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SCHWARZ TYPE LEMMAS AND THEIR APPLICATIONS IN BANACH

SPACES

SHAOLIN CHEN, HIDETAKA HAMADA, SAMINATHAN PONNUSAMY, AND

RAMAKRISHNAN VIJAYAKUMAR

ABSTRACT. The main purpose of this paper is to develop some methods to in-

vestigate the Schwarz type lemmas of holomorphic mappings and pluriharmonic

mappings in Banach spaces. Initially, we extend the classical Schwarz lemmas

of holomorphic mappings to Banach spaces, and then we apply these extensions

to establish a sharp Bloch type theorem for pluriharmonic mappings on homo-

geneous unit balls of Cn and to obtain some sharp boundary Schwarz type lem-

mas for holomorphic mappings in Banach spaces. Furthermore, we improve and

generalize the classical Schwarz lemmas of planar harmonic mappings into the

sharp forms of Banach spaces, and present some applications to sharp boundary

Schwarz type lemmas for pluriharmonic mappings in Banach spaces. Addition-

ally, using a relatively simple method of proof, we prove some sharp Schwarz-

Pick type estimates of pluriharmonic mappings in JB∗-triples, and the obtained

results provide the improvements and generalizations of the corresponding re-

sults in [9].

CONTENTS

1. Preliminaries 2

2. Schwarz type lemmas of holomorphic mappings and their applications 5

3. Schwarz type lemmas of pluriharmonic mappings and their applications 9

4. Proofs of the main results 16

Part I 16

The proof of Theorem 2.2 17





Part II 24





2010 Mathematics Subject Classification. Primary 32A30, 32U05, 32K05; Secondary 30C80,

31C10, 32M15.

Keywords. Banach space, bounded symmetric domain, harmonic function, holomorphic map-

ping, pluriharmonic mapping, Schwarz type lemma

1

http://arxiv.org/abs/2110.02767v1

SCHWARZ TYPE LEMMAS AND THEIR APPLICATIONS IN BANACH SPACES 2




5. A concluding remark 29

6. Acknowledgments 29

References 29

1. PRELIMINARIES

It is well known that the Schwarz lemma has become a crucial theme in lots

of branches of mathematical research for more than a hundred years to date. We

refer the reader to [1, 5, 10, 21, 37, 38, 40, 52, 59, 61] for more details on this

topic. This paper continues the study of the classical Schwarz lemmas of holomor-

phic mappings and harmonic mappings (or complex-valued harmonic functions).

First, we extend the classical Schwarz lemmas of holomorphic mappings to Ba-

nach spaces, and then we use the obtained results to establish a sharp Bloch type

theorem for pluriharmonic mappings on homogeneous unit balls of Cn and ob-

tain sharp boundary Schwarz type lemmas for holomorphic mappings in Banach

spaces. In addition, we improve and generalize the classical Schwarz lemmas of

planar harmonic mappings into the sharp forms of Banach spaces, and obtain some

applications to sharp boundary Schwarz type lemmas for pluriharmonic mappings

in Banach spaces. At last, we use a relatively simple method to prove some sharp

Schwarz-Pick type estimates of pluriharmonic mappings in JB∗-triples, and the ob-

tained results are also the improvements and generalizations of the corresponding

known results.

In order to state our main results, we need to recall some basic definitions and

introduce some necessary terminologies.

Let Cn be the complex space of dimension n (n ≥ 1), and ‖·‖e be the Euclidean

norm on Cn. For real or complex Banach spaces X and Y with norm ‖ · ‖X and

‖ · ‖Y , respectively, let L(X,Y ) be the space of all continuous linear operators

from X into Y with the standard operator norm

‖A‖ = supx∈X\0

‖Ax‖Y‖x‖X

,

where A ∈ L(X,Y ). Then L(X,Y ) is a Banach space with respect to this norm.

Denote by X∗ the dual space of the real or complex Banach space X. For x ∈X\0, let

T (x) = lx ∈ X∗ : lx(x) = ‖x‖X and ‖lx‖X∗ = 1.

Then the well known Hahn-Banach theorem implies that T (x) 6= ∅.

Let ψ be a mapping of a domain Ω ⊂ X into a real or complex Banach space

Y , where X is a complex Banach space. We say that ψ is differentiable at z ∈ Ω


if there exists a bounded real linear operator Dψ(z) : X → Y such that

lim‖τ‖X→0+

‖ψ(z + τ)− ψ(z) −Dψ(z)τ‖Y‖τ‖X

= 0.

Here Dψ(z) is called the Frechet derivative of ψ at z. If Y is a complex Banach

space and Dψ(z) is bounded complex linear for each z ∈ Ω, then ψ is said to be

holomorphic on Ω. Also, for a differentiable mapping ψ : Ω → Y and for a point

z0 ∈ Ω which satisfies one of the following conditions:

(i) ψ(z0) = 0;

(ii) ψ(z0) 6= 0 and ‖ψ(z)‖Y is differentiable at z = z0,

we define

|∇‖ψ‖Y (z0)| = sup‖β‖X=1

limR∋t→0+

|‖ψ(z0 + tβ)‖Y − ‖ψ(z0)‖Y |t

.

As in the proof of [60, eq.(3.1)], we obtain the following result.

Proposition 1.1.

(1.1) |∇‖ψ‖Y (z0)| = ‖Dψ(z0)‖ if ψ(z0) = 0;

sup‖β‖X=1

∣∣lψ(z0)(Dψ(z0)β)∣∣ if ψ(z0) 6= 0,

where lψ(z0) ∈ T (ψ(z0)).

Let Ω be a domain in a complex Banach space X. A mapping f of Ω into a real

or complex Banach space Y is said to be pluriharmonic if the restriction of l(f(·))to every holomorphic curve is harmonic for any l ∈ Y ∗ (cf. [9, 18, 32, 50, 51]).

In particular, if Ω is a balanced domain, for a pluriharmonic mapping f : Ω → Yand w ∈ ∂Ω, then we let

Λf (0;w) = sup|ϕ[f,w, lu]ζ(0)| : lu ∈ T (u), ‖u‖Y = 1+sup|ϕ[f,w, lu]ζ(0)| : lu ∈ T (u), ‖u‖Y = 1,

where

ϕ[f,w, lu](ζ) = lu (f (ζw)) , ζ ∈ U

and U is the open unit disk of the complex plane C. We note that the inequal-

ity Λf (0;w) ≤ 4π always holds for pluriharmonic mappings f : Ω → Y with

‖f(x)‖Y < 1 for all x ∈ Ω by the harmonic Schwarz lemma on the unit disk. If

Y = Cn or Y = ℓ2, where

ℓ2 =

z = (z1, z2, . . . ) : zj ∈ C,

∞∑

j=1

|zj |2 <∞

,

then the mapping f = h+g is pluriharmonic on Ω, where h and g are holomorphic

in Ω. In this case,

Λf (0;w) = ‖Dh(0)w‖Y +∥∥∥Dg(0)w

∥∥∥Y, w ∈ ∂Ω.

Furthermore, if X = Y = Cn and Ω is a simply connected domain, then f :

Ω → Cn is pluriharmonic if and only if it has a representation f = h + g, where


h and g are holomorphic in Ω. This representation is unique if g(0) = 0 (cf.

[18, 53, 57]). For a pluriharmonic mapping f = h + g : Ω → Cn, it is an

elementary exercise to see that the real Jacobian determinant of f can be written as

det Jf = det

[Dh DgDg Dh

]

and if h is locally biholomorphic in Ω, then the determinant of Jf has the form

det Jf = |detDh|2 det(I −Dg[Dh]−1Dg[Dh]−1

),

where I is the identity operator (see [18]).

For an n×m complex matrix A = (aij), the Frobenius norm of A is defined as

follows:

‖A‖F =

√√√√n∑

i=1

m∑

j=1

|aij |2.

Then we have

(1.2) ‖A‖2F ≤ m‖A‖2,where

‖A‖ = supξ∈Cm\0

‖Aξ‖e‖ξ‖e

.

Let Ω be a domain in Cm. For a pluriharmonic mapping f : Ω → C

n, let

∇f(z) =(∂f

∂x1(z),

∂f

∂y1(z), . . . ,

∂f

∂xm(z),

∂f

∂ym(z)

),

where z = (z1, . . . , zm) ∈ Ω and zj = xj + iyj for j = 1, . . . ,m.

Definition 1.2. A complex Banach space X is called a JB∗-triple if there exists a

triple product ·, ·, · : X3 → X which is conjugate linear in the middle variable,

but linear and symmetric in the other variables, and satisfies

(i) a, b, x, y, z = a, b, x, y, z − x, b, a, y, z + x, y, a, b, z;

(ii) the map aa : x ∈ X 7→ a, a, x ∈ X is hermitian with nonnegative

spectrum;

(iii) ‖a, a, a‖X = ‖a‖3X ;for a, b, x, y, z ∈ X.

Let Ω be a domain in a complex Banach space X. Denote by Aut(Ω) the set of

biholomorphic automorphisms of Ω. A domain Ω ⊂ X is said to be homogeneous

if for each x, y ∈ Ω, there exists some mapping f ∈ Aut(Ω) such that f(x) = y.

It is known that every bounded symmetric domain in a complex Banach space

X is homogeneous. Conversely, the open unit ball BX of X admits a symmetry

s(z) = −z at 0 and if BX is homogeneous, then BX is a symmetric domain. It is

well known that the Euclidean unit ball in Cn, the polydisc in C

n and the classical

Cartan domains are bounded symmetric domains in Cn. Banach spaces with a


homogeneous open unit ball are precisely the JB∗-triples. We refer to [13, 15, 44,

45, 46] for more details of JB∗-triples and bounded symmetric domains.

Let X be a JB∗-triple. For every z, w ∈ X, the Bergman operator B(z, w) ∈L(X) is defined by

B(z, w)(·) = I − 2zw + z, w, ·, w, z,where zw(x) = z, w, x. Let BX be the unit ball of X. Then, for each a ∈ BX ,

the Mobius transformation ga defined by

(1.3) ga(z) = a+B(a, a)1/2(I + za)−1z,

is a biholomorphic automorphism of BX with ga(0) = a, ga(−a) = 0, g−a = g−1a

and Dga(0) = B(a, a)1/2. By [34, Corollary 3.6], we have

(1.4) ‖Dgz0(0)−1‖ = ‖Dg−z0(z0)‖ =1

1− ‖z0‖2X.

Given JB*-triples X1, . . . , Xn, we can form the ℓ∞-sum X = X1 ⊕ · · · ⊕Xn

which becomes a JB*-triple equipped with the coordinatewise triple product:

x, y, z = (x1, y1, z1, · · · , xi, yi, zi, · · · , xn, yn, zn)for x = (xi), y = (yi), z = (zi) ∈ X. Let BXj

be the open unit ball of Xj for

j = 1, . . . , n. Then their product BX1× · · · × BXn is the open unit ball of BX of

the JB∗-triple X. Let gj,aj (aj ∈ BXj) be the Mobius transformation of BXj

for

j = 1, . . . , n. Then, for a = (a1, . . . , an) ∈ BX ,

ga(z) = (g1,a1(z1), . . . , gn,an(zn)), z = (z1, . . . , zn) ∈ BX ,

is the Mobius transformation of BX .

2. SCHWARZ TYPE LEMMAS OF HOLOMORPHIC MAPPINGS AND THEIR

APPLICATIONS

The classical Schwarz lemma states that every holomorphic mapping f of the

unit disk U into itself with f(0) = 0 satisfies |f(z)| ≤ |z| for all z ∈ U. Moreover,

unless f is a rotation, one has the strict inequality |f ′(0)| < 1, and f maps each

disk Ur := z : |z| < r < 1 into a strictly smaller one. Lindelof removed the

assumption “origin is a fixed point” and improved the classical Schwarz lemma of

holomorphic mappings into the following form.

Theorem A. ([35, Proposition 2.2.2]) Let f be a holomorphic mapping of U into

itself. Then, for z ∈ U, the following sharp estimate

|f(z)| ≤ |f(0)|+ |z|1 + |f(0)| |z|

holds.

Remark 2.1. By the maximum modulus principle, Theorem A continuous to hold

for holomorphic mapping f from U into U.

In particular, if the origin is a fixed point of f , then Osserman obtained a better

estimate as follows.


Theorem B. ([47, Lemma 2]) Let f be a holomorphic mapping of U into itself

with f(0) = 0. Then, for z ∈ U, the following sharp estimate

(2.1) |f(z)| ≤ |z| |f′(0)|+ |z|

1 + |f ′(0)| |z|holds.

As an application, by using (2.1), Osserman [47] established a version of the

boundary Schwarz lemma which is as follows.

Theorem C. Let f be a holomorphic mapping of U into itself with f(0) = 0. If fis holomorphic at b ∈ ∂U (or more generally, if f is differentiable at b ∈ ∂U) and

|f(b)| = 1, then |f ′(b)| ≥ 2/(1 + |f ′(0)|). Moreover, the inequality is sharp.

In the following, we extend Theorems A and B to Banach spaces, and then

we apply the obtained results to study the Bloch type Theorem and the boundary

Schwarz type lemmas.

Theorem 2.2. Suppose that BX and BY are the unit balls of the complex Banach

spaces X and Y , respectively. Let f : BX → BY be a holomorphic mapping.

Then

‖f(z)‖Y ≤ ‖f(0)‖Y + ‖z‖X1 + ‖f(0)‖Y ‖z‖X

for z ∈ BX .

This estimate is sharp with equality possible for each value of ‖f(0)‖Y and for

each z ∈ BX .


spaces X and Y , respectively. Let f : BX → BY be a holomorphic mapping with

f(0) = 0. Then

‖f(z)‖Y ≤ ‖Df(0)‖ + ‖z‖X1 + ‖Df(0)‖‖z‖X

‖z‖X ≤ ‖z‖X for z ∈ BX .

The first estimate is sharp with equality possible for each value of ‖Df(0)‖ and

for each z ∈ BX .

We use B to denote the homogeneous unit ball of X = Cn. It is easy to see that

B is the unit ball of a finite dimensional JB∗-triple X. Let k ∈ [0, 1) be a constant.

Denote by PH (k) the set of all pluriharmonic mappings f = h+ g of B into Cn

with h(0) = g(0) = 0 and

‖ωf‖ ≤ k,

where h is locally biholomorphic in B, g is holomorphic in B, ωf = Dg[Dh]−1

and

‖ωf‖ = supz∈B,ξ∈Cn\0

‖ωf (z)ξ‖e‖ξ‖e

.

For n ≥ 2, f = h + g ∈ PH (k) is a quasiregular mapping if and only if h is

a quasiregular mapping (see [11]). In particular, if n = 1, then f ∈ PH (k) is a

K-quasiregular mapping, where K = (1 + k)/(1 − k) (cf. [12, 17, 56, 58]).


Denote by P a set of mappings from B into Cn. For a mapping f ∈ P and a

point a ∈ B, we write Bf (a) as the radius of the largest univalent Euclidean ball

centered at f(a) in f(B). Here a univalent ball in f(B) centered at f(a) means

that f maps an open subset of B containing the point a univalently onto this ball.

Let

B = inff∈P

supa∈B

Bf(a).

If B > 0 is finite, then we call B the Bloch type constant of the set P . One of

the long standing open problems of determining the precise value of Bloch type

constant of holomorphic mappings with one variable has attracted much attention

(see [3, 4, 20, 36, 42]). For holomorphic mappings of several complex variables,

the Bloch type constant does not exist unless one considers the class of functions

under certain constraints. For example, consider fk(z) = (kz1, z2/k, z3, . . . , zn)for k ∈ N = 1, 2, . . ., where n ≥ 2 and z is in the Euclidean unit ball Bn of

Cn. It is easy to see that each fk is univalent and |fk(0)| = detDfk(0) − 1 = 0.

Moreover, each fk(Bn) contains no ball with radius bigger than 1/k. Hence, there

does not exist an absolute constant r0 which can work for all k ∈ N such that

z ∈ Cn : ‖z‖e < r0 is contained in fk(B

n). For more details on studies of the

Bloch type constant of holomorphic mappings with several complex variables, we

refer to the works of Chen and Gauthier [6], Fitzgerald and Gong [19], Graham

and Varolin [22], Hamada [24], Takahashi [54], and Wu [59]. On the studies of the

Bloch type constant for the class of pluriharmonic mappings, we refer to [8, 26].

In the following, for f = h+ g ∈ PH (k), we will use Theorems 2.2 and 2.3

to investigate the ratio Bf/Bh and give a sharp estimate. For the related studies

of the planar harmonic mappings, see [8, 10, 20].

Theorem 2.4. For k ∈ [0, 1), let f = h+ g ∈ PH (k). Then, for z ∈ B,

(2.2) 1− k ≤ Bf (z)

Bh(z)≤ µk

(‖ωf (z)‖k

)≤ µk(1) = 1 + k,

where

µk(x) = 1 + k

[1

x+

(1− 1

x2

)log(1 + x)

]for x ∈ (0, 1],

µk(x) = limx→0+

µk(x) = 1 +k

2for x = 0.

Moreover, the left hand of (2.2) is sharp for all z ∈ B, and the right hand of

(2.2) is asymptotically sharp when k tends to 0.

In [59], Wu generalized the classical Schwarz lemma of holomorphic mappings

to higher dimension. Burns and Krantz [5] established a new version Schwarz

lemma at the boundary, and obtained a new rigidity result for holomorphic map-

pings. Later, Huang [30] further strengthened the result of Burns-Krantz for holo-

morphic mappings with an interior fixed point. Recently, Liu and Tang [40] ob-

tained a Schwarz lemma at the boundary of holomorphic mappings from a Levi

strongly pseudoconvex domain into itself. See [21, 23, 31, 55] for more details on


this line. In the following, we will apply Theorem 2.3 to establish a new version

Schwarz lemma at the boundary, which is a generalization of Theorem C.

Theorem 2.5. Let BX and BY be the unit balls of the complex Banach spaces

X and Y , respectively. Suppose that f is a holomorphic mapping of BX into

BY . If f(0) = 0 and f is holomorphic at b ∈ ∂BX (or more generally, if f is

differentiable at b ∈ ∂BX) with ‖f(b)‖Y = 1, then

‖Df(b)b‖Y ≥ 2

1 + ‖Df(0)‖ .

This inequality is sharp with equality possible for each value of ‖Df(0)‖.

In particular, if we replace BX andBY by a balanced domain and a finite dimen-

sional bounded symmetric domain in Theorem 2.5, respectively, then we obtain a

better estimate (cf. [27]). Before we present the next result, let us recall some

definitions.

Let BY be a bounded symmetric domain realized as the open unit ball of a finite

dimensional JB∗-triple Y . We recall a constant c(BY ) defined in [25]. Let h0 be

the Bergman metric on BY at 0 and let

c(BY ) =1

2sup

x,y∈BY

|h0(x, y)|.

It follows from [14, Ineq. (2.3)] that

dimY + r

2≤ c(BY ) ≤ dimY,

where r is the rank of Y .

An element x in a JB∗-triple Y is called a tripotent if x satisfies x, x, x = x.

If two tripotents x and y satisfy 2xy = 0, then x and y are said to be orthogo-

nal. Obviously, orthogonality is a symmetric relation. A tripotent x is said to be

maximal if any tripotent which is orthogonal to x is 0.

Theorem 2.6. Suppose that G is a balanced domain in a complex Banach space

X and BY is a bounded symmetric domain realized as the open unit ball of a finite

dimensional JB∗-triple Y . Let Γ ⊂ ∂BY be the set of maximal tripotents of Y ,

and let f : G → BY be a holomorphic mapping. Also let f be holomorphic at

z = α ∈ ∂G and f(α) = β ∈ Γ.

(i) We have

(2.3)1

2c(BY )h0(Df(α)α, β) ≥

2∣∣∣1− 1

2c(BY )h0(f(0), β)∣∣∣2

1−∣∣∣ 12c(BY )h0(f(0), β)

∣∣∣2+ ‖Df(0)α‖Y

,

where h0 is the Bergman metric on BY at 0.

(ii) Moreover, if f(0) = 0, then we have

(2.4)1


2

1 + ‖Df(0)α‖Y.


(iii) In particular, if G = BX is the unit ball of X, then the inequalities (2.3)and (2.4) are sharp with equality possible for each values of

a =1

2c(BY )h0(f(0), β), b =

1

2c(BY )h0(Df(0)α, β)

with |b| ≤ 1− |a|2.

We remark that Theorem 2.6 is an improvement and generalization of [39, The-

orem 3.1] and [41, Theorem 3.1] (cf. [62, Theorem 1.5]). By using arguments

similar to those in the proof of Theorem 2.6, we have the following theorem (cf.

[62, Theorem 1.5]). We omit the proof.


X and BH is unit ball of a complex Hilbert space H with inner product 〈·, ·〉. Let

f : G→ BH be a holomorphic mapping. If f is holomorphic at z = α ∈ ∂G and

f(α) = β ∈ ∂BH , then

(2.5) 〈Df(α)α, β〉 ≥ 2 |1− 〈f(0), β〉|2

1− |〈f(0), β〉|2 + ‖Df(0)α‖H.

Moreover, if f(0) = 0, then we have

(2.6) 〈Df(α)α, β〉 ≥ 2

1 + ‖Df(0)α‖H.

In particular, if G = BX is the unit ball of X, then the inequalities (2.5) and

(2.6) are sharp with equality possible for each values of 〈f(0), β〉 and ‖Df(0)α‖Hwith ‖Df(0)α‖H ≤ 1− |〈f(0), β〉|2

The proofs of Theorems 2.2∼2.6 will be presented in part I of Section 4.

3. SCHWARZ TYPE LEMMAS OF PLURIHARMONIC MAPPINGS AND THEIR

APPLICATIONS

Heinz in his classical paper [28] showed that the following version of Schwarz

Lemma for harmonic mappings: If f is a harmonic mapping of U into itself with

f(0) = 0, then

(3.1) |f(z)| ≤ 4

πarctan |z|

for z ∈ U. In 2011, Chen and Gauthier generalized (3.1) into the following form.

Theorem D. ([8, Theorem 4]) Let f be a pluriharmonic mapping of the Euclidean

unit ball Bn into the Euclidean unit ball Bm such that f(0) = 0, where m is a

positive integer. Then, for all z ∈ Bn,

‖f(z)‖e ≤4

πarctan ‖z‖e.

This estimate is sharp.

Hamada and Kohr [26] extended Theorem D to pluriharmonic mappings of the

unit ball of a complex Banach space X into the unit ball Bna of Cn with respect to

an arbitrary norm ‖ · ‖a on Cn as follows.


Theorem E. ([26, Theorem 4.1]) Let BX be the unit ball of a complex Banach

space X, Bna be the unit ball of Cn with respect to an arbitrary norm ‖ · ‖a on Cn

and f : BX → Bna be a pluriharmonic mapping such that f(0) = 0. Then, for

z ∈ BX , the following sharp inequality

‖f(z)‖a ≤4

πarctan ‖z‖X

holds.

We remove the assumption “f(0) = 0” in Theorem E and obtain the following

result.


spaces X and Y , respectively, and f : BX → BY is a pluriharmonic mapping.

Then ∥∥∥∥f(z)−1− ‖z‖2X1 + ‖z‖2X

f(0)

∥∥∥∥Y

≤ 4

πarctan ‖z‖X for z ∈ BX .

In particular, if f(0) = 0, then this estimate is sharp.

In particular, if f(0) = 0 in Theorem 3.1, then we have a better estimate as

follows (cf. [62, Theorem 1.7]).

Theorem 3.2. Assume the hypothesis of Theorem 3.1, and in addition let f(0) = 0.Then we have

‖f(z)‖Y ≤ 4

πarctan

( ‖z‖X + π4Λf (0;w)

1 + π4Λf (0;w)‖z‖X

‖z‖X)

≤ 4

πarctan ‖z‖X for z ∈ BX ,

where w = z/‖z‖X .

Let f be a one-to-one harmonic mapping of U onto itself with f(0) = 0 which

is C1 up to the boundary. By using (3.1), Heinz [28, Ineq. (15)] proved that, for

any θ ∈ [0, 2π],

(3.2) |fζ(eiθ)|+ |fζ(eiθ)| ≥2

π.

In the following, we extend (3.2) into the following forms.

First, we will apply Theorem 3.1 and the Harnack principle to establish a new

version Schwarz lemma at the boundary for pluriharmonic mappings.

Theorem 3.3. Assume the hypotheses of Theorem 3.1. In addition, let f be differ-

entiable at b ∈ ∂BX with ‖f(b)‖Y = 1. Then we have

‖Df(b)b‖Y ≥ max

2

π− ‖f(0)‖Y ,

1− ‖f(0)‖Y2

.

Next, we will apply Theorem 3.2 to establish a new version Schwarz lemma at

the boundary for pluriharmonic mappings. For the proof, it suffices to use argu-

ments similar to those in the proof of Theorem 2.5. We omit the proof.

Theorem 3.4. Assume the hypothesis of Theorem 3.3. Then, if f(0) = 0, we have

‖Df(b)b‖Y ≥ 4

π

1

1 + π4Λf (0; b)

.


In particular, we consider the cases such that BY is a bounded symmetric do-

main in Cn or is the complex Hilbert ball (cf. [62, Theorems 1.8 and 1.12]). In

these cases, the domain of the definition of the mapping f can be generalized to a

balanced domain in a complex Banach space.


X and BY is a bounded symmetric domain realized as the open unit ball of a finite

dimensional JB∗-triple Y = Cn. Let Γ ⊂ ∂BY be the set of maximal tripotents of

Y , and let f : G → BY be a pluriharmonic mapping with f(0) = 0. Assume that

f is differentiable at z = α ∈ ∂G and f(α) = β ∈ Γ. Let

ϕ(ζ) =1

2c(BY )h0(f(ζα), β), ζ ∈ U,

where h0 is the Bergman metric on BY at 0. Then ϕ is harmonic mapping of U

into itself with ϕ(0) = 0 and we have

(3.3) Re(ϕζ(1) + ϕζ(1)

)≥ 4

π

1

1 + π4Λf (0, α)

,

where “Re” denotes the real part of a complex number.

In particular, if f = h+g, where h and g are holomorphic on G, and f satisfies

the above assumptions, then we have

(3.4)1

2c(BY )Re(h0(Dh(α)α +Dg(α)α, β)

)≥ 4

π

1

1 + π4Λf (0, α)

.

If G = BX is the unit ball of X, then the inequalities (3.3) and (3.4) are sharp.

If BY is the complex Hilbert ball, then we have the following result. We omit

the proof, since it is similar to that in the proof of Theorem 3.5.


X and BH is the unit ball of a complex Hilbert space H with inner product 〈·, ·〉.Let f : G → BH be a pluriharmonic mapping with f(0) = 0. Assume that f is

differentiable at z = α ∈ ∂G and f(α) = β ∈ ∂BH . Let

ϕ(ζ) = 〈f(ζα), β〉, ζ ∈ U.

Then ϕ is a harmonic mapping of U into itself with f(0) = 0 and

(3.5) Re(ϕζ(1) + ϕζ(1)

)≥ 4

π

1

1 + π4Λf (0, α)

.

In particular, if H = ℓ2 and f = h + g, where h and g are holomorphic on G,

and f satisfies the above assumptions, then we have

(3.6) Re〈Dh(α)α +Dg(α)α, β〉 ≥ 4

π

1

1 + π4Λf (0, α)

.

The inequalities (3.5) and (3.6) are sharp.


The classical Schwarz-Pick lemma states that an analytic function f of U into

itself satisfies

(3.7) |f ′(z)| ≤ 1− |f(z)|21− |z|2 , z ∈ U.

Chu et al. [15, Lemma 3.12] generalized (3.7) to holomorphic mappings between

the unit balls of JB∗-triples (cf. [2], [6], [26]).

In 1989, Colonna established an analogue of the Schwarz-Pick lemma for planar

harmonic mappings.

Theorem F. ([16, Theorems 3 and 4]) Let f be a harmonic mapping of U into

itself. Then, for z ∈ U,

|fz(z)| + |fz(z)| ≤4

π

1

1− |z|2 .

This estimate is sharp, and all the extremal functions are

f(z) =2γ

πarg

(1 + φ(z)

1− φ(z)

),

where |γ| = 1 and φ is a conformal automorphism of U.

For real valued harmonic functions in U, Kalaj and Vuorinen [33] and Chen [7,

Theorem 1.2] obtained a better estimate as follows.

Theorem G. Let f : U → (−1, 1) be a real-valued harmonic function. Then the

following inequalities hold:

(1) |∇f(z)| ≤ 4

π

1− |f(z)|21− |z|2 , for z ∈ U. This inequality is sharp for each

z ∈ U.

(2) |∇f(z)| ≤ 4

π

cos(π2 f(z))

1− |z|2 , for z ∈ U. This inequality is sharp for each

z ∈ U.

Chen and Gauthier [8] generalized Theorem F to pluriharmonic mappings be-

tween the Euclidean unit balls. Later, Hamada and Kohr [26] generalized Theorem

F to pluriharmonic mappings from the unit ball of a JB∗-triple into the unit ball Bnaof Cn with respect to an arbitrary norm on C

n.

For mappings with values in higher dimensional spaces, Pavlovic [48, 49] showed

that the inequality (3.7) does not hold for analytic functions f of U into Bn, where

n ≥ 2 is an integer. However, Pavlovic proved the following Schwarz-Pick type

lemma for analytic functions f of U into Bn:

|∇‖f(z)‖e| ≤1− ‖f(z)‖2e1− |z|2 , z ∈ U,

where ∇‖f(z)‖e denotes the gradient of ‖f‖e.In [63], Zhu established a Schwarz-Pick type estimate for pluriharmonic map-

pings f of Bn into itself as follows.


Theorem H. ([63, Theorem 1.1]) For n ≥ 1, let f be a pluriharmonic mapping of

Bn into itself. Then the following inequality

|∇‖f(z)‖e| ≤4√n

π

1

(1− ‖z‖e)holds for all z ∈ B

n.

In [9], Chen and Hamada improved Theorem H into the sharp form for plurihar-

monic mappings of Bn into the unit ball of the Minkowski space. In the following,

we will present a relatively simple method of proof to improve Theorem H and [9,

Theorem 2.2] to pluriharmonic mappings from the unit ball BX of a JB∗-triple Xto the unit ball of a complex Banach space Y .

Theorem 3.7. Let BX be a bounded symmetric domain realized as the unit ball

of a JB∗-triple X. Let BY be the unit ball of a complex Banach space Y . Also,

let f : BX → BY be a pluriharmonic mapping. Let z0 ∈ BX be a point which

satisfies one of the following conditions:

(i) f(z0) = 0;

(ii) f(z0) 6= 0 and ‖f(z)‖Y is differentiable at z = z0.

Then

(3.8) |∇‖f‖Y (z0)| ≤4

π

1

1− ‖z0‖2X.

The estimate (3.8) is sharp for each z0 ∈ BX .

In particular, if f is a pluriharmonic mapping of BX into the unit ball BY of a

real Banach space Y , then we have a better estimate as follows. Here we omit the

proof because it suffices to use Theorem F instead of Theorem G and use arguments

similar to those in the proof of Theorem 3.7.


of a JB∗-triple X. Let BY be the unit ball of a real Banach space Y . Also, let

f : BX → BY be a pluriharmonic mapping. Let z0 ∈ BX be a point which

satisfies one of the following conditions:

(i) f(z0) = 0;

(ii) f(z0) 6= 0 and ‖f(z)‖Y is differentiable at z = z0.

Then we have the following estimates.

(1) |∇‖f‖Y (z0)| ≤4

π

1− ‖f(z0)‖2Y1− ‖z0‖2X

.

(2) |∇‖f‖Y (z0)| ≤4

π

cos(π2 ‖f(z0)‖Y )1− ‖z0‖2X

.

The above estimates are sharp for each z0 ∈ BX .

We remark that Theorem 3.8 is an improvement and generalization of [9, Theo-

rem 2.3].

In [63], Zhu established some other Schwarz-Pick type estimates as follows.


Theorem I. ([63, Theorem 1.2]) Let f = h+ g be a pluriharmonic self-mapping

of the unit ball Bn of Cn, where h and g are holomorphic mappings of Bn into Cn.

Then

(3.9) ‖Dh(z)ej‖2e + ‖Dg(z)ej‖2e ≤1− ‖f(z)‖2e(1− ‖z‖e)2

, j = 1, . . . , n,

where e1, . . . , en is the usual orthonormal basis of Cn, and

(3.10) ‖∇f(z)‖2F ≤ 2n(1− ‖f(z)‖2e)(1− ‖z‖e)2

.

Theorem J. ([63, Theorem 1.3]) Let f = h+ g be a pluriharmonic self-mapping

of Bn, where h and g are holomorphic mappings of Bn into Cn. Assume that

h is locally biholomorphic on Bn and ‖ωf‖ ≤ k < 1 holds on B

n. Let K =(1 + k)/(1 − k). Then the following inequalities hold for all z ∈ B

n:

(3.11) ‖Dh(z)‖ + ‖Dg(z)‖ ≤ 2K

K + 1

√n(1− ‖f(z)‖2e)1− ‖z‖e

,

(3.12) |∇‖f‖e(z)| ≤√n(‖Dh(z)‖ + ‖Dg(z)‖)

and

(3.13) ‖∇f(z)‖F ≤√

2n(‖Dh(z)‖2 + ‖Dg(z)‖2).Chen and Hamada [9] generalized and improved Theorem I into the following

sharp form on pluriharmonic mappings of the polydisc Un in C

n into the Euclidean

unit ball Bm in Cm.

Theorem K. For n ≥ 1, let f = (f1, . . . , fm) : Un → B

m be a pluriharmonic

mapping, where m is a positive integer. Then, for z = (z1, . . . , zn) ∈ Un, we have

(3.14)

m∑

j=1

n∑

k=1

(∣∣∣∣∂fj(z)

∂zk

∣∣∣∣2

+

∣∣∣∣∂fj(z)

∂zk

∣∣∣∣2)

≤ 1− ‖f(z)‖2e(1− ‖z‖2∞)2

,

where ‖z‖∞ = max1≤j≤n |zj |. Moreover, the inequality (3.14) is sharp for each

z ∈ Un with |z1| = · · · = |zn|.

In the following, we will generalize and improve (3.12) into the following form.

Moreover, we only need to assume that f is pluriharmonic.

Proposition 3.9. Let Ω be a domain in a complex Banach spaceX, and f = h+g :Ω → C

n be a pluriharmonic mapping, where h and g are holomorphic mappings

of Ω into the Euclidean space Cn. Then

|∇‖f‖e(z)| ≤ ‖Dh(z)‖ + ‖Dg(z)‖, z ∈ Ω.

Proof. For a pluriharmonic mapping f = h+ g : Ω → Cn, we have (see e.g. [26,

p.638]),

‖Df(z)‖ ≤ ‖Dh(z)‖ + ‖Dg(z)‖, z ∈ Ω.

Then, by (1.1), we have

|∇‖f‖e(z)| ≤ ‖Df(z)‖ ≤ ‖Dh(z)‖ + ‖Dg(z)‖, z ∈ Ω.


This completes the proof.

Let X be the ℓ∞-sum X = X1 ⊕ · · · ⊕ Xm of JB*-triples X1, . . . , Xm and

let BX = BX1× · · · × BXm be the unit ball of X. By applying Theorem K, we

obtain the following improvement of (3.9) to pluriharmonic mappings of the unit

ball BX = BX1× · · · × BXm into the Euclidean unit ball Bn of Cn.

Theorem 3.10. Let X be the ℓ∞-sum X = X1 ⊕ · · · ⊕Xm of JB*-triples X1, . . . ,

Xm and BX = BX1× · · · × BXm be the unit ball of X. Also, let f = h + g :

BX → Bn be a pluriharmonic mapping, where h and g are holomorphic mappings

of BX into Cn. Then we have

(3.15)

m∑

j=1

(‖Dh(z)wj‖2e + ‖Dg(z)wj‖2e

)≤ 1− ‖f(z)‖2e

(1− ‖z‖2X )2

for all wj ∈ Xj with ‖wj‖Xj= 1 (1 ≤ j ≤ m), where wj = Ij(wj) and

Ij : Xj → X is the natural inclusion mapping for j = 1, . . . ,m.

Moreover, the inequality (3.15) is sharp for each z = (z1, . . . , zm) ∈ BX with

‖z1‖X1= · · · = ‖zm‖Xm .

In particular, for BX = Bk1 × · · · × B

kp , we have the following generalization

of (3.14) to pluriharmonic mappings of BX into Bn, where B

kj are the Euclidean

unit balls in Ckj for j ∈ 1, . . . , p.

Corollary 3.11. Let BX = Bk1 × · · · ×B

kp . Also, let f = h+ g : BX → Bn be a

pluriharmonic mapping, where h and g are holomorphic mappings of BX into Cn.

Then

(3.16)

n∑

j=1

m∑

k=1

(∣∣∣∣∂fj(z)

∂zk

∣∣∣∣2

+

∣∣∣∣∂fj(z)

∂zk

∣∣∣∣2)

≤ κ(1 − ‖f(z)‖2e)(1− ‖z‖2X )2

,

where m = k1 + · · ·+ kp and κ = maxk1, · · · , kp.

Proof. By using the natural inclusion map ζj ∈ Bkj → (ζj, 0) ∈ B

κ, we may

assume that κ = k1 = · · · = kp. Let e1, . . . , eκ be the usual orthonormal basis of

Cκ and let ej,l = Ij(el), where Ij : Xj → X is the natural inclusion mapping.

Then we have ‖eν‖Xj= 1 for each ν with 1 ≤ ν ≤ κ and each j with 1 ≤ j ≤ p.

By (3.15), we have

p∑

j=1

(‖Dh(z)ej,l‖2e + ‖Dg(z)ej,l‖2e

)≤ 1− ‖f(z)‖2e

(1− ‖z‖2X )2, 1 ≤ l ≤ κ,

which gives that

n∑

j=1

m∑

k=1

(∣∣∣∣∂fj(z)

∂zk

∣∣∣∣2

+

∣∣∣∣∂fj(z)

∂zk

∣∣∣∣2)

=κ∑

l=1

p∑

j=1

(‖Dh(z)ej,l‖2e + ‖Dg(z)ej,l‖2e

)

≤ κ(1− ‖f(z)‖2e)(1− ‖z‖2X)2

.



The following proposition is a generalization of (3.13) to pluriharmonic map-

pings of a domain Ω in the Euclidean space Cm into the Euclidean space C

n. Note

that we only need to assume that f is pluriharmonic.

Proposition 3.12. Let Ω be a domain in the Euclidean space Cm. Let f = h+ g :

Ω → Cn be a pluriharmonic mapping, where h and g are holomorphic mappings

from Ω to the Euclidean space Cn. Then, we have

(3.17) ‖∇f(z)‖F ≤√

2m(‖Dh(z)‖2 + ‖Dg(z)‖2).Proof. The inequality (3.17) follows from the relation

(3.18) ‖∇f(z)‖2F = 2(‖Dh(z)‖2F + ‖Dg(z)‖2F )and the inequality (1.2). This completes the proof.

The following corollary is a generalization of (3.10) to pluriharmonic mappings

from BX = Bk1 × · · · × B

kp to Bn. Note that (3.19) an improvement of (3.10). In

particular, the following result holds for BX = Bm or for BX = U

m.

Corollary 3.13. Assume the hypotheses of Corollary 3.11. Then

(3.19) ‖∇f(z)‖2F ≤ 2κ(1 − ‖f(z)‖2e)(1− ‖z‖2X)2

,

where κ = maxk1, · · · , kp.

Proof. The inequality (3.16) and the relation (3.18) imply (3.19). This completes

the proof.

The following theorem is a generalization of (3.11) to pluriharmonic mappings

from the unit ball BX of a finite dimensional JB∗-triple X to the Euclidean unit

ball Bn of Cn, where n = dimX. Note that the condition ‖ωf‖ < 1 in Bn implies

that f is a sense-preserving and locally univalent mapping in Bn (see [26, p.637]).

Also, the condition (3.20) is also an improvement of (3.11).


of a JB∗-triple X with dimX = n < ∞. Also, let f = h + g : BX → Bn be a

pluriharmonic mapping, where h and g are holomorphic mappings of BX into Cn.

Assume that h is locally biholomorphic in BX and ‖ωf‖ ≤ k < 1 holds in BX .

Then, for all z ∈ BX ,

(3.20) ‖Dh(z)‖ + ‖Dg(z)‖ ≤ 2K√2(K2 + 1)

√1− ‖f(z)‖2e1− ‖z‖2X

,

where K = (1 + k)/(1 − k).

The proofs of Theorems 3.1∼3.14 will be given in part II of Section 4.

4. PROOFS OF THE MAIN RESULTS

Part I. Schwarz type lemmas of holomorphic mappings and their applications.


The proof of Theorem 2.2. Let z be any fixed point in BX\0. Without loss

of generality, we assume that f(z) 6= 0. Let η = z/‖z‖X ∈ ∂BX . For any fixed

b ∈ ∂BY , let

F (ξ) = lb(f(ξη)), ξ ∈ U,

where lb ∈ T (b). Then F is a holomorphic mapping of U into U. By Remark 2.1,

we have

(4.1) |F (ξ)| ≤ |F (0)| + |ξ|1 + |F (0)| |ξ|

for ξ ∈ U. Elementary calculation leads to

(4.2) |F (0)| = |lb(f(0))| ≤ ‖lb‖Y ∗‖f(0)‖Y = ‖f(0)‖Y .Since, for any fixed a ∈ [0, 1), the function (x) = (a+x)/(1+ax) is increasing

with respect to the variable x ∈ [0,∞), by (4.1) and (4.2), we see that

(4.3) |lb(f(ξη))| ≤‖f(0)‖Y + |ξ|1 + ‖f(0)‖Y |ξ|

.

Finally, by letting ξ = ‖z‖X and b = f(z)/‖f(z)‖Y in (4.3), we obtain

‖f(z)‖Y ≤ ‖f(0)‖Y + ‖z‖X1 + ‖f(0)‖Y ‖z‖X

.

Next, we show the sharpness part. If there is a point z0 ∈ BX such that

‖f(z0)‖Y = 1, then ‖f(z)‖Y = 1 for all z ∈ BX . In this case, the sharpness

part is obvious. In the following, we assume that ‖f(z)‖Y < 1 for all z ∈ BX . For

any fixed point z0 ∈ BX\0, let lw0∈ T (w0) be fixed, where w0 = z0/‖z0‖X .

For each fixed b ∈ ∂BY and any fixed a ∈ [0, 1), let

f(z) =a+ lw0

(z)

1 + alw0(z)

b, z ∈ BX .

Then f is a holomorphic mapping from BX into BY and by taking z = z0, we

obtain

‖f(z0)‖Y =

∣∣∣∣a+ lw0

(z0)

1 + alw0(z0)

∣∣∣∣ =‖f(0)‖Y + ‖z0‖X1 + ‖f(0)‖Y ‖z0‖X

,

which completes the proof.

The proof of Theorem 2.3. Let z be any fixed point in BX\0. Without loss

of generality, we assume that f(z) 6= 0. Let η = z/‖z‖X ∈ ∂BX . For any fixed

b ∈ ∂BY , let

F (ξ) = lb(f(ξη)), ξ ∈ U,

where lb ∈ T (b). Then F is a holomorphic mapping of U into itself with F (0) = 0.

By Theorem B, we have


(4.4) |F (ξ)| ≤ |F ′(0)| + |ξ|1 + |F ′(0)| |ξ| |ξ| for ξ ∈ U.

Elementary calculation leads to

|F ′(0)| = |lb(Df(0)η)| ≤ ‖lb‖Y ∗‖Df(0)η‖Y ≤ ‖Df(0)‖,which together with (4.4) yields that

(4.5) |lb(f(ξη))| ≤‖Df(0)‖ + |ξ|1 + ‖Df(0)‖ |ξ| |ξ|.

Finally, by letting ξ = ‖z‖X and b = f(z)/‖f(z)‖Y in (4.5), we obtain

‖f(z)‖Y ≤ ‖Df(0)‖+ ‖z‖X1 + ‖Df(0)‖‖z‖X

‖z‖X .

Next, we show the sharpness part. For any fixed point z0 ∈ BX\0, let lw0∈

T (w0) be fixed, where w0 = z0/‖z0‖X . For any fixed b ∈ ∂BY and any fixed

a ∈ [0, 1], let

f(z) = lw0(z)

a+ lw0(z)

1 + alw0(z)

b, z ∈ BX .

Then f is a holomorphic mapping from BX into BY with f(0) = 0 and by letting

z = z0, we have

‖f(z0)‖Y = |lw0(z0)|

∣∣∣∣a+ lw0

(z0)

1 + alw0(z0)

∣∣∣∣ = ‖z0‖X‖Df(0)‖+ ‖z0‖X1 + ‖Df(0)‖‖z0‖X

,


The proof of Theorem 2.4. We only need to prove (2.2) for k ∈ (0, 1), since

Bf/Bh ≡ 1 for k = 0. In the following, we assume that k ∈ (0, 1) and we divide

the proof into five steps.

Step 1. For f = h + g ∈ PH (k), we claim that Bf (0) = d(0, ∂f(B))and Bh(0) = d(0, ∂h(B)), where d(0, ∂f(B)) and d(0, ∂h(B)) denote the Eu-

clidean distances from 0 to ∂f(B) and ∂h(B), respectively. We only need to

prove Bf (0) = d(0, ∂f(B)) because the proof of another one is similar. From the

definition of Bf (0), we see that Bf (0) is equal either to the Euclidean distance

from f(0) to a boundary point of f(B) or to the Euclidean distance from f(0) to a

critical value of f . In the following, we will show that the critical value of f does

not exist. Since ‖ωf‖ ≤ k, we see that

det Jf = |detDh|2 det(I −Dg[Dh]−1Dg[Dh]−1

)6= 0.

Consequently, f is locally univalent in B. Let V be a subdomain of B such that

f(V ) = w : ‖w‖e < Bf (0). If there exist z0 ∈ ∂V and z1 ∈ V such

that f(z0) = f(z1), then, by the condition that detJf 6= 0 in B, there exist

neighbourhoods U0(z0) of z0, U1(z1) of z1, and U2(f(z0)) of f(z0) = f(z1) such

that f maps U0(z0) and U1(z1) onto U2(f(z0)) univalently, respectively. This


contradicts the fact that f is univalent in V . Therefore, the critical points do not

exist. Hence Bf (0) = d(0, ∂f(B)). Without loss of generality, we assume that

there exists a boundary point ξ0 of f(B) such that ξ0 ∈ w ∈ Cn : ‖w‖e =

Bf(0). Let ℓξ0 = f−1([0, ξ0)) be the preimage of the semi-open segment [0, ξ0)with the starting point 0 in the ball B. Then

(4.6) Bf (0) = ‖ξ0‖e =∫

ℓξ0

‖df(z)‖e = infγ

∫

γ‖df(z)‖e ,

where the minimum is taken over all smooth paths [0, 1) ∋ t 7→ γ(t) ∈ B with

γ(0) = 0 and limt7→1− ‖γ(t)‖X = 1. Similarly, we can also assume that

Bh(0) = ‖ξ1‖e =∫

ℓξ1

‖dh(z)‖e = infγ

∫

γ‖dh(z)‖e ,

where ξ1 is a boundary point of h(B) such that ξ1 ∈ w ∈ Cn : ‖w‖e = Bh(0)

and the simple smooth curve ℓξ1 = h−1([0, ξ1)) is the preimage of the semi-open

segment [0, ξ1) under the mapping h. For t ∈ [0, 1), let ℓξ0 := ℓξ0(t) = f−1(ξ0t)and ℓξ1 := ℓξ1(t) = h−1(ξ1t).

Step 2. We first establish the lower bound of Bf (0)/Bh(0). Differentiation of

the equation f−1(f(z)) = z yields the following two formulas:

Df−1Dh+Df−1Dg = I

and

Df−1Dg +Df−1Dh = O,

which imply that

(4.7) Df−1 = [Dh]−1 (I − ωfωf )−1

and

(4.8) Df−1 = −[Dh]−1 (I − ωfωf )−1 ωf ,

where f−1(w) = (σ1(w), . . . , σn(w))′ and Df−1(w) =

(∂σj∂wk

)n×n

for k, j ∈1, . . . , n. Then, by (4.7) and (4.8), we have

∥∥∥∥DhDf−1 ξ0

‖ξ0‖e

∥∥∥∥e

+

∥∥∥∥DhDf−1 ξ0

‖ξ0‖e

∥∥∥∥e

≤∥∥∥(I − ωfωf )

−1∥∥∥

+∥∥∥(I − ωfωf )

−1 ωf

∥∥∥

≤ 1

1− ‖ωf‖2+

‖ωf‖1− ‖ωf‖2

=1

1− ‖ωf‖.

Consequently,


Bh(0) =

∫ 1

0‖dh(ℓξ1(t))‖e ≤

∫ 1

0‖dh(ℓξ0(t))‖e

=

∫ 1

0

∥∥(Dh(ℓξ0(t))Df−1(ξ0t)ξ0 +Dh(ℓξ0(t))Df−1(ξ0t)ξ0

)dt∥∥e

≤ ‖ξ0‖e∫ 1

0

(∥∥∥∥Dh(ℓξ0(t))Df−1(ξ0t)

ξ0‖ξ0‖e

∥∥∥∥e

+

∥∥∥∥Dh(ℓξ0(t))Df−1(ξ0t)

ξ0‖ξ0‖e

∥∥∥∥e

)dt

≤ Bf(0)

∫ 1

0

dt

1− ‖ωf (ℓξ0(t))‖,

which gives that

(4.9)Bf (0)

Bh(0)≥ 1∫ 10

dt1−‖ωf (ℓξ0 (t))‖

≥ 1− k.

Step 3. In this step, we will give the lower bound of Bf (z)/Bh(z) for all

z ∈ B. Since B is homogeneous, we see that, for any fixed ζ ∈ B, there exists a

φ ∈ Aut(B) such that φ(0) = ζ. For z ∈ B, let

(4.10) F (z) = f(φ(z)) − f(φ(0)) = H(z) +G(z),

where H(z) = h(φ(z)) − h(φ(0)) and G(z) = g(φ(z)) − g(φ(0)). Then H(0) =G(0) = 0 and

‖ωF (z)‖ = ‖DG(z)[DH(z)]−1‖ = ‖ωf (φ(z))‖ ≤ k,

which imply that F ∈ PH (k).By (4.9) and (4.10), we have

(4.11) BF (0) = Bf (ζ) ≥ (1− k)BH(0).

Note that BH(0) = Bh(ζ), which, together with (4.11), implies that

(4.12) Bf (ζ) ≥ (1− k)Bh(ζ).

Next we prove that (4.12) is sharp for all ζ ∈ B. Let

R = inf‖z‖e : z ∈ ∂B.Then there exists a point z0 ∈ ∂B such that ‖z0‖e = R. Let U be a unitary

transformation of Cn such that Uz0 is a pure imaginary vector in Cn. For z ∈ B,

let

f(z) = h(z) + g(z) = Uz + kUz,

where k ∈ [0, 1) is a constant. Then Bh(0) = R. Also, f is univalent on B and

Bf(0) = R(1− k), which gives that Bf (0)/Bh(0) = 1− k. In the following, we

will show that Bf (ζ)/Bh(ζ) = 1− k for all ζ ∈ B.


For any fixed ζ ∈ B, let

F (z) = f(φ(z)) − f(φ(0)) = H(z) +G(z),

where φ ∈ Aut(B) with φ(0) = ζ . Then

Bf (ζ) = BF (0) = (1− k)BH(0) = (1− k)Bh(ζ).

Step 4. Now we estimate the upper bound of Bf (0)/Bh(0). From (4.6) and

the relation Dh(ℓξ1(t))ℓ′ξ1(t) = ξ1, we have

Bf (0) =

∫ 1

0‖df(ℓξ0(t))‖e ≤

∫ 1

0‖df(ℓξ1(t))‖e

=

∫ 1

0

∥∥∥(Dh(ℓξ1(t))ℓ

′ξ1(t) +Dg(ℓξ1(t))ℓ

′ξ1(t))dt∥∥∥e

=

∫ 1

0

∥∥∥(ξ1 +Dg(ℓξ1(t))[Dh(ℓξ1(t))]

−1Dh(ℓξ1(t))ℓ′ξ1(t))dt∥∥∥e

=

∫ 1

0

∥∥∥(ξ1 + ωf (ℓξ1(t))ξ1

)dt∥∥∥e

≤ ‖ξ1‖e∫ 1

0(1 + ‖ωf (ℓξ1(t))‖) dt

= Bh(0)

(1 +

∫ 1

0‖ωf (ℓξ1(t))‖dt

).(4.13)

Applying Theorem 2.2 to ωf/k, we have

(4.14)‖ωf (z)‖

k≤ ‖z‖X +

‖ωf (0)‖k

1 +‖ωf (0)‖

k ‖z‖Xfor z ∈ B, where X = C

n. Since h−1(w‖ξ1‖e) biholomorphically maps Bn onto

some subdomain of B with h−1(0) = 0, by Theorem 2.3, we see that ‖ℓξ1(t)‖X ≤t. Consequently, by (4.13) and (4.14), we have

Bf(0) ≤ Bh(0)

(1 + k

∫ 1

0

‖ωf (0)‖ + k‖ℓξ1(t)‖Xk + ‖ωf (0)‖ ‖ℓξ1(t)‖X

dt

)(4.15)

≤ Bh(0)µk

(‖ωf (0)‖k

).

For any fixed k ∈ [0, 1), it is not difficult to see that µk(x) is an increasing function

of x ∈ (0, 1].Step 5. At last, we will establish the upper bound of Bf (z)/Bh(z) for all

z ∈ B. For any fixed ζ ∈ B, let φ ∈ Aut(B) with φ(0) = ζ. It follows from (4.10)

and (4.15) that

Bf (ζ) = BF (0) ≤ µk

(‖ωf (φ(0))‖k

)BH(0),(4.16)


where F and H are defined in (4.10). Note that

(4.17) BH(0) = Bh(ζ).

It follows from (4.16) and (4.17) that

(4.18) Bf (ζ) ≤ µk

(‖ωf (ζ)‖k

)Bh(ζ) ≤ µk(1)Bh(ζ) = (1 + k)Bh(ζ).

Furthermore, the estimate in (4.18) is asymptotically sharp because

limk→0+

µk(x) = 1.

The proof of the theorem is finished.

The proof of Theorem 2.5. The triangle inequality leads to

(4.19)

∥∥∥∥f(z)− f(b)

‖z‖X − ‖b‖X

∥∥∥∥Y

≥ 1− ‖f(z)‖Y1− ‖z‖X

for z ∈ BX . It follows from Theorem 2.3 that

1− ‖f(z)‖Y1− ‖z‖X

≥ 1 + ‖z‖X1 + ‖Df(0)‖‖z‖X

,

which, together with (4.19), implies that

(4.20)

lim infz→b

∥∥∥∥f(z)− f(b)

‖z‖X − ‖b‖X

∥∥∥∥Y

≥ lim infz→b

1 + ‖z‖X1 + ‖Df(0)‖‖z‖X

=2

1 + ‖Df(0)‖ .

Elementary calculations lead to

∥∥∥∥f(z)− f(b)

‖z‖X − ‖b‖X

∥∥∥∥Y

=

∥∥∥∥f(z)− f(b)−Df(b)(z − b) +Df(b)(z − b)

‖z‖X − ‖b‖X

∥∥∥∥Y

≤∥∥∥∥f(z)− f(b)−Df(b)(z − b)

‖z‖X − ‖b‖X

∥∥∥∥Y

+

∥∥∥∥Df(b)(z − b)

‖z‖X − ‖b‖X

∥∥∥∥Y

.(4.21)

If we replace z in (4.21) by Zj = (1 − 1/j)b, where j ∈ 1, 2, . . ., then we have

‖Zj‖X − ‖b‖X = −‖Zj − b‖X and

∥∥∥∥f(Zj)− f(b)

‖Zj‖X − ‖b‖X

∥∥∥∥Y

≤∥∥∥∥f(Zj)− f(b)−Df(b)(Zj − b)

‖Zj − b‖X

∥∥∥∥Y

+

∥∥∥∥Df(b)(

Zj − b

‖Zj − b‖X

)∥∥∥∥Y

=

∥∥∥∥f(Zj)− f(b)−Df(b)(Zj − b)

‖Zj − b‖X

∥∥∥∥Y

+ ‖Df(b)b‖Y ,


which gives that

(4.22) lim infj→+∞

∥∥∥∥f(Zj)− f(b)

‖Zj‖X − ‖b‖X

∥∥∥∥Y

≤ ‖Df(b)b‖Y .

The desired result follows from (4.20) and (4.22).

For a given value ‖Df(0)‖ = r ∈ [0, 1], the sharpness part follows from the

mapping

f(z) =lb(z) + r

1 + rlb(z)lb(z)y, z ∈ BX ,

where lb ∈ T (b), and y ∈ ∂BY is arbitrary.

Theorem L. ([62, Theorem 1.1]) Let f be a holomorphic self-mapping of U. If fis holomorphic at z = 1 with f(1) = 1, then

f ′(1) ≥ 2|1− f(0)|21− |f(0)|2 + |f ′(0)| .

This estimate is sharp with equality possible for each value of f(0) and |f ′(0)|with |f ′(0)| ≤ 1− |f(0)|2. The extreme function is

f(z) =γA(z) + f(0)

1 + γf(0)A(z),

where γ = (1− f(0))/(1 − f(0)) and A(z) = z((1− |f(0)|2)z + |f ′(0)|)/((1 −|f(0)|2) + |f ′(0)|z).The proof of Theorem 2.6. For ζ ∈ U, let

F (ζ) =1

2c(BY )h0(f(ζα), β),

where f : G→ BY is a holomorphic mapping with f(α) = β. By the assumption,

and Loos [43, Theorem 6.5], we have F (1) = 1. Then F is a holomorphic mapping

of U into itself such that F is holomorphic at ζ = 1 and F (1) = 1. Elementary

computations lead to

F ′(1) =1

2c(BY )h0(Df(α)α, β) and F ′(0) =

1

2c(BY )h0(Df(0)α, β),

which, together with Theorem L, yield that

1


2∣∣∣1− 1

2c(BY )h0(f(0), β)∣∣∣2

1−∣∣∣ 12c(BY )h0(f(0), β)

∣∣∣2+ ‖Df(0)α‖Y

,

where we have used the inequality |F ′(0)| ≤ ‖Df(0)α‖Y . Next, we prove the

sharpness part. Since G = BX is the unit ball of X, for any holomorphic function

ϕ of U into itself and for any lα ∈ T (α), the mapping f(z) = ϕ(lα(z))β is a

holomorphic mapping of BX into BY . Then, it follows from Theorem L that there

exists a holomorphic mapping of BX into BY with f(α) = β such that

h0(f(ζα), β) = 2c(BY )a+ ǫτ(ζ)

1 + ǫaτ(ζ),


where

a =1

2c(BY )h0(f(0), β), b =

1

2c(BY )h0(Df(0)α, β)

and

ǫ =1− a

1− a, τ(ζ) = ζ

(1− |a|2)ζ + |b|(1− |a|2) + |b|ζ .

The proof of this theorem is finished.

Part II. Schwarz type lemmas of pluriharmonic mappings and their applications.

The proof of Theorem 3.1. Let z ∈ BX \0 be fixed. Without loss of generality,

we may assume that a = f(z)− 1−‖z‖2X1+‖z‖2

X

f(0) 6= 0. Let w = z/‖z‖X ∈ ∂BX and

let u ∈ ∂BY be arbitrarily fixed. Since, for each lu ∈ T (u),

ϕ(ζ) = lu(f(ζw)), ζ ∈ U,

is a harmonic mapping in U such that ϕ(U) ⊆ U, we obtain from [48, Theorem

3.6.1] (or [29, Theorem 1]) that, for all ζ ∈ U,∣∣∣∣lu(f(ζw))−

1− |ζ|21 + |ζ|2 lu(f(0))

∣∣∣∣ ≤4

πarctan|ζ|.

Especially, let ζ = ‖z‖X . Since ζw = z, we have

|lu(a)| =∣∣∣∣lu(f(z)− 1− ‖z‖2X

1 + ‖z‖2Xf(0)

)∣∣∣∣ ≤4

πarctan‖z‖X .

Finally, if u = a/‖a‖Y , then we obtain∥∥∥∥f(z)−

1− ‖z‖2X1 + ‖z‖2X

f(0)

∥∥∥∥Y

≤ 4

πarctan‖z‖X .

Next, we prove the sharpness part. For any fixed point z0 ∈ BX\0, let lw0∈

T (w0) be fixed, where w0 = z0/‖z0‖X . It follows from [48, Theorem 3.6.1] (or

[29, Theorem 1]) that there exists a harmonic mapping Φ of U into itself with

Φ(0) = 0 such that

|Φ(lw0(z0))| =

4

πarctan |lw0

(z0)|.

For any fixed b ∈ ∂BY , let

f(z) = Φ(lw0(z))b, z ∈ BX .

Then f : BX → BY is harmonic and

‖f(z0)‖ = |Φ(lw0(z0))| =

4

πarctan |lw0

(z0)|

=4

πarctan ‖z0‖X ,



Theorem M. ([62, Theorem 1.7]) Let f be a harmonic mapping of U into itself

with f(0) = 0. Then, for z ∈ U, we have

|f(z)| ≤ 4

πarctan

( |z|+ π4 (|fz(0)| + |fz(0)|)

1 + π4 (|fz(0)| + |fz(0)|)|z|

|z|).

The proof of Theorem 3.2. For any fixed z ∈ BX\0, letw = z/‖z‖X ∈ ∂BX .

Without loss of generality, we assume f(z) 6= 0. Let u ∈ ∂BY be arbitrarily fixed.

Since, for each lu ∈ T (u), the function ϕ defined by

ϕ(ζ) = lu(f(ζw))

is a harmonic mapping of U into itself with ϕ(0) = 0, we obtain from Theorem M

that, for all ζ ∈ U,

(4.23) |lu(f(ζw))| = |ϕ(ζ)| ≤ 4

πarctan

(|ζ|+ π

4 (|ϕζ(0)|+ |ϕζ(0)|)1 + π

4 (|ϕζ(0)| + |ϕζ(0)|)|ζ||ζ|).

By the definition of Λf (0;w), we have

|ϕζ(0)| + |ϕζ(0)| ≤ Λf (0;w),

which, together with (4.23), implies that

|lu(f(ζw))| ≤4

πarctan

( |ζ|+ π4Λf (0;w)

1 + π4Λf (0;w)|ζ|

|ζ|).(4.24)

By letting ζ = ‖z‖X in (4.24), we have

|lu(f(z))| ≤4

πarctan

( ‖z‖X + π4Λf (0;w)

1 + π4Λf (0;w)‖z‖X

‖z‖X).

Finally, if u = f(z)/‖f(z)‖Y , then we get the desired result.

The proof of Theorem 3.3. It follows from Theorem 3.1 that

1− ‖f(z)‖Y1− ‖z‖X

≥ 1− 4π arctan ‖z‖X1− ‖z‖X

− 1 + ‖z‖X1 + ‖z‖2X

‖f(0)‖Y .

By using arguments similar to those in the proof of Theorem 2.5, we obtain

‖Df(b)b‖Y ≥ lim infz→b

(1− 4

π arctan ‖z‖X1− ‖z‖X

− 1 + ‖z‖X1 + ‖z‖2X

‖f(0)‖Y)

(4.25)

=2

π− ‖f(0)‖Y .

Next, let lf(b) ∈ T (f(b)) and

p(ζ) = 1− Re(lf(b)(f(ζb))

), ζ ∈ U.


Then p is a positive real valued harmonic function on U with p(1) = 0. By Har-

nack’s inequality, we have

1− r

1 + rp(0) ≤ p(ζ), r = |ζ| < 1.

Therefore, we have

1

1 + rp(0) ≤ p(r)− p(1)

1− r, 0 < r < 1.

Letting r → 1−, we have

(4.26)

1− ‖f(0)‖Y2

≤ 1− Re(lf(b)(f(0))

)

2≤ Re

(lf(b)(Df(b)b)

)≤ ‖Df(b)b‖Y .

Then combining (4.25) and (4.26) gives the desired result. This completes the

proof.

The proof of Theorem 3.5. By the definition of c(BY ) and the assumption on f ,

we see that ϕ is a harmonic mapping of U into itself with ϕ(0) = 0. Also, ϕ is

differentiable at ζ = 1 and ϕ(1) = 1. Then, by [62, Theorem 1.8], we have

Re(ϕζ(1) + ϕζ(1)

)≥ 4

π

1

1 + π4 (|ϕζ(0)|+ |ϕζ(0)|)

≥ 4

π

1

1 + π4Λf (0;α)

,

which implies that (3.3). The mapping f(z) = ψ(lα(z))β gives the sharpness,

where

ψ(ζ) =2

πarctan

2Re(ζ)

1− |ζ|2 , ζ ∈ U.


The proof of Theorem 3.7. First, we consider the case z0 = 0. By Proposition

1.1, it suffices to show that

‖Df(0)‖ ≤ 4

πif f(0) = 0;

sup‖β‖X=1

∣∣lf(0)(Df(0)β)∣∣ ≤ 4

πif f(0) 6= 0.

(i) If f(0) = 0, then let F (ζ) = l(f(ζβ)) for ζ ∈ U, where β ∈ X with

‖β‖X = 1 and l ∈ Y ∗ with ‖l‖Y ∗ = 1 are arbitrarily fixed. Then F : U → U is

harmonic. By applying Theorem F to the harmonic mapping F , we have

|l(Df(0)β)| ≤ 4

π.

Since β ∈ X with ‖β‖X = 1 and l ∈ Y ∗ with ‖l‖Y ∗ = 1 are arbitrary, we obtain

that

‖Df(0)‖ ≤ 4

π.


(ii) If f(0) 6= 0, then let F (ζ) = lf(0)(f(ζβ)) for ζ ∈ U, where β ∈ X with

‖β‖X = 1 is arbitrarily fixed. Then F : U → U is harmonic. By applying

Theorem F to the harmonic mapping F , we have

|lf(0)(Df(0)β)| ≤4

π.

Therefore, we have proved (3.8) in the case z0 = 0.

Next, we consider the case z0 6= 0. Let gz0 ∈ Aut(BX) be the Mobius transfor-

mation of BX defined by (1.3). By Proposition 1.1, we have

|∇‖f‖Y (z0)| ≤ |∇‖f gz0‖Y (0)| · ‖Dgz0(0)−1‖X .Since f gz0 satisfies the assumptions of the theorem for z0 = 0, by applying (3.8)

in the case z0 = 0 and using (1.4), we obtain that

|∇‖f‖Y (z0)| ≤4

π

1

1− ‖z0‖2X.

Finally, we will show that the estimate (3.8) is sharp. Let z0 ∈ BX \0 be fixed

and let w0 = z0/‖z0‖X . It follows from Theorem F that there exists a harmonic

mapping φ of U into itself such that φ(‖z0‖X) ∈ R \ 0 and

|φζ(‖z0‖X)|+ |φζ(‖z0‖X)| =4

π

1

1− ‖z0‖2X.

For any fixed lw0∈ T (w0) and any fixed a ∈ ∂BY , let

f(z) = φ(lw0(z))a, z ∈ BX .

Then f is a pluriharmonic mapping from BX into BY . Moreover,

|∇‖f‖Y (z0)| = sup‖β‖X=1

|φζ(‖z0‖X)lw0(β) + φζ(‖z0‖X)lw0

(β)|

= |φζ(‖z0‖X)|+ |φζ(‖z0‖X)|

=4

π

1

1− ‖z0‖2X.

If z0 = 0, then for arbitrary w0 ∈ ∂BX , by using the above argument, we have

|∇‖f‖Y (0)| =4

π.


The proof of Theorem 3.10. First, we show that

(4.27)

m∑

j=1

(‖Dh(0)wj‖2e + ‖Dg(0)wj‖2e

)≤ 1− ‖f(0)‖2e

for all wj ∈ Xj with ‖wj‖Xj= 1 (1 ≤ j ≤ m). Indeed, let wj ∈ Xj with

‖wj‖Xj= 1 (1 ≤ j ≤ m) be fixed and let F (ζ1, . . . , ζm) = f(ζ1w1, . . . , ζmwm)

for ζ = (ζ1, . . . , ζm) ∈ Um. Applying Theorem K to F , we obtain (4.27).

Next, let z ∈ BX \ 0 and w = (w1, . . . , wm) ∈ X with ‖wj‖Xj= 1 for

j = 1, . . . ,m be fixed. Let gz be the Mobius transformation defined by (1.3).


Applying (4.27) to the pluriharmonic mapping f gz = hgz+g gz and the unit

vector [Dgz(0)]−1w/‖[Dgz(0)]−1w‖X , we have

∑mj=1


)

‖[Dgz(0)]−1w‖2X≤ 1− ‖f(z)‖2e.

Therefore, by using (1.4), we have

m∑

j=1


)≤ ‖[Dgz(0)]−1w‖2X(1−‖f(z)‖2e) ≤

1− ‖f(z)‖2e(1− ‖z‖2X )2

as desired.

Next, we prove the sharpness part. Let a = (a1, . . . , am) ∈ BX \ 0 with

‖a1‖X1= · · · = ‖am‖Xm be arbitrarily fixed. For z ∈ BX , let

f(z) = (f1(z), f2(z), . . . , fn(z)) ∈ Bn,

where

f1(z) =−‖a1‖X1

+ la(z)

1− ‖a1‖X1la(z)

,

and fj(z) ≡ 0 for j ∈ 2, . . . , n. Then this mapping gives the sharpness at z = a.


The proof of Theorem 3.14. Let w ∈ X with ‖w‖X = 1 be fixed. Since

‖Dg(z)w‖e = ‖ωf (z)Dh(z)w‖e ≤ k‖Dh(z)w‖e for all z ∈ BX ,

there exists a function ηw(z) ∈ [0, k] such that ‖Dg(z)w‖e = ηw(z)‖Dh(z)w‖e .

Then, by (3.15) with m = 1, we have

‖Dh(z)w‖e ≤1√

1 + η2w(z)

√1− ‖f(z)‖2e1− ‖z‖2X

and

‖Dg(z)w‖e ≤ηw(z)√1 + η2w(z)

√1− ‖f(z)‖2e1− ‖z‖2X

,

which, together with the monotonicity of function χ(t) = (1 + t)/√1 + t2 for

t ∈ [0, 1), give that

‖Dh(z)‖ + ‖Dg(z)‖ ≤ 1 + ηw(z)√1 + η2w(z)

√1− ‖f(z)‖2e1− ‖z‖2X

≤ 1 + k√1 + k2

√1− ‖f(z)‖2e1− ‖z‖2X

=2K√

2(K2 + 1)

√1− ‖f(z)‖2e1− ‖z‖2X

.



5. A CONCLUDING REMARK

Let Bj be the unit ball of a complex Hilbert space Hj for j = 1, 2, respectively.

Note that if f is C1 at z0 ∈ ∂B1 with values in H2, then the adjoint operator

Df(z0)∗ is defined by

Re (〈Df(z0)∗w, z〉H1) = Re (〈w,Df(z0)z〉H2

) for z ∈ H1, w ∈ H2,

where 〈·, ·〉Hjis the inner product of Hj , j = 1, 2. The following result was

obtained in [21, Proposition 1.8].

Proposition 5.1. Let Bj be the unit ball of a complex Hilbert spaceHj for j = 1, 2,

respectively. Let f : B1 → B2 be a pluriharmonic mapping. Assume that f is of

class C1 at some point z0 ∈ ∂B1 and f(z0) = w0 ∈ ∂B2. Then there exists a

constant λ ∈ R such that Df(z0)∗w0 = λz0. Moreover, λ ≥ 1−Re

(〈f(0),w0〉

)

2 > 0.

By using Proposition 5.1 and the arguments similar to those in the proof of

Theorem 3.3, we obtain a better estimate:

λ ≥ max

2

π− ‖f(0)‖H2

,1− Re

(〈f(0), w0〉

)

2

.

6. ACKNOWLEDGMENTS

The first author (Shaolin Chen) was partly supported by NNSF of China (No.

12071116); the second author (Hidetaka Hamada) was partially supported by JSPS

KAKENHI Grant Number JP19K03553.

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S. CHEN, COLLEGE OF MATHEMATICS AND STATISTICS, HENGYANG NORMAL UNIVER-

SITY, HENGYANG, HUNAN 421002, PEOPLE’S REPUBLIC OF CHINA; HUNAN PROVINCIAL KEY

LABORATORY OF INTELLIGENT INFORMATION PROCESSING AND APPLICATION, 421002, PEO-

PLE’S REPUBLIC OF CHINA.

Email address: [email protected]


H. HAMADA, FACULTY OF SCIENCE AND ENGINEERING, KYUSHU SANGYO UNIVERSITY,

3-1 MATSUKADAI 2-CHOME, HIGASHI-KU, FUKUOKA 813-8503, JAPAN.


S. PONNUSAMY, DEPARTMENT OF MATHEMATICS, INDIAN INSTITUTE OF TECHNOLOGY

MADRAS, CHENNAI-600 036, INDIA.

Email address: [email protected], [email protected]

R. VIJAYAKUMAR, DEPARTMENT OF MATHEMATICS, INDIAN INSTITUTE OF TECHNOLOGY

MADRAS, CHENNAI-600 036, INDIA.


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