,ξ,β))≤k!(1+|〈β,ξ〉|(1-|ξ|2+|〈β,ξ〉|2)1/2)k-1[(1-|ξ|2)+|〈β,ξ〉|2(1-|ξ|2)2]kσk.
Note the definition of Carathéodory metric and FcBn(z,β)=(1-|z|2+|〈β,z〉|2)/(1-|z|2)2 in [11], we can get
(3.21)FcBX(f(z),Dk(f,z,β))≤k!(1+|〈β,z〉|(1-|z|2+|〈β,z〉|2)1/2)k-1[FcBn(z,β)]k.
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.Remark 3.4.
IfBX=Bm, then Hf(z)(Dk(f,z,β),Dk(f,z,β))=FcBm(f(z),Dk(f,z,β)) and Hz(β,β)=FcBm(z,β). Thus, the Theorem 3.3 reduces to Theorem A established by Dai et al. [9].



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