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

Abstract

Let 𝐵_𝑋 be the unit ball in a complex Banach space 𝑋. Assume 𝐵_𝑋 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 𝐵_𝑋 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 𝑓(𝑧) which is bounded by one in the unit disk 𝐷⊂ℂ satisfies the following inequlity 𝑓^′(𝑧≤𝑓(𝑧)1−)²1−|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: 𝑓^(𝑘)(𝑧≤𝑓(𝑧)𝑘!1−)²1−|𝑧|^2𝑘(1+|𝑧|)^𝑘−1,𝑧∈𝐷,𝑘≥1. (1.2)

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 ℂ^𝑛, respectively. Recently, Chen and Liu in [8] obtained estimates of high-order derivatives for all the bounded holomorphic functions on the unit ball of ℂ^𝑛. 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 𝑓(𝑧) is holomorphic mapping from 𝐵^𝑛 to 𝐵^𝑚. Then for any multiindex 𝑘≥1 and 𝛽∈ℂ^𝑛⧵{0}, 𝐻_𝑓(𝑧)𝐷^𝑘𝑓,𝑧,𝛽,𝐷^𝑘𝑓,𝑧,𝛽≤𝑘!1+(|⟨𝛽,𝑧⟩|)((1−|𝑧|²)|𝛽|²+|⟨𝛽,𝑧⟩|²)^1/2𝑘−1𝐻_𝑧𝛽,𝛽^𝑘, (1.3) where 𝐷^𝑘(𝑓,𝑧,𝛽)=∑_|𝛼|=𝑘(𝑘!/𝛼!)(𝜕𝑓^𝑘(𝑧)/𝜕𝑧^𝛼1₁𝜕𝑧^𝛼2₂⋯𝜕𝑧^𝛼𝑁_𝑁)𝛽^𝛼 and 𝐻_𝑧(𝛽,𝛽) is the Bergman metric on 𝐵^𝑛.

In this paper, we will extend Theorem A to holomorphic mappings from the unit ball 𝐵^𝑛 to 𝐵_𝑋 associated with the Carathéodory metric. In particular, when 𝐵_𝑋=𝐵^𝑚, 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 𝐵_𝑋.

Throughout this paper, the symbol 𝑋 is used to denote a complex Banach space with norm ||⋅||, and 𝐵_𝑋={𝑧∈𝑋∶||𝑧||<1} is the unit ball in 𝑋. Let ℂ^𝑛 be the space of 𝑛 complex variables 𝑧=(𝑧₁,…,𝑧_𝑛)^′ with the Euclidean inner product ⟨𝑧,𝑤⟩=∑^𝑛_𝑗=1𝑧_𝑗𝑤_𝑗, where the symbol ’ stands for the transpose of vector or matrix. The unit ball of ℂ^𝑛 is always written by 𝐵^𝑛. It is well known that if 𝑓 is a holomorphic mapping from 𝐵_𝑋 into 𝑋, then the following well-known expansion 𝑓𝑦=^∞_𝑛=01𝐷𝑛!^𝑛𝑓(𝑥)𝑦−𝑥^𝑛 (1.4) holds for all 𝑦 in some neighborhood of 𝑥∈𝐵_𝑋, where 𝐷^𝑛𝑓(𝑥) means the 𝑛th Fréchet derivative of 𝑓 at the point 𝑥, and 𝐷^𝑛𝑓(𝑥)𝑦−𝑥^𝑛=𝐷^𝑛𝑓(𝑥)𝑦−𝑥,𝑦−𝑥,…,𝑦−𝑥. (1.5) Furthermore, 𝐷^𝑛𝑓(𝑥) is a bounded symmetric 𝑛-linear mapping from ∏^𝑛_𝑗=1𝑋 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,…,𝛼_𝑛)∈ℕ^𝑛 is an 𝑛-tuple of nonnegative integers, |𝛼|=𝛼₁+𝛼₂+⋯+𝛼_𝑛, 𝛼!=𝛼₁!⋯𝛼_𝑛!, 𝑧^𝛼=𝑧^𝛼1₁⋯𝑧^𝛼𝑛_𝑛.

Let 𝐾(𝑧,𝑧) be the Bergman kernel function. Then the Bergman metric 𝐻_𝑧(𝛽,𝛽) can be defined as 𝐻_𝑧𝛽,𝛽=^𝑛_𝑗,𝑘=1𝜕²log𝐾(𝑧,𝑧)𝜕𝑧_𝑗𝜕𝑧_𝑘𝑢_𝑗𝑢_𝑘, (1.6) where 𝑧∈Ω,𝑢=(𝑢₁,𝑢₂,…,𝑢_𝑛)∈ℂ^𝑛. It is well known that 𝐻_𝑧(𝛽,𝛽)=(1−|𝑧|²+|⟨𝛽,𝑧⟩|²)/(1−|𝑧|²)² in [9].

Let 𝐹^𝐵𝑋_𝑐(𝑧,𝜉) be the infinitesimal form of Carathéodory metric of domain 𝐵_𝑋. By the definition of the Carathéodory metric [11], we have for any 𝜉∈𝑋, 𝐹^𝐵𝑋_𝑐(𝑧,𝜉)=sup𝐷𝑓(𝑧)𝜉∶𝑓∈𝐻𝐵_𝑋,𝐵_𝑋,𝑓(𝑧)=0, (1.7) where 𝐻(𝐵_𝑋,𝐵_𝑋) denotes the family of holomorphic mappings which map 𝐵_𝑋 into 𝐵_𝑋.

2. Some Lemmas

In order to prove the main results, we need the following lemmas. Let 𝐵_𝑋 be the unit ball in a complex Banach space 𝑋, and 𝐵_𝑋 is homogeneous.

Lemma 2.1 (see [11]). If 𝑓∈𝐻(𝐵_𝑋,𝐵_𝑋), then 𝐹^𝐵𝑋_𝑐𝑓(𝑧),𝐷𝑓(𝑧)𝜉≤𝐹^𝐵𝑋_𝑐(𝑧,𝜉),𝑧∈𝐵_𝑋,𝜉∈𝑋. (2.1) In particular, when 𝑓 is biholomorphic mapping, then 𝐹^𝐵𝑋_𝑐(𝑓(𝑧),𝐷𝑓(𝑧)𝜉)=𝐹^𝐵𝑋_𝑐(𝑧,𝜉).

Lemma 2.2 (see [12]). Consider the following: 𝐹^𝐵𝑋_𝑐(0,𝜉)=𝜉,𝜉∈𝑋. (2.2)

Lemma 2.3. Let 𝑓∈𝐻(𝐷,𝐵_𝑋). Then 𝑓 can be written with the following 𝑛-variable power series given by 𝑓(𝑧)=^∞_𝑗=0𝑎_𝑗𝑧^𝑗,𝑧∈𝐷. (2.3) Then the following holds 𝐹^𝐵𝑋_𝑐𝑎₀,𝑎_𝑘≤1 (2.4) for any integer 𝑘≥0.

Proof. For the fixed 𝑘, we define 𝑓_𝑘(𝑧)=^𝑘_𝑗=1𝑓𝑒^{𝑖(2𝜋𝑗/𝑘)}𝑧𝑘. (2.5) Then 𝑓_𝑘∈𝐻(𝐷,𝐵_𝑋). It is clear that 1𝑘^𝑘_𝑗=1𝑒^{𝑖(2𝜋𝑗𝑙/𝑘)}=1,if𝑙≡0(mod𝑘),0,otherwise. (2.6) From the power series expansion of the holomorphic function 𝑓, we get 𝑓_𝑘(𝑧)=1𝑘^𝑘_𝑗=1𝑎₀+^∞_𝑙=1𝑒^{𝑖(2𝜋𝑗𝑙/𝑘)}_|𝛼|=𝑙𝑎_𝛼𝑧^𝛼=𝑎₀+^∞_𝑙=1𝑎_𝑙𝑘𝑧^𝑙𝑘. (2.7) In terms of the homogeneity of 𝐵_𝑋, we can take Ψ∈Aut(𝐵_𝑋) and Ψ(𝑎₀)=0, then Ψ∘𝑓_𝑘∈𝐻(𝐷,𝐵_𝑋). This implies that Ψ∘𝑓_𝑘(𝑧)=Ψ𝑎₀+^∞_𝑙=1𝑎_𝑙𝑘𝑧^𝑙𝑘=Ψ𝑎₀+𝐷Ψ𝑎₀^∞_𝑙=1𝑎_𝑙𝑘𝑧^𝑙𝑘+𝐷²Ψ𝑎₀^∞_𝑙=1𝑎_𝑙𝑘𝑧^𝑙𝑘𝑎+⋯=𝐷Ψ₀𝑎_𝑘𝑧^𝑘+𝐷Ψ𝑎₀𝑎_2𝑘𝑧^2𝑘+𝐷Ψ𝑎₀𝑎_3𝑘𝑧^3𝑘+⋯. (2.8) By making use of the orthogonality, we obtain 𝐷Ψ𝑎₀𝑎_𝛼𝑧^𝛼=12𝜋₀^2𝜋Ψ∘𝑓_𝑘𝑧𝑒^𝑖𝜃𝑒^{−𝑖𝛼𝜃}𝑑𝜃. (2.9) Hence, 𝐷Ψ𝑎₀𝑎_𝛼𝑧^𝛼≤12𝜋₀^2𝜋Ψ∘𝑓_𝑘𝑧𝑒^𝑖𝜃𝑒^{−𝑖𝛼𝜃}𝑑𝜃≤1. (2.10) This implies the following inequality 𝐷Ψ𝑎₀𝑎_𝛼|𝑧|^𝛼≤1 (2.11) holds for any 𝑧∈𝐷. Thus, 𝐷Ψ𝑎₀𝑎_𝛼𝑧^𝛼≤1 (2.12) holds for any 𝑧∈𝐷. It means that ||𝐷Ψ(𝑎₀)(𝑎_𝛼)||≤1.
By Lemmas 2.1 and 2.2, we obtain 𝐹^𝐵𝑋_𝑐𝑎₀,𝑎_𝛼=𝐹^𝐵𝑋_𝑐0,𝐷Ψ𝑎₀𝑎_𝛼=𝐷Ψ𝑎₀𝑎_𝛼≤1, (2.13) which is the desired result.

3. Main Results

Theorem 3.1. Let 𝑓∶𝐷→𝐵_𝑋 be a holomorphic mapping. Then the following inequality 𝐹^𝐵𝑋_𝑐𝑓(𝑧),𝑓^(𝑘)(𝑧(|𝑧)≤𝑘!1+|)^𝑘−11−|𝑧|^2𝑘 (3.1) holds for 𝑘≥1 and 𝑧∈𝐷.

Proof. Let 𝑔(𝜉) be a holomorphic function on 𝐷 defined by 𝑔(𝜉)=𝑓𝑧+𝜉1+𝑧𝜉,𝜉∈𝐷. (3.2) Then 𝑔 can be written as a power series as follows: 𝑔(𝜉)=^∞_𝑗=0𝑎_𝑗𝜉^𝑗. (3.3) In order to obtain Theorem 3.1, we need to prove the following equality: 𝑓^(𝑘)(𝑧)=𝑘!1−|𝑧|^2𝑘_𝑗=0𝑗𝑎𝑘−1_𝑘−𝑗𝑧^|𝑗|. (3.4) Let 0<𝑟<1 such that 𝐷(𝑧,𝑟)⊂𝐷, the Cauchy integral formula shows that 𝑓(𝑧)=12𝜋𝑖_|𝑤|=𝑟𝑓(𝑤)𝑤−𝑧𝑑𝑤. (3.5) Thus, 𝑓^(𝑘)(𝑧)=𝑘!2𝜋𝑖_|𝑤|=𝑟𝑓(𝑤)(𝑤−𝑧)^𝑘+1𝑑𝑤. (3.6) Let 𝑤=(𝑧+𝜉)/(1+𝑧𝜉). Then 𝑑𝑤=|𝑧|𝑑𝜉1−²(1+𝑧𝜉)²|𝑧|,𝑤−𝑧=𝜉1−²(1+𝑧𝜉)². (3.7) Substituting (3.7) into (3.6), we get 𝑓^(𝑘)(𝑧)=𝑘!2𝜋𝑖1−|𝑧|^{2𝑘|(𝑧+𝜉)/(1+𝑧̄𝜉)|=𝑟}𝑔(𝜉)(1+𝑧𝜉)^𝑘−1𝜉^𝑘+1=𝑑𝜉𝑘!1−|𝑧|^{2𝑘𝑘−1𝑗=0}𝑗𝑎𝑘−1_𝑘−𝑗𝑧^𝑗, (3.8) which prove the equality (3.4).
From Lemma 2.3, we have for any integer 𝑘≥1, 𝐹^𝐵𝑋_𝑐𝑎₀,𝑎_𝑘≤1. (3.9) This implies that 𝐹^𝐵𝑋_𝑐𝑓(𝑧),𝑓^(𝑘)(𝑧)≤𝐹^𝐵𝑋_𝑐𝑎₀,𝑘!1−|𝑧|^{2𝑘𝑘−1𝑗=0}𝑗𝑎𝑘−1_𝑘−𝑗|𝑧|^𝑗≤𝑘!1−|𝑧|^2𝑘(1+|𝑧|)^𝑘−1 (3.10) which completes the desired result.

Remark 3.2. If 𝐵_𝑋=𝐷, then the inequality (3.1) reduces to 𝑓^(𝑘)(𝑧𝑓(𝑧)≤𝑘!1−)²1−|𝑧|^2𝑘(1+|𝑧|)^𝑘−1 (3.11) which coincides with the Theorem 1.1 of Dai and Pan [6] in one complex variable.

Theorem 3.3. Let 𝑓∶𝐵^𝑛→𝐵_𝑋 be a holomorphic mapping. Then the following inequality 𝐹^𝐵𝑋_𝑐𝑓(𝑧),𝐷^𝑘𝑓,𝑧,𝛽≤𝑘!1+𝛽,𝑧1−|𝑧|²𝛽²+𝛽,𝑧^21/2𝑘−1𝐹^𝐵𝑛_𝑐𝑧,𝛽^𝑘 (3.12) holds for 𝑘≥1, 𝛽∈ℂ^𝑛⧵{0} and 𝑧∈𝐵^𝑛.

Proof. For any fixed 𝑘≥1,𝛽∈𝜕𝐵^𝑛, and 𝜉∈𝐵^𝑛. Define the following disk: Δ=𝜆∈ℂ∶𝜉+𝜆𝛽²<1. (3.13) Notice that ⟨𝛽,𝜉−⟨𝜉,𝛽⟩𝛽⟩=0. Hence, 𝜉+𝜆𝛽²=𝜆+𝜉,𝛽𝛽+𝜉−𝜉,𝛽𝛽²=𝜆+𝜉,𝛽²+𝜉−𝜉,𝛽𝛽²<1. (3.14) That is, 𝜆+𝜉,𝛽<1−𝜉−𝜉,𝛽𝛽²=1−𝜉²+𝜉,𝛽². (3.15) Set 𝜎=1−|𝜉|²+|⟨𝜉,𝛽⟩|². For the fixed 𝜉 and 𝛽, we define 𝑔(𝜔)=𝑓𝜉+𝜔𝜎−𝜉,𝛽𝛽,𝜔∈𝐷. (3.16) Then 𝑔(𝜔) 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 𝐹^𝐵𝑋_𝑐𝑔𝜔^′,𝑔^(𝑘)𝜔^′𝜔≤𝑘!1+^′𝑘−11−𝜔^′2𝑘, (3.17) which holds for 𝑘≥1. Since 𝑔(𝜔^′)=𝑓(𝜉), and 𝜔^′=𝛽,𝜉1−𝜉²+𝜉,𝛽²,1−𝜔^′2=1−𝜉²1−𝜉²+𝜉,𝛽². (3.18) In terms of the chain rule, we have 𝑔^(𝑘)𝜔^′=_|𝛼|=𝑘𝑘!𝛼!𝜕𝑓^𝑘(𝜉)𝜕𝑧^𝛼1₁𝜕𝑧^𝛼2₂⋯𝜕𝑧^𝛼𝑁_𝑁𝜎𝛽^𝛼=𝜎^𝑘_|𝛼|=𝑘𝑘!𝛼!𝜕𝑓^𝑘(𝜉)𝜕𝑧^𝛼1₁𝜕𝑧^𝛼2₂⋯𝜕𝑧^𝛼𝑁_𝑁𝛽^𝛼=𝜎^𝑘𝐷^𝑘𝑓,𝜉,𝛽. (3.19) Hence, 𝐹^𝐵𝑋_𝑐𝑓(𝜉),𝜎^𝑘𝐷^𝑘𝑓,𝜉,𝛽≤𝑘!1+𝛽,𝜉(1−𝜉²+𝛽,𝜉²)^1/2𝑘−1(1−𝜉²)+𝛽,𝜉²1−𝜉^22𝑘𝜎^𝑘. (3.20) Note the definition of Carathéodory metric and 𝐹^𝐵𝑛_𝑐(𝑧,𝛽)=(1−|𝑧|²+|⟨𝛽,𝑧⟩|²)/(1−|𝑧|²)² in [11], we can get 𝐹^𝐵𝑋_𝑐𝑓(𝑧),𝐷^𝑘𝑓,𝑧,𝛽≤𝑘!1+𝛽,𝑧(1−|𝑧|²+𝛽,𝑧²)^1/2𝑘−1𝐹^𝐵𝑛_𝑐𝑧,𝛽^𝑘. (3.21) This gives the proof of the case 𝑧=𝜉 and 𝛽∈𝜕𝐵_𝑛. For general vector 𝛽∈ℂ^𝑛⧵{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. If 𝐵_𝑋=𝐵^𝑚, then 𝐻_𝑓(𝑧)(𝐷^𝑘(𝑓,𝑧,𝛽),𝐷^𝑘(𝑓,𝑧,𝛽))=𝐹^𝐵𝑚_𝑐(𝑓(𝑧),𝐷^𝑘(𝑓,𝑧,𝛽)) and 𝐻_𝑧(𝛽,𝛽)=𝐹^𝐵𝑚_𝑐(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).