schwarz-pick estimates for holomorphic mappings with values in … · 2014. 4. 25. · abstract and...

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2012, Article ID 647972, 9 pagesdoi:10.1155/2012/647972

Research ArticleSchwarz-Pick Estimates for Holomorphic Mappingswith Values in Homogeneous Ball

Jianfei Wang

Department of Mathematics, Zhejiang Normal University, Zhejiang, Jinhua 321004, China

Correspondence should be addressed to Jianfei Wang, [email protected]

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 CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

Let BX be the unit ball in a complex Banach space X. Assume BX is homogeneous. Thegeneralization of the Schwarz-Pick estimates of partial derivatives of arbitrary order is establishedfor holomorphic mappings from the unit ball Bn to BX associated with the Caratheodory 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)whichis bounded by one in the unit disk D ⊂ C satisfies the following inequlity

∣∣f ′(z)

∣∣ ≤ 1 − ∣∣f(z)∣∣2

1 − |z|2(1.1)

at each point z of D. Ruscheweyh in [1] firstly obtained best-possible estimates of higherorder derivatives of bounded holomorphic functions on the unit disk in 1985. Recently, a lotof attention (see Ghatage et al. [2], MacCluer et al. [3], Avkhadiev and Wirths [4], Ghatageand Zheng [5], Dai and Pan [6]) has been paid to the Schwarz-Pick estimates of high-orderderivative estimates in one complex variable. The best result is given as follows:

∣∣∣f (k)(z)

∣∣∣ ≤

k!(

1 − ∣∣f(z)∣∣2)

(

1 − |z|2)k

(1 + |z|)k−1, 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 higherdimensions. Anderson et al. [7] gave Schwarz-Pick estimates of derivatives of arbitrary orderof 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 boundedholomorphic functions on the unit ball of C

n. Later, Dai et al. in [9, 10] generalized thehigh order Schwarz-Pick estimates for holomorphic mappings between unit balls in complexHilbert space. Their main result is expressed as follows.

Theorem A. Suppose f(z) is holomorphic mapping from Bn to Bm. Then for any multiindex k ≥ 1and β ∈ C

n \ {0},

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

,

(1.3)

where Dk(f, z, β) =∑

|α|=k(k!/α!)(∂fk(z)/∂zα1

1 ∂zα22 · · · ∂zαN

N )βα and Hz(β, β) is the Bergmanmetric on Bn.

In this paper, we will extend Theorem A to holomorphic mappings from the unit ballBn to BX associated with the Caratheodory metric. In particular, when BX = Bm, our resultcoincides with Theorem A. Furthermore, our result shows that the high-order Schwarz-Pickestimates 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 withnorm || · ||, and BX = {z ∈ X : ||z|| < 1} is the unit ball in X. Let C

n be the space of n complexvariables z = (z1, . . . , zn)′ with the Euclidean inner product 〈z,w〉 =

∑nj=1 zjwj , 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 followingwell-known expansion

f(

y)

=∞∑

n=0

1n!Dnf(x)

((

y − x)n)

(1.4)

holds for all y in some neighborhood of x ∈ BX , where Dnf(x) means the nth Frechetderivative of f at the point x, and

Dnf(x)((

y − x)n) = Dnf(x)

(

y − x, y − x, . . . , y − x)

. (1.5)

Furthermore, Dnf(x) is a bounded symmetric n-linear mapping from∏n

j=1X into X. For adomain Ω ∈ X, a mapping f : Ω → X is called to be biholomorphic if f(Ω) is a domain; theinverse f−1 exists and is holomorphic on f(Ω). Let Aut(Ω) denote the set of biholomorphicmappings 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 = zα11 · · · zαn

n .

Abstract and Applied Analysis 3

Let K(z, z) be the Bergman kernel function. Then the Bergman metric Hz(β, β) can bedefined as

Hz

(

β, β)

=n∑

j,k=1

∂2 logK(z, z)∂zj∂zk

ujuk, (1.6)

where z ∈ Ω, u = (u1, u2, . . . , un) ∈ Cn. It is well known thatHz(β, β) = (1−|z|2+ |〈β, z〉|2)/(1−

|z|2)2 in [9].Let FBX

c (z, ξ) be the infinitesimal form of Caratheodory metric of domain BX . By thedefinition of the Caratheodory metric [11], we have for any ξ ∈ X,

FBXc (z, ξ) = sup

{∣∣Df(z)ξ

∣∣ : f ∈ H(BX, BX), f(z) = 0

}

, (1.7)

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. Let BX be the unit ball in acomplex Banach space X, and BX is homogeneous.

Lemma 2.1 (see [11]). If f ∈ H(BX, BX), then

FBXc

(

f(z), Df(z)ξ) ≤ FBX

c (z, ξ), z ∈ BX, ξ ∈ X. (2.1)

In particular, when f is biholomorphic mapping, then FBXc (f(z), Df(z)ξ) = FBX

c (z, ξ).

Lemma 2.2 (see [12]). Consider the following:

FBXc (0, ξ) = ‖ξ‖, ξ ∈ X. (2.2)

Lemma 2.3. Let f ∈ H(D,BX). Then f can be written with the following n-variable power seriesgiven by

f(z) =∞∑

j=0

ajzj , z ∈ D. (2.3)

Then the following holds

FBXc (a0, ak) ≤ 1 (2.4)

for any integer k ≥ 0.


Proof. For the fixed k, we define

fk(z) =k∑

j=1

f(

ei(2πj/k)z)

k. (2.5)

Then fk ∈ H(D,BX). It is clear that

1k

k∑

j=1

ei(2πjl/k) =

{

1, if l ≡ 0 (modk),0, otherwise.

(2.6)

From the power series expansion of the holomorphic function f , we get

fk(z) =1k

⎛

⎝

k∑

j=1

⎛

⎝a0 +∞∑

l=1

ei(2πjl/k)∑

|α|=laαz

α

⎞

⎠

⎞

⎠

= a0 +∞∑

l=1

alkzlk.

(2.7)

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

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 + · · · .

(2.8)

By making use of the orthogonality, we obtain

DΨ(a0)(aα)zα =12π

∫2π

0

(

Ψ ◦ fk)(

zeiθ)

e−iαθdθ. (2.9)

Hence,

‖DΨ(a0)(aα)zα‖ ≤ 12π

∫2π

0

∥∥∥

(

Ψ ◦ fk)(

zeiθ)

e−iαθ∥∥∥dθ ≤ 1. (2.10)

This implies the following inequality

‖DΨ(a0)(aα)‖|z|α ≤ 1 (2.11)


holds for any z ∈ D. Thus,

‖DΨ(a0)(aαzα)‖ ≤ 1 (2.12)

holds for any z ∈ D. It means that ||DΨ(a0)(aα)|| ≤ 1.By Lemmas 2.1 and 2.2, we obtain

FBXc (a0, aα) = FBX

c (0, DΨ(a0)(aα)) = ‖DΨ(a0)(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

FBXc

(

f(z), f (k)(z))

≤ k!(1 + |z|)k−1(

1 − |z|2)k (3.1)

holds for k ≥ 1 and z ∈ D.

Proof. Let g(ξ) be a holomorphic function on D defined by

g(ξ) = f

(z + ξ

1 + zξ

)

, ξ ∈ D. (3.2)

Then g can be written as a power series as follows:

g(ξ) =∞∑

j=0

ajξj . (3.3)

In order to obtain Theorem 3.1, we need to prove the following equality:

f (k)(z) =k!

1 − |z|2k∑

j=0

(k − 1j

)

ak−jz|j|. (3.4)

Let 0 < r < 1 such that D(z, r) ⊂ D, the Cauchy integral formula shows that

f(z) =1

2πi

∫

|w|=r

f(w)w − z

dw. (3.5)


Thus,

f (k)(z) =k!2πi

∫

|w|=r

f(w)

(w − z)k+1dw. (3.6)

Let w = (z + ξ)/(1 + zξ). Then

dw

dξ=

1 − |z|2(1 + zξ)2

, w − z = ξ1 − |z|2(1 + zξ)2

. (3.7)

Substituting (3.7) into (3.6), we get

f (k)(z) =k!

2πi(

1 − |z|2)k

∫

|(z+ξ)/(1+zξ)|=r

g(ξ)(1 + zξ)k−1

ξk+1dξ

=k!

(

1 − |z|2)k

k−1∑

j=0

(k − 1j

)

ak−jzj ,

(3.8)

which prove the equality (3.4).From Lemma 2.3, we have for any integer k ≥ 1,

FBXc (a0, ak) ≤ 1. (3.9)

This implies that

FBXc

(

f(z), f (k)(z))

≤ FBXc

⎛

⎜⎝a0,

k!(

1 − |z|2)k

k−1∑

j=0

(k − 1j

)

ak−j |z|j⎞

⎟⎠

≤ k!(

1 − |z|2)k

(1 + |z|)k−1(3.10)

which completes the desired result.

Remark 3.2. If BX = D, then the inequality (3.1) reduces to

∣∣∣f (k)(z)

∣∣∣ ≤ k!

1 − ∣∣f(z)∣∣2(

1 − |z|2)k

(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 : Bn → BX be a holomorphic mapping. Then the following inequality

FBXc

(

f(z), Dk(f, z, β)) ≤ k!

⎛

⎜⎝1 +

∣∣⟨

β, z⟩∣∣

((

1 − |z|2)∣∣β∣∣2 +∣∣⟨

β, z⟩∣∣2)1/2

⎞

⎟⎠

k−1[

FBn

c

(

z, β)]k

(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:

Δ ={

λ ∈ C :∣∣ξ + λβ

∣∣2< 1}

. (3.13)

Notice that 〈β, ξ − 〈ξ, β〉β〉 = 0. Hence,

∣∣ξ + λβ

∣∣2 =

∣∣(

λ +⟨

ξ, β⟩)

β + ξ − ⟨ξ, β⟩β∣∣2

=∣∣λ +

⟨

ξ, β⟩∣∣2 +∣∣ξ − ⟨ξ, β⟩β∣∣2 < 1.

(3.14)

That is,

∣∣λ +

⟨

ξ, β⟩∣∣ <

√

1 − ∣∣ξ − ⟨ξ, β⟩β∣∣2 =√

1 − |ξ|2 + ∣∣⟨ξ, β⟩∣∣2. (3.15)

Set σ =√

1 − |ξ|2 + |〈ξ, β〉|2. For the fixed ξ and β, we define

g(ω) = f(

ξ +(

ωσ − ⟨ξ, β⟩)β), 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

FBXc

(

g(

ω′), g(k)(ω′))

≤ k!(1 + |ω′|)k−1(

1 − |ω′|2)k

, (3.17)

which holds for k ≥ 1. Since g(ω′) = f(ξ), and

∣∣ω′∣∣ =

∣∣⟨

β, ξ⟩∣∣

√

1 − |ξ|2 + ∣∣⟨ξ, β⟩∣∣2, 1 − ∣∣ω′∣∣2 =

1 − |ξ|2

1 − |ξ|2 + ∣∣⟨ξ, β⟩∣∣2. (3.18)

In terms of the chain rule, we have

g(k)(ω′) =∑

|α|=k

k!α!

∂fk(ξ)∂zα1

1 ∂zα22 · · · ∂zαN

N

(

σβ)α = σk

∑

|α|=k

k!α!

∂fk(ξ)∂zα1

1 ∂zα22 · · · ∂zαN

N

βα = σkDk(f, ξ, β)

.

(3.19)


Hence,

FBXc

(

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

⎤

⎥⎦

k

σk.

(3.20)

Note the definition of Caratheodory metric and FBn

c (z, β) = (1 − |z|2 + |〈β, z〉|2)/(1 − |z|2)2 in[11], we can get

FBXc

(

f(z), Dk(f, z, β)) ≤ k!

(

1 +

∣∣⟨

β, z⟩∣∣

(1 − |z|2 + ∣∣⟨β, z⟩∣∣2)1/2

)k−1[

FBn

c

(

z, β)]k

. (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 thesame result, which completes the proof of the Theorem 3.3.

Remark 3.4. If BX = Bm, then Hf(z)(Dk(f, z, β), Dk(f, z, β)) = FBm

c (f(z), Dk(f, z, β)) andHz(β, β) = FBm

c (z, β). Thus, the Theorem 3.3 reduces to Theorem A established by Dai etal. [9].

Acknowledgments

The author cordially thanks the referees’ thorough reviewing with useful suggestions andcomments 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 NationalNatural Science Foundation of China (nos. 11001246, 11101139), NSF of Zhejiang province(nos. Y6110260, Y6110053, and LQ12A01004), and Zhejiang Innovation Project (no. T200905).

References

[1] St. Ruscheweyh, “Two remarks on bounded analytic functions,” Serdica, vol. 11, no. 2, pp. 200–202,1985.

[2] P. Ghatage, J. Yan, and D. Zheng, “Composition operators with closed range on the Bloch space,”Proceedings of the American Mathematical Society, vol. 129, no. 7, pp. 2039–2044, 2001.

[3] B. D. MacCluer, K. Stroethoff, and R. Zhao, “Generalized Schwarz-Pick estimates,” Proceedings of theAmerican Mathematical Society, vol. 131, no. 2, pp. 593–599, 2003.

[4] F. G. Avkhadiev and K.-J. Wirths, “Schwarz-Pick inequalities for derivatives of arbitrary order,”Constructive Approximation, vol. 19, no. 2, pp. 265–277, 2003.

[5] P. Ghatage and D. Zheng, “Hyperbolic derivatives and generalized Schwarz-Pick estimates,”Proceedings of the American Mathematical Society, vol. 132, no. 11, pp. 3309–3318, 2004.

[6] S. Dai and Y. Pan, “Note on Schwarz-Pick estimates for bounded and positive real part analyticfunctions,” Proceedings of the American Mathematical Society, vol. 136, no. 2, pp. 635–640, 2008.

[7] J. M. Anderson, M. A. Dritschel, and J. Rovnyak, “Schwarz-Pick inequalities for the Schur-Agler classon the polydisk and unit ball,” Computational Methods and Function Theory, vol. 8, no. 1-2, pp. 339–361,2008.

[8] Z. H. Chen and Y. Liu, “Schwarz-Pick estimates for bounded holomorphic functions in the unit ballof Cn,” Acta Mathematica Sinica, vol. 26, no. 5, pp. 901–908, 2010.


[9] S. Dai, H. Chen, and Y. Pan, “The Schwarz-Pick lemma of high order in several variables,” MichiganMathematical Journal, vol. 59, no. 3, pp. 517–533, 2010.

[10] S. Dai, H. Chen, and Y. Pan, “The high order Schwarz-Pick lemma on complex Hilbert balls,” ScienceChina, vol. 53, no. 10, pp. 2649–2656, 2010.

[11] S. G. Krantz, Function Theory of Several Complex Variables, John Wiley & Sons, New York, NY, USA,1982.

[12] S. Gong, Convex and Starlike Mappings in Several Complex Variables, vol. 435 of Mathematics and itsApplications (China Series), Kluwer Academic Publishers, Dordrecht, The Netherlands, 1998.

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