Invariant Pseudodistances and Pseudometrics in Complex Analysis in Several Variables

Invariant Pseudodistances and Pseudometrics in Complex Analysis in Several Variables

Thomas ¨Onskog

Abstract. At present time invariant pseudodistances and pseudometrics pose an important tool in complex analysis in several variables. This thesis is mainly devoted to giving a thorough definition of these objects and in particular to Schwarz-Pick systems such as the Carath´eodory and Kobayashi pseudodistances.

Most basic properties, such as continuity and boundary behaviour, of the Carath´eodory and Kobayashi pseudo-distances and pseudometrics are investigated. As an application of the theory, the thesis is concluded with a proof of the biholomorphic inequivalence between the unit ball and unit polydisc in Cⁿ.

Contents

1. Introduction 3

1.1. Preliminaries 3

1.2. The Schwarz and Schwarz-Pick lemmata and automorphisms of D 6

1.3. Pseudodistances and pseudometrics 8

1.4. The Poincar´e distance and Poincar´e metric 12

section2. Schwarz-Pick systems15

2.1. Determining σ_Dn and σ_Bn for Schwarz-Pick systems 15

3. The Carath´eodory pseudodistance and pseudometric 17

3.1. The Carath´eodory pseudodistance 17

3.2. The Carathodory-Reiffen pseudometric 19

3.3. Comparison between CΩ, eCΩand cⁱ_Ω 23

4. The Kobayashi pseudodistance and pseudometric 31

4.1. The Kobayashi pseudodistance 31

4.2. The Kobayashi-Royden pseudometric 35

4.3. Comparison between KΩand kⁱ_Ω 37

5. Comparison between the distances and metrics 41

5.1. When do all Schwarz-Pick systems coincide? 41

5.2. Boundary behaviour on strictly pseudoconvex domains 41

6. The biholomorphic inequivalence of Bn and Dn. 47

References 51

1. Introduction

The Riemann mapping theorem states that any simply connected domain in the complex plane, other than the entire plane itself, allows an biholomorphic mapping onto the unit disc. This theorem constitutes one of the most extraordinary and astonishing results of complex analysis in one variable and implies that a very large number of domains in C are in fact equivalent up to biholomorphic mappings.

In the beginning of the 20th century it, however, became clear, due to Poincar´e, that in higher dimension even such elementary and symmetric simply connected domains as the unit ball and the unit polydisc may not be biholomorphically mapped onto each other. In the hope of being able to classify domains that could be connected by a biholomorphic mapping, it became interesting to investigate properties that were unaffected – or invariant – by biholomorphic mappings.

In the end of the thirties, C. Carath´eodory discovered that the set of bounded, holomorphic functions on a given domain in Cⁿcould be used to define a pseudodistance, that would be invari- ant under biholomorphic mappings. This pseudodistance, nowadays known as the Carath´eodory pseudodistance, has, during the course of the century, been supplemented by a pseudometric with similar properties.

Some thirty years ago, S. Kobayashi commenced the study of another important object – the set of analytic discs, that is holomorphic mappings from the unit disc into a given domain in Cⁿ – on which he defined another pseudodistance that was invariant under biholomorphic mapping. The work of Kobayashi resulted in the Kobayashi pseudodistance and the corresponding Kobayashi- Royden pseudometric.

This thesis is intended to be an introduction to the field of invariant pseudodistances and pseudometrics in Cⁿ for those equipped with at least an undergraduate course on complex analysis.

The first section includes all the definitions and basic theorems necessary for the introduction of invariant pseudodistances and pseudometrics as well as a survey of the Poincar´e distance and metric on the unit disc.

In the three subsequent sections the notion of a Schwarz-Pick system in general and the Carath´eodory and Kobayashi pseudodistances in particular are defined. The two latter pseudodis- tances are thereafter, in the fifth section, being compared to each other with regard to boundary behaviour and degeneracy etc. Concluding the thesis, a short proof, using the Kobayashi pseu- dodistance, of the biholomorphic inequivalence between the unit ball and unit polydisc in Cⁿ is carried out.

1.1. Preliminaries. Before beginning the actual thesis, some notation and basic results will be stated in this first section in order to simplify the understanding of later chapters. Those familiar with analysis in several complex variables, may, without loss, continue directly to Section 1.2.

1.1.1. Notation. The one-dimensional complex plane is denoted by C and the n-dimensional com- plex space consisting of n-tuples (z₁, . . . , z_n), where each z_i is a complex number, is denoted by Cⁿ. Now consider a subset Ω of Cⁿ, where n³1. The boundary of Ω is written ∂Ω and the closure of Ω, Ω ∪ ∂Ω, is written ¯Ω. If Ω is open and connected it will be called a domain.

The tangent space of Ω at a point p ∈ Ω, that is the set of tangent vectors ξ at p, is denoted by Tp(Ω) and its elements are written (p; ξ). For domains Ω ⊂ Cⁿ, Tp(Ω) may be identified with Cⁿ. The union of the tangent spaces for all points p ∈ Ω, that is ∪p∈ΩTp(Ω), is known as the tangent bundle and denoted as T (Ω).

The Euclidean norm of a point z ∈ C is given by |z| = (z¯z)¹². The rules |zw| = |z||w| and

|¯z| = |z|, where z and w are arbitrary points in C, easily follow from the definition. Similarly the Euclidean norm of a point (z₁, . . . , z_n) ∈ Cⁿ is defined as

(z1, . . . , zn)

=Xⁿ

i=1

ziz¯i

¹₂

=Xⁿ

i=1

|zi|²¹₂ .

An open disc with center p and radius r in the complex plane is denoted as B(p, r) and corresponds to the set

B(p, r) =z ∈ C; |z − p| < r .

The unit disc in C, that is the open disc with center at the origin and unit radius, is written D.

In higher dimension an open ball with center (p₁, . . . , p_n) and radius r represents the set B (p₁, . . . , p_n), r
=n

(z₁, . . . , z_n) ∈ Cⁿ;

(z₁, . . . , z_n) − (p₁, . . . , p_n) < ro

, whereas an open polydisc with center (p1, . . . , pn) and radius r corresponds to the set

D (p1, . . . , pn), r
= (z1, . . . , zn) ∈ Cⁿ; |zi− pi| < r ,

Thus the unit disc is in higher dimension usually replaced by either the unit ball Bn

Bn =n

(z₁, . . . , z_n) ∈ Cⁿ;

(z₁, . . . , z_n) < 1o

, or the unit polydisc Dⁿ

Dn =(z1, . . . , z_n) ∈ Cⁿ; |z_i| < 1 for 1 ≤ i ≤ n .

Consider a complex-valued function f : Ω → C, where Ω is a domain in C. Now separating, if possible, the real and imaginary parts as z = x + iy and f (z) = u(z) + iv(z) = u(x, y) + iv(x, y), we call f holomorphic or analytic on Ω if the partial derivatives of u and v with respect to x and y are continuous and satisfy the Cauchy-Riemann equations

∂u

∂x =∂v

∂y and ∂u

∂y = −∂v

∂x, for all z ∈ Ω. Using the partial differential operators

∂

∂z =1 2

 ∂

∂x +1 i

∂

∂y



and ∂

∂ ¯z =1 2

 ∂

∂x −1 i

∂

∂y

 , the Cauchy-Riemann equations can be written more compactly as

∂f

∂ ¯z = 0.

If instead Ω ⊂ Cⁿ a function f : Ω → C is holomorphic if it is holomorphic in each of the n variables separately, that is if

∂f

∂ ¯zi

= 0 for 1 ≤ i ≤ n.

Finally a function with multivariable image, f = (f1, . . . , fn), is holomorphic if all the functions fi are holomorphic. The set of holomorphic functions from Ω1 ⊂ Cⁿ to Ω2 ⊂ C^m is denoted H(Ω1, Ω2).

A biholomorphic function f : Ω1→ Ω2is a bijective, holomorphic function. In finite dimension, such as Cⁿ, it can be shown that if f is biholomorphic, then f⁻¹ must be holomorphic ([12], pages 86-88). A biholomorphic function f : Ω → Ω is in complex analysis known as an automorphism on Ω. The set of automorphisms on Ω is written as Aut(Ω). Aut(Ω) acts transitively if for all z, w ∈ Ω, there exists a function f ∈ Aut(Ω), such that f (z) = w.

1.1.2. Some basic results. The following lemmata are found in most elementary texts on complex analysis in Cⁿ and will henceforth often be used without reference.

Lemma 1.1. (Cauchy-Schwarz inequality) ([6], page 98)

Let z = (z₁, . . . , z_n) and w = (w₁, . . . , w_n) be arbitrary points in Cⁿ. Then

n

X

i=1

|ziw_i| ≤Xⁿ

i=1

|zi|²¹₂Xⁿ

i=1

|wi|²¹₂

= kzkkwk.

Lemma 1.2. (Maximum modulus principle) ([15], pages 165-166)

A non-constant holomorphic function on a bounded domain Ω in C with image in C, which is continuous on ¯Ω, attains its maximum modulus on the boundary of Ω.

or similarly

A holomorphic function on a bounded domain Ω ⊂ C with image in C, which achieves its maximum modulus in the interior of Ω is constant on Ω.

 Lemma 1.3. (Generalized Cauchy integral formula) ([10], page 31)

Let w = (w1, . . . , wn) ∈ Cⁿ and r > 0. If f is continuous on ¯D(w, r) and holomorphic on D(w, r), then for any z = (w1, . . . , wn) ∈ D(w, r)

f (z) = 1 (2πi)ⁿ

Z

|ζn−wn|=r

· · · Z

|ζ1−w1|=r

f (ζ1, . . . , ζn)

(ζ1− z1) · · · (ζn− zn)dζ1· · · dζn.

 This formula is used in the proof of the following important lemma.

Lemma 1.4. (Cauchy estimates in Cⁿ) ([13], page 9)

Let z = (z1, . . . , zn) be an arbitrary point in Cⁿ. If f is holomorphic on D(z, r) and has maximum modulus M on D(z, r), then

∂^αf

∂z^α₁¹· · · ∂z^αnⁿ

(z) 

≤ α1! · · · αn!

r^α M where αi∈ N and α = α1+ · · · + αn.

If instead f only is holomorphic on the open ball B(z, r) with maximum modulus M on B(z, r), we can use the Cauchy estimate on the polydisc D(z,^√r_n) ⊂ B(z, r) instead, that is

∂^αf

∂z₁^α1· · · ∂zn^αn

(z)  

≤ α₁! · · · α_n!

r^α M n^α2, where αi∈ N and α = α1+ · · · + αn.

 Lemma 1.5. (Taylor theorem in Cⁿ) ([13], page 16 and [10], page 77)

Let w = (w1, . . . , wn) be an arbitrary point in Cⁿ. If f is holomorphic on D(w, r) ⊂ Cⁿ, then for all z ∈ D(w, r),

f (z) =

∞

X

i=0

X

α=i

∂^αf

∂z^α₁¹· · · ∂zn^αn

(w)(z1− w1)^α1· · · (zn− wn)^αn

α₁! · · · α_n! , where αi∈ N and α = α1+ · · · + αn. The first sum can be terminated at any positive number, causing the emergence of a rest term

f (z) =

k

X

i=0

X

α=i

∂^αf

∂z₁^α1· · · ∂zn^αn

(w)(z₁− w₁)^α1· · · (z_n− w_n)^αn

α1! · · · αn! + O kz − wk^k+1
, where αi∈ N and α = α1+ · · · + αn.

 Lemma 1.6. (Chain rule in Cⁿ) ([13], page 20-21)

Let Ω1 ⊂ Cⁿ. Suppose f : Ω1 → Ω2 ⊂ C^m and g : Ω2 → Ω3 ⊂ C^l are differentiable functions.

Differentiating g ◦ f : Ω1→ Ω3 then yields

∂(g ◦ f )_k

∂zj

=

m

X

i=1

 ∂g

∂wi

◦ f_k∂f_i

∂zj

+ ∂g

∂ ¯wi

◦ f_k∂ ¯f_i

∂zj

 , and

∂(g ◦ fk)k

∂ ¯zj

=

m

X

i=1

 ∂g

∂wi

◦ fk

∂fi

∂ ¯zj

+∂g

∂ ¯wi

◦ f∂ ¯fi

∂ ¯zj

 ,

for all 1 ≤ j ≤ n and 1 ≤ k ≤ l. If f and g are holomorphic, then so is g ◦ f , and the chain rule reduces to

∂(g ◦ f )k

∂z_j =

m

X

i=1

∂g

∂z_i ◦ fk

∂fi

∂z_j for all 1 ≤ j ≤ n and 1 ≤ k ≤ l.

 Lemma 1.7. (Montel theorem) ([7], page 369)

Any family F of holomorphic functions on a domain Ω ⊂ Cⁿ, which, for some finite C > 0, satisfies

f (z)

< C for all f ∈ F and z ∈ Ω,

is normal, that is for any sequence {fn} ⊂ F , there exists a subsequence {fn_k} converging uniformly on compact subsets of Ω to some function f ∈ H(Ω, C).

 We conclude with two results from the theory of integration.

Lemma 1.8. (H¨older inequality) ([6], page 96)

Let p and q belong to [1, ∞], such that ¹_p +¹_q = 1. If f and g satisfy R

X|f |^pdµ
¹_p

< ∞ and R

X|g|^qdµ
¹_q

< ∞ for some measurable set X in a measure space with measure µ, then Z

X

|f g|dµ ≤Z

X

|f |^pdµ¹_pZ

X

|g|^qdµ¹_q .

 Lemma 1.9. (Lebesgue dominated convergence theorem ([14], page 26)

Let X be a measurable set in a measure space with measure µ. Suppose furthermore that fn is a sequence of complex measurable functions on X, such that f (x) = limn→∞fn(x) exists for all x ∈ X. If there is an integrable function g, satisfying 

fn(x)

≤ g(x) for all n ∈ N and x ∈ X, then f is integrable and

n→∞lim Z

X

fndµ = Z

X

f dµ.

 1.2. The Schwarz and Schwarz-Pick lemmata and automorphisms of D. We shall begin by studying the classical Schwarz lemma of one complex variable, which gives crucial knowledge of the structure of holomorphic mappings on the unit disc. Schwarz’s lemma will be used repeatedly in many different contexts throughout the thesis.

Proposition 1.10. (Schwarz’s lemma) If f ∈ H(D, D) and f (0) = 0 then

(1)  f (z)

≤ |z| for all z ∈ D, (2) 

f⁰(0) ≤ 1.

If either f (z)

= |z| for some nonzero z ∈ D or f⁰(0)

= 1, then f (z) = e^iθz for some θ ∈ R and all z ∈ D.

Proof: Since f is holomorphic and f (0) = 0, we have lim

z→0,z6=0

f (z)

z = lim

z→0,z6=0

f (z) − f (0)

z − 0 = f⁰(0), showing that the function

g(z) =

( f (z)

z , z 6= 0 f⁰(0), z = 0

,

is holomorphic on D. Applying the first form of the maximum modulus principle to g on the disc D=z : |z| ≤ 1 −  for  > 0 yields

g(z)

≤ max

z∈∂D

g(z)

= max

z∈∂D

f (z)  1 −  ≤ 1

1 − .

Letting  → 0⁺ gives g(z)

≤ 1 on D, which implies that f (z)

≤ |z| for all z ∈ D as desired. To prove (2), evaluate the inequality 

g(z)

≤ 1 at the origin. By the definition of g(z), this means that

f⁰(0) ≤ 1.

If now  f (z)

 = |z| for some nonzero z in the unit disc or if  f⁰(0)

 = 1, then  g(z)

 = 1 for some point in the interior of D. Thus by the second form of the maximum modulus principle, g(z) is constant and therefore equal to some unimodular constant, e^iθ, where θ is a fixed number in [0, 2π). We can conclude that f (z) = g(z)z = e^iθz for all z ∈ D.

 Before moving on to the Schwarz-Pick lemma, where the f (0) = 0 criterion has been dropped, we consider the automorphisms of the unit disc, Aut(D). For all a ∈ D the M¨obius transformation

φa(z) = z − a 1 − ¯az.

is a holomorphic function on the disc with the property φ_a(a) = 0. Investigating φ_a(λ)

, where

|λ| = 1, yields

φa(λ) =

λ − a 1 − ¯aλ

  =

   1

¯λ λ − a 1 − ¯aλ

  =

λ − a λ − ¯¯ a   = 1,

so it is clear that φ_a is a holomorphic map from the unit disc into itself. In addition a straightfor- ward calculation shows that φ_a is onto. Thus since the property φ_−a◦ φ_a(z) = z, for all z ∈ D, easily can be deduced from the definition of M¨obius transformations, it follows that φais an invert- ible or bijective function. Hence M¨obius transformations are automorphisms on the unit disc. It is easily shown that the functions f (z) = e^iθz discussed in Schwarz’s lemma also belong to Aut(D).

Now consider an arbitrary function f ∈ Aut(D). Let g = φf (0)◦f . Then since g is a composition of two functions in Aut(D), it belongs to Aut(D). The functions in Aut(D) are invertible, so g⁻¹ is also an automorphism of the unit disc. Noting that g(0) = 0 and (g⁻¹)(0) = 0 allows us to use the Schwarz lemma on both g and its inverse. The lemma yields

g⁰(0)

≤ 1 and 

(g⁻¹)⁰(0) = 1

g⁰(0) 

≤ 1, so we can conclude that

g⁰(0)

= 1. By the equality part of Schwarz’s lemma, g(z) = e^iθz for some θ ∈ R and by using the definition of g it follows that f = φ−f (0)◦g, so since f was taken arbitrarily, this shows that all members of Aut(D) are actually compositions of a M¨obius transformation and a rotation.

Noteworthy in the context of Aut(D) is also that this set acts transitively. An arbitrary point z ∈ D can be mapped to the arbitrary point w ∈ D by the automorphism h = φ−w◦ φz.

We shall now use the Schwarz lemma to prove the more generalized Schwarz-Pick lemma, which gives further information about the properties of the automorphisms of the unit disc.

Proposition 1.11. (Schwarz-Pick lemma) If f ∈ H(D, D) then

(1)   

f (z) − f (w) 1 − f (w)f (z)

  ≤

z − w 1 − ¯wz

for all z, w ∈ D, (2)

f⁰(z)  1 −

f (z) 

2 ≤ 1

1 − |z|² for all z ∈ D.

If f ∈ Aut(D), then both inequalities are equalities. Conversely, if equality holds in (1) for some z 6= w or if equality holds in (2) for some z ∈ D, then f ∈ Aut(D).

Proof: Let g = φ_{f (w)}◦ f ◦ φ_−w or equivalently g(z) = φ_{f (w)} f (_{1+ ¯}^z+w_wz)
. Then g ∈ H(D, D) and g(0) = φ_{f (w)}f (w) = 0. By Schwarz’s lemma

g(λ)

≤ |λ| for all λ ∈ D and in particular 

g φ_w(z)
  ≤

φ_w(z)

 or 

φ_{f (w)} f (z)
  ≤

φ_w(z) .

Writing out the last inequality gives   

f (z) − f (w) 1 − f (w)f (z)

  ≤

z − w 1 − ¯wz

  ,

which proves (1). Using (1), it is possible to prove (2) by means of the following manipulation f⁰(z)

 1 −

f (z) 

2 = lim

w→z,w6=z

  

f (z) − f (w) z − w

1

|1 − f (w)f (z)|



≤ lim

w→z,w6=z

1

|1 − ¯wz| = 1 1 − |z|². Suppose now that f ∈ Aut(D). Then f⁻¹∈ Aut(D). Repeated use of (1) yields

z − w 1 − ¯wz

  =

f⁻¹ f (z)
− f⁻¹ f (w)
1 − f⁻¹ f (w)
f⁻¹ f (z)

  ≤

f (z) − f (w) 1 − f (w)f (z)

  ≤

z − w 1 − ¯wz

  , so we must have equality. Similarly for (2), we get

1 1 − |z|² =

(f⁻¹◦ f )⁰(z)  1 −

f⁻¹(f (z)) 

2 ≤

f⁰(z)  1 −

f (z) 

2 ≤ 1

1 − |z|², so again equality holds for f ∈ Aut(D).

If conversely equality holds in (1) for some z 6= w then 

g φ_w(z)
 =

φ_w(z)

. By the equality part of Schwarz’s lemma, this yields g(λ) = e^iθλ for some θ ∈ R and λ ∈ D. Thus g ∈ Aut(D) and by the definition of g, f = φ_{−f (w)}◦ g ◦ φw∈ Aut(D) as desired. If we instead have equality in (2), we let w = z in the definition of g and use the chain rule and the fact that φ⁰_a(z) = _(1−¯^1−|a|_az)²2 to obtain

g⁰(0)  =

φ⁰_{f (z)} f ◦ φ_−z(0)
    

f⁰ φ_−z(0)
  

φ⁰_−z(0) 

=  

φ⁰_{f (z)} f (z)
  

f⁰(z)

 1 − |z|²

= 1 −

f (z) 

2

1 − f (z)f (z)
² f⁰(z)

 1 − |z|²

=

f⁰(z)  1 −

f (z) 

2 1 − |z|²
= 1.

Using the equality part of Schwarz’s lemma, it follows that g is a rotation, which gives g ∈ Aut(D).

Thus f = φ_{−f (w)}◦ g ◦ φw∈ Aut(D).

 Now equipped with some knowledge of the structure and behaviour of holomorphic functions on the unit disc, we are ready to define metrics and distances on domains in Cⁿ. The basic properties of these will partly be determined with the aid of the lemmata above.

1.3. Pseudodistances and pseudometrics. As noted in the introduction, one of the main reasons to study distances in complex analysis is in order to determine whether there exist any biholomorphic mappings from one domain into another. In section 1.2 we saw that, from the unit disc to itself, biholomorphic mappings exist and are given by compositions of a M¨obius transfor- mation and a rotation. In C, the Riemann mapping theorem states that there are biholomorphic mappings between the unit disc and any simply connected domain other than the entire plane ([9], pages 21-24). In Cⁿ, n > 1, no similar relations have been found. In addition Poincar´e proved, as we shall se in section 6 using one of the invariant pseudometrics discussed below, that there are no biholomorphic mappings between the unit polydisc, Bn, and the unit ball, Dn.

At places where confusion cannot arise, we will from now on use the letter z for both the one-dimensional point z in C and the multi-dimensional point (z¹, . . . , zn) in Cⁿ and so forth.

1.3.1. Pseudodistances. First of all we need a formal definition of the distances – or pseudodis- tances – that will be used in this thesis.

Definition 1.12. A function σ : Ω × Ω → [0, ∞), where Ω ⊂ Cⁿ is a pseudodistance if (1) σ(z, w) = σ(w, z) for all z, w ∈ Ω,

(2) σ(z, w) ≤ σ(z, v) + σ(v, w) for all z, v, w ∈ Ω.

If in addition σ(z, w) = 0 if and only if z = w, σ is called a distance.

 Most pseudodistances studied in the following text are not distances, since, on some domains, they may give a zero distance between distinct points. At these points we say that the pseudodis- tance degenerate.

Now consider the two domains Ω1and Ω2 and appoint to these the two pseudodistances σ1and σ2, respectively, with the property of preserving lengths under biholomorphic mappings. A study of such invariant pseudodistances can give crucial information about the existence of biholomorphic mappings. The pseudodistances that will be studied in the following sections will be contractions, that is they are decreasing under holomorphic mappings. This means that if h : Ω1 → Ω2 is holomorphic, then

σ2 h(z), h(w)
≤ σ1(z, w).

If then h is biholomorphic, h⁻¹ : Ω2→ Ω1 is also holomorphic and σ₁ h⁻¹(z), h⁻¹(w)
≤ σ2(z, w).

Conclusively for biholomorphic mappings we get σ1(z, w) = σ1



h⁻¹ h(z)
, h⁻¹ h(w)


≤ σ2 h(z), h(w)
≤ σ1(z, w),

so σ2 h(z), h(w)
= σ1(z, w). Using the contrapositive result, we can conclude that if there exists a pair of pseudodistances σ1 on Ω1 and σ2 on Ω2, decreasing under holomorphic mappings, such that for all holomorphic functions h : Ω1 → Ω2, there exists some pair of points z, w ∈ Ω1, such that σ2 h(z), h(w)
6= σ1(z, w), then no biholomorphic mappings from Ω1to Ω2 can exist.

1.3.2. Pseudometrics and integrated forms. It is also possible to define a distance from a differ- ential geometric point of view by constructing a metric on the tangent bundle. At all points of a domain, the metric appoints a certain positive value (or length) to all tangent vectors.

Definition 1.13. A function α : T (Ω) → [0, ∞), where Ω ⊂ Cⁿ, is called an infinitesimal Finsler pseudometric if

(1) α(p; tv) = |t|α(p; v) for all (p; v) ∈ T (Ω) and t ∈ C, (2) α is upper semicontinuous.

If in addition α(p; v) > 0 for all (p; v) ∈ T (Ω) where v 6= 0, then α is called an infinitesimal Finsler metric.

 By integrating the metric over some class of curves connecting two points a pseudodistance can be constructed on the domain. These, so-called admissible, curves in Ω, over which we may integrate, are hereafter always considered to be parametrized curves with piecewise continuous derivative. If α is an infinitesimal Finsler pseudometric and γ : [a, b] → Ω is an admissible curve, we can define an integral Lα(γ) as

L_α(γ) = Z b

a

α γ(t), γ⁰(t)
dt.

L_α(γ) is dependent on the choice of curve. We can, however, use L_α(γ) to construct a curve- independent pseudodistance on Ω.

Definition 1.14. The integrated form, αⁱ, of an infinitesimal Finsler pseudometric, α, is given by αⁱ(z, w) = infLα(γ); γ is an admissible curve between z and w .

 We must check if the integrated form actually determines a pseudodistance. The non-negativity of α forces αⁱ to be non-negative and the fact that an admissible curve between z and w is also an admissible curve between w and z makes αⁱ symmetric. To check the triangle inequality,

αⁱ(z, w) ≤ αⁱ(z, v) + αⁱ(v, w),

let γ_zvⁿ and γ_vwⁿ be sequences of admissible curves, for which lim_n→∞L_α(γ_zvⁿ) = αⁱ(z, v) and lim_n→∞ L_α(γ_vwⁿ ) = αⁱ(v, w). Now γ_zvⁿ + γ_vwⁿ defines an admissible curve from z to w. Letting n → ∞, we get a curve from z to w, with length αⁱ(z, v) + αⁱ(v, w). There might, however, exist admissible curves γ between z and w which make L_α(γ) even smaller, so

αⁱ(z, w) = infLα(γ); γ is an admissible curve between z and w ≤ αⁱ(z, v) + αⁱ(v, w), as desired. Thus the integrated form is a pseudodistance.

We will next show that under holomorphic mappings the integrated form behaves similarly as the corresponding pseudometric. First, however, we need a technical definition.

Definition 1.15. Consider a holomorphic function f : Ω1 → Ω2, where Ω1⊂ Cⁿ and Ω2⊂ C^m. The push-forward of f at a point z ∈ Ω1is the function f_∗^z: Tz(Ω1) → Tf (z)(Ω2) defined as

f_∗^z(z; ξ) =Xⁿ

i=1

∂f₁

∂zi

(z)ξ_i, . . . ,

n

X

i=1

∂f_m

∂zi

(z)ξ_i .

If m = 1, this naturally reduces to

f_∗^z(z; ξ) =

n

X

i=1

∂f

∂z_i(z)ξi.

The push-forward can also be defined on the entire tangent bundle and will, in that case, be written as f_∗: T (Ω₁) → T (Ω₂). Henceforth we shall denote f_∗(z; ξ) by f_∗(z)ξ.

 The push-forward can be used in order to define what is meant by a contraction, or decrease under holomorphic mappings, in the context of metrics.

Definition 1.16. If α₁and α₂ are infinitesimal Finsler pseudometrics on Ω₁⊂ Cⁿ and Ω₂⊂ C^m respectively, then a continuously differentiable function f : Ω1→ Ω2 is called a contraction if

α₂ f (z); f_∗(z)ξ
≤ α1(z; ξ), for all (z; ξ) ∈ T (Ω1).

 In the next proposition we will show that the property of being a contraction in the metric sense implies being a contraction in the distance sense. Here we denote by (Ω, σ) the domain Ω equipped with the pseudodistance σ and by (Ω, α) the domain Ω equipped with the pseudometric α.

Proposition 1.17. If the holomorphic function f : (Ω₁, α₁) → (Ω₂, α₂) is a contraction, then so is f : (Ω₁, αⁱ₁) → (Ω₂, αⁱ₂).

Proof: Suppose that Ω₁⊂ Cⁿ and Ω₂⊂ C^m. Let γ be an admissible curve γ : [a, b] → Ω₁between the arbitrary points z and w in Ω1. Then f ◦γ is an admissible curve (f ◦γ) : [a, b] → Ω2connecting

f (z) and f (w). The definitions and the chain rule for holomorphic functions yield αⁱ₂ f (z), f (w)

≤ Lα₂(f ◦ γ) = Z b

a

α2 (f ◦ γ)(t), (f ◦ γ)⁰(t)
dt =

= Z b

a

α2



f γ(t)
,Xⁿ

i=1

∂f1

∂z_i γ(t)
γ_i⁰(t), . . . ,

n

X

i=1

∂fm

∂z_i γ(t)
γ_i⁰(t) dt =

= Z b

a

α2

f γ(t)
, f∗ γ(t)
γ⁰(t) dt ≤

Z b a

α1 γ(t), γ⁰(t)
dt = Lα₁(γ).

Now taking an infimum over the set of admissible curves γ : [a, b] → Ω₁, we get

αⁱ₂(f (z), f (w)) ≤ infLα₁(γ); γ is an admissible curve between z and w = αⁱ₁(z, w), which shows the desired conclusion.

 1.3.3. Inner pseudodistances. The method used above to create pseudodistances from infinitesimal Finsler pseudometrics can also be used to define new pseudodistances by integration of a continuous pseudodistance. The pseudodistance that arises from this operation will be called inner.

To define the inner pseudodistance between two points z and w in Ω ⊂ Cⁿ, choose an admissible curve γ : [a, b] → Ω and create a partition of [a, b] such that a = x0≤ x1≤ · · · ≤ xn= b, γ(x0) = z and γ(xn) = w. Then choose finer and finer partitions, satisfying sup_1≤i≤nxi− xi−1 = ∆ → 0.

We define

Lσ(γ) = lim

∆→0 n

X

i=1

σ(xi, xi−1),

where σ is the continuous pseudodistance used to create the inner pseudodistance. The continuity of σ assures that L_σ(γ) is well defined.

Definition 1.18. If σ is a continuous pseudodistance on a domain Ω in Cⁿ and z, w ∈ Ω, we define the inner pseudodistance as

σ(z, w) = infe Lσ(γ); γ is an admissible curve between z and w , where Lσ(γ) is defined as above.

 Using the same argument as in the case of the integrated form, it is clear that the inner pseu- dodistance actually determines a pseudodistance.

Proposition 1.19. For all continuous pseudodistances σ on a domain Ω ∈ Cⁿ,σe³σ.

Proof: Take two arbitrary points z, w ∈ Ω and an arbitrary admissible curve γ : [a, b] → Ω between these points. Choose a partition of [a, b] as in the definition of Lσ(γ). Repeated use of the triangular inequality of σ then yields

σ(z, w) ≤ σ z, γ(x1)) + σ(γ(x1), γ(x2)) + · · · + σ(γ(xn−1), w
=

n

X

i=1

σ(xi, xi−1).

Again choosing partitions such that ∆ → 0, the continuity of σ gives σ(z, w) ≤ L_σ(γ) for all admissible curves γ between z and w. Now taking an infimum over all such curves γ, yields σ(z, w) ≤eσ(z, w)

 A pseudodistance satisfyingσ = σ is said to be inner.e

1.4. The Poincar´e distance and Poincar´e metric. When studying the simplest domain in the complex plane, the unit disc, we were in section 1.2 able to characterize all elements in the set of automorphisms. Following the line of arguing about the connection between biholomorphic mappings and invariant pseudometrics in section 1.3.1, it would be interesting to determine which distances that are invariant under the mappings of Aut(D).

1.4.1. The M¨obius distance. Guided by the first part of the Schwarz-Pick lemma, we define the M¨obius pseudodistance, which, as desired, is decreasing under holomorphic mappings and invariant under the automorphisms of the unit disc.

Definition 1.20. The M¨obius pseudodistance between two points z and w in the unit disc is defined as

m_D(z, w) =   

z − w 1 − wz

  .

 First note that the non-negativity and finiteness for arbitrary z, w ∈ D are obvious. The symmetry easily follows from the fact that |1 − wz| = |1 − wz| = |1 − wz|. In order to prove that the definition actually determines a pseudodistance it remains only to show the triangle inequality

m_D(z, w) ≤ m_D(z, v) + m_D(v, w).

The function m_Dis invariant under Aut(D), which acts transitively on D. Thus there exists a unique automorphism h, such that h(v) = 0 and h(w) = a, where a ∈ [0, 1). Putting h(z) = b ∈ D \ {0, a}, the inequality reduces to m_D(b, a) ≤ m_D(b, 0) + m_D(0, a) or equivalently

|b − a|

|1 − ab| ≤ |b| + a.

For the given values of a and b, 1 ≤ 1 + a|b| < 2, so the inequality above follows if we can show that _|1−ab|^|b−a| ≤ _1+a|b|^|b|+a. Putting b = ce^iθ, the required inequality turns into

ce^iθ− a 1 − ace^iθ  

≤ c + a

1 + ac for all θ ∈ [0, 2π).

Squaring and evaluating the left-hand side yields 

ce^iθ− a 1 − ace^iθ   

2

= c²− 2ac cos θ + a²

1 − 2ac cos θ + a²c² =: g(θ).

Differentiating and putting the derivative

g⁰(θ) = 2ac(1 − a²)(1 − c²) sin θ (1 − 2ac cos θ + a²c²)² ,

equal to zero, reveals that the critical points of g are 0 and π. At these points the value of g is

a−c 1−ac

2

and _1+ac^a+c
2

respectively. Since _1−ac^a−c
2

≤ _1+ac^a+c
2

for all a, c ∈ [0, 1), g(θ) ≤ _1+ac^a+c
2

for all θ ∈ [0, 2π). Hence the triangle inequality for the M¨obius function is satisfied for all points in D.

Since m_D(z, w) = 0 implies |z − w| = 0, that is z = w, the M¨obius pseudodistance is in fact a distance.

One drawback of the M¨obius distance is that equality in the triangle inequality only appears if the point v above is taken to be either z or w. We would like to determine another distance with the same invariance properties as the M¨obius distance, but for which there exist several points v where the triangle inequality is reduced to an equality. This is done by first constructing a suitable metric.

1.4.2. The Poincar´e metric. By performing the following calculation, it is clear that the M¨obius distance locally equals the Euclidean distance multiplied with some location-dependent factor.

lim

w→z,w6=z

m_D(z, w)

|z − w| = lim

w→z,w6=z

z − w 1 − wz

|z − w| = lim

w→z,w6=z

1

|1 − wz| = 1

1 − |z|². (1.1) This factor can be used to define an infinitesimal Finsler pseudometric.

Definition 1.21. The Poincar´e metric at a point (z; ξ) in the tangent bundle of D is defined as β_D(z; ξ) = |ξ|

1 − |z|².

 Obviously β_D(z; λξ) = |λ|β_D(z; ξ). The Poincar´e metric is continuous. Let |z − w| → 0 and

|ξ − ν| → 0. The continuity is then proved by the following calculation β_D(z; ξ) − β_D(w; ν) = |ξ|

1 − |z|² − |ν|

1 − |w|² = |ξ| 1 − |w|²
− |ν| 1 − |z|²
1 − |z|²

1 − |w|²

= 1 − |w|²

|ξ| − |ν|
+ |ν| |z|²− |w|²
1 − |z|²

1 − |w|²
→ 0.

Since β_D(z; ξ) > 0 for all (z; ξ), such that ξ 6= 0, the Poincar´e metric is in fact an infinitesimal Finsler metric.

Using the second part of the Schwarz-Pick lemma, it follows that the functions of H(D, D) are contractions in the Poincar´e metric

β_D f (z); f_∗(z)ξ
=

f⁰(z)ξ  1 −

f (z) 

2 ≤ ξ

1 − |z|² = β_D(z; ξ).

Proposition 1.17 then implies that these functions will be contractions also when measured by the integrated form of the infinitesimal Poincar´e metric. Accordingly the pseudodistance created by the integrated form will be invariant under the functions in Aut(D). It is therefore interesting to investigate the properties of this integrated pseudodistance.

1.4.3. The Poincar distance. In order to calculate the integrated form, let γ be an admissible curve in D joining the origin and the real point r ∈ (0, 1). Let γr denote the real part of γ and γ_r+ the curve max{γ_r, 0}. Then both γ_rand γ_r+ are admissible curve as well. The integral L_β

D(γ) can be calculated in the following way

Lβ_D(γ) = Z b

a

β_D γ(t); γ⁰(t)
dt = Z b

a

γ⁰(t)  1 −

γ(t) 

2dt³ Z b

a

γ_r⁰(t)  1 −

γr(t) 

2dt³ Z b

a

γ⁰_r+(t)  1 −

γr₊(t) 

2dt =

= Z b

a

γ_r⁰₊(t)

1 − γ_r+(t)² =u = γr₊(t), du = γ⁰_r+(t)dt = Z r

0

du

1 − u² = tanh⁻¹(r).

Considering the curve γ(t) = rt, 0 ≤ t ≤ 1, we get L_β

D(γ) = Z 1

0

rdt

1 − r²t² = [u = rt, du = rdt] = Z r

0

du

1 − u² = tanh⁻¹(r),

so β_Dⁱ(0, r) = infLβ_D(γ); γ is an admissible curve between 0 and r = tanh⁻¹(r). For two arbi- trary points z and w in D, the invariance of β_Dⁱ under the automorphisms of D yields

β_Dⁱ(z, w) = βⁱ_D 0, φ_z(w)
= βⁱ_D 0,

φ_z(w) 



= tanh⁻¹ φ_z(w)



= tanh⁻¹ m(z, w)
.

so the integrated form of the Poincar´e metric is the inverse tangens hyperbolicus of the M¨obius distance.

Definition 1.22. For the arbitrary points z, w ∈ D, the Poincar´e distance is defined as ρ_D(z, w) = β_Dⁱ(z, w) = tanh⁻¹

z − w 1 − wz

  .

 Since f (x) = tanh⁻¹(x) is a strictly increasing and well defined function on [0, 1), the non- negativity, finiteness and symmetry of the Poincar´e distance follow directly from the corresponding properties of the M¨obius distance. The triangle inequality, however, poses a more difficult problem.

Due to its construction, the Poincar´e distance obviously inherits the property of being invariant under the elements of Aut(D) from the M¨obius distance. Thus using the same automorphism h as in the proof of the triangle inequality for the M¨obius distance, the triangle inequality of the Poincar´e distance is reduced to

tanh⁻¹  

b − a 1 − ab

≤ tanh⁻¹ |b|
+ tanh⁻¹(a).

Using the fact that tanh⁻¹(x) = ¹₂log(^1+x_1−x), the right-hand side of the inequality can be rewritten as

tanh⁻¹ |b|
+ tanh⁻¹(a) = 1 2

"

log1 + |b|

1 − |b|



+ log1 + a 1 − a



#

= 1

2log(1 + |b|)(1 + a) (1 − |b|)(1 − a)



=

= 1

2log

1 + _1+a|b|^|b|+a 1 − _1+a|b|^|b|+a

!

= tanh⁻¹ |b| + a 1 + a|b|

 .

It is thus necessary to show that tanh⁻¹ _|1−ab|^|b−a|
≤ tanh⁻¹ _1+a|b|^|b|+a
, which, since f (x) = tanh⁻¹(x) is strictly increasing, reduces to proving that _|1−ab|^|b−a| ≤ _1+a|b|^|b|+a. This inequality was, however, validated above when we proved the triangle inequality for the M¨obius distance. Consequently the triangle inequality must hold for the Poincar´e distance as well.

Finally, since the M¨obius function defines a distance, so does the Poincar´e distance.

Investigating the properties of the Poincar´e distance, it is easily seen that the distance between the origin and a point approaching the boundary of the unit disc goes to infinity, so in a way the Poincare distance pushes the boundary of the unit disc infinitely far away. Hence the unit disc equipped with the Poincar´e distance is quite similar to the complex plane equipped with the Euclidean distance. It can also be shown that the unit disc with the Poincar´e distance is a complete metric space ([9], pages 55-56).

The Poincar´e metric and Poincar´e distance are, as we shall see in later chapters, useful tools in the investigation of the unit disc. Two of the many important applications are the fairly elementary proofs of the Little and Great Picard theorems utilizing the Poincar´e metric ([9], page 75-81 and [7], pages 11-12 respectively).

2. Schwarz-Pick systems

The Poincar´e distance defined in the previous section has the suitable property of being distance decreasing under holomorphic mappings. In the following chapters, we will study some other pseudodistances with this property. The studied systems will mainly be of the type defined in the definition below.

Definition 2.1. A system, {σ_Ω}, which assigns a pseudodistance σ_Ωto every domain Ω in Cⁿ is called a Schwarz-Pick system if the following properties hold

(1) the pseudodistance assigned to D is the Poincar´e distance,

(2) if σ1 and σ2are pseudodistances assigned to Ω1and Ω2 respectively and f ∈ H(Ω1, Ω2), then

σ₂ f (z), f (w)
≤ σ1(z, w).

We shall call pseudodistances assigned by a Schwarz-Pick system Schwarz-Pick pseudodistances.

 Before turning to specific Schwarz-Pick systems, it can be interesting to prove that in some domains the distance between points are the same for all Schwarz-Pick systems. The definition above is in these cases strong enough to uniquely determine the distance.

2.1. Determining σ_Dn and σ_Bn for Schwarz-Pick systems. For domains in Cⁿ with a very high symmetry, such as the unit polydisc, Dⁿ, and the unit ball, Bⁿ, it is possible to find explicit formulae for the pseudodistance assigned to the domain by any Schwarz-Pick system. This is done by using holomorphic functions from the domain into the unit disc and vice versa.

Example 2.2. A Schwarz-Pick pseudodistance from the origin to a point z = (z1, . . . , zn) ∈ Dⁿ equals

σ_Dn(0, z) = supρ_D(0, z1), . . . , ρ_D(0, zn) . For a point w = (w₁, . . . , w_n) ∈ Dn, define a norm as kwk =

(w₁, . . . , w_n)

0= sup |w₁|, . . . , |wn|
.

It follows that kwk⁰ < 1 for all elements of Dn. Consider the function ϕ : D → Cⁿ defined as ϕ(λ) = _kzk^λz0. For all λ ∈ D

ϕ(λ)

0= kλzk⁰

kzk⁰ = |λ|kzk⁰

kzk⁰ = |λ| < 1.

so ϕ(λ) is a holomorphic map with image in Dn. The Schwarz-Pick pseudodistances are decreasing under holomorphic mappings and satisfy σ_D= ρ_D. Thus

σ_Dn(0, z) = σ_Dn

ϕ(0), ϕ kzk⁰


≤ σ_D 0, kzk⁰
= ρ_D 0, kzk⁰
=

= supρ_D(0, z1), . . . , ρ_D(0, zn) . (2.1) Now define the functions ψi : Dⁿ → C as ψi(z) = zi for 1 ≤ i ≤ n. These functions are holomorphic in all n dimensions separately, so they are holomorphic maps. The images lie in D, since if kzk⁰ < 1, then by definition of the norm |zi| < 1 for all 1 ≤ i ≤ n. Now using that Schwarz-Pick pseudodistances are decreasing under holomorphic mappings and satisfy σ_D = ρ_D, we get

supρ_D(0, z1), . . . , ρ_D(0, zn)

= supσ_D(0, z1), . . . , ρ_D(0, zn) =

= supn

σ_D ψ1(0), ψ1(z)
, . . . , σ_D ψn(0), ψn(z)
o

≤

≤ σ_Dn(0, z). (2.2)

Combining Equations (2.1) and (2.2), we can conclude that

σ_Dn(0, z) = supρ_D(0, z1), . . . , ρ_D(0, zn) , which is the desired result.

 The elements in the group of automorphisms of the unit polydisc Dⁿ can be derived to have the appearance

φa1,...,an(z1, . . . , zn) = e^iθ1φa1(zσ⁰(1)), . . . , e^iθnφan(zσ⁰(n))
,

where θ1, . . . , θn ∈ [0, 2π), a1, . . . , an ∈ D, the functions φa_i are M¨obius transformations and σ⁰ is a permutation on {1, . . . , n} ([7], page 379). Hereby follows that for every (z1, . . . , zn) ∈ Dⁿ, there exists a mapping φz₁,...,z_n in Aut(Dⁿ) satisfying φz₁,...,z_n(z1, . . . , zn) = 0. Hence the Schwarz-Pick pseudodistance between two arbitrary points z = (z1, . . . , zn) and w = (w1, . . . , wn) in Dⁿ equals

σ_Dn(z, w) = σ_Dn φ_z1,...,zn(z), φ_z1,...,zn(w)
= σ_Dn

0, φ_z1(w₁), . . . , φ_zn(w_n)


=

= supn

ρ_D 0, φz₁(w1)
, . . . , ρ_D 0, φz_n(wn)
o

= supρ_D(z1, w1), . . . , ρ_D(zn, wn) . Example 2.3. A Schwarz-Pick pseudodistance from the origin to a point (z₁, . . . , z_n) ∈ Bn is equal to

σ_Bn(0, z) = ρ_D 0, kzk
.

The calculations will be similar to those used in Example 2.2 but utilizing slightly different holo- morphic functions. Define the function ϕ : D → Cⁿ as ϕ(λ) = _kzk^λz. For all λ ∈ D

ϕ(λ)

= kλzk

kzk = |λ|kzk

kzk = |λ| < 1,

so this is a holomorphic map with image in Bn. The Schwarz-Pick pseudodistances are decreasing under holomorphic mappings and satisfy σ_D= ρ_D. Hence

σ_Bn(0, z) = σ_Bn

ϕ(0), ϕ kzk


≤ σ_D 0, kzk
= ρ_D 0, kzk
. (2.3) For all w = (w₁, . . . , w_n) ∈ Bn define a function ψ : Bn→ C as ψ(w) = ^{w1z¯1+···w}_kzk ^nz¯n. Since ψ(w) is holomorphic in all n dimensions separately, it is a holomorphic map. Using the Cauchy-Schwarz inequality, we get

ψ(w)

 = |w₁z¯₁+ · · · w_nz¯_n|

kzk ≤

Pn

j=1|wjz¯j|

kzk ≤

Pn

j=1|wj|²¹₂ Pn

j=1|¯zj|²¹₂

kzk = kwkk¯zk

kzk =

= kwk < 1.

so the image of ψ(w) lies in D. Schwarz-Pick pseudodistances are decreasing under holomorphic mappings and satisfy σ_D= ρ_D, which gives

ρ_D 0, kzk
= σ_D 0, kzk
= σ_D ψ(0), ψ(z)
≤ σ_Bn(0, z). (2.4) Finally combining Equations (2.3) and (2.4), we can conclude that

σ_Bn(0, z) = ρ_D 0, kzk
, which is the desired result.

 The group of automorphisms of the unit ball acts transitively so it is possible, as in the case of Dn, to explicitly calculate the distance in a Schwarz-Pick system between two arbitrary points in Bn. Since the calculation and result are quite tedious, they have been omitted ([7], page 380).

3. The Carath´eodory pseudodistance and pseudometric

3.1. The Carath´eodory pseudodistance. The regularity of holomorphic mappings shown in the Schwarz and Schwarz-Pick lemmata makes it possible to define pseudodistances by using certain properties of holomorphic mappings. This was first done in 1927 in the paper “ ¨Uber eine spezielle Metrik, die in der Theorie der analytischen Funktionen auftritt” by C. Carath´eodory.

Definition 3.1. For a domain Ω ⊂ Cⁿ, the Carath´eodory pseudodistance is defined as CΩ(z, w) = supn

ρ_D f (z), f (w)
; f ∈ H(Ω, D)o , where z and w are arbitrary points in Ω.

The Poincar´e distance is a non-negative real-valued function and by the definition above so is C_Ω. The property C_Ω(z, w) = C_Ω(w, z) also immediately follows from the symmetry of the Poincar´e distance, while the triangle inequality follows from taking a supremum of the following inequality, which is merely the triangle inequality of the Poincar´e distance. The supremum is taken over the set H(Ω, D).

ρ_D f (z), f (w)
≤ ρ_D f (z), f (v)
+ ρ_D f (v), f (w)
.

Finally we show the finiteness of the Carath´eodory pseudodistance by noting that for every z ∈ Ω there exists a real number r, such that B(z, r) ⊂ Ω. Taking a point x ∈ B(z, r), x 6= z, we can define the function φ : D → Ω as φ(λ) = z + rkx−zk^x−z λ. Then, for every function f ∈ H(Ω, D), f ◦ φ ∈ H(D, D) and, using the definition and properties of the Poincar´e metric, we get

ρ_D f (z), f (x)
= ρ_D



(f ◦ φ)(0), (f ◦ φ)kx − zk r



≤ ρ_D

0,kx − zk r



= tanh⁻¹kx − zk r

 , where the lattest expression is finite, since kx − zk < r. For two arbitrary points z, w ∈ Ω, we can, as Ω is connected, find an r⁰ > 0 and a finite chain {z = x₀, x₁, . . . , x_N = w} of points in Ω, such that x_i+1∈ B(xi, r⁰) for all 0 ≤ i ≤ N − 1. Hence, by the triangle inequality for ρ_D,

ρ_D f (z), f (w)
≤

N

X

i=1

tanh⁻¹kxi+1− xik r⁰



< ∞,

for all f ∈ H(Ω, D). The upper bound of ρD f (z), f (w)
is independent of the choice of holomorphic function f and thus unaffected by taking a supremum over H(Ω, D). It follows that C^Ω(z, w) is finite for all z, w ∈ Ω and Definition 3.1 gives a pseudodistance.

The Carath´eodory pseudodistance can be degenerate and is therefore in general not a dis- tance. In C for example Liouville’s theorem states that all functions f in H(C, D) are constant so ρ_D(f (z), f (w)) = 0 for all z, w ∈ C. Consequently CC(z, w) = 0 for all choices of z and w in C. This reasoning is expanded to more general domains in the next proposition, where we have denoted the set of bounded holomorphic functions on a domain Ω by H^∞(Ω).

Proposition 3.2. For a domain Ω ⊂ Cⁿ, the following holds for the Carath´eodory pseudodistance on Ω,

(1) CΩ≡ 0 if and only if H^∞(Ω) only consists of constant functions,

(2) CΩis a distance if and only if H^∞(Ω) separates points in Ω, that is for any two points z 6= w in Ω there exists an f ∈ H^∞(Ω) such that f (z) 6= f (w).

Proof: (1) Assume that H^∞(Ω) only consists of constant functions. Since ρ_D is a distance, ρ_D f (z), f (w)
= 0 for any two points z, w ∈ Ω and f ∈ H(Ω, D). Consequently C^Ω≡ 0.

If conversely C_Ω ≡ 0, then ρ_D f (z), f (w)
= 0 for all holomorphic functions f : Ω → D and points z, w ∈ Ω. Since ρ_Dis a distance, f (z) = f (w) for these points and functions. By a rescaling we can conclude that all functions in H^∞(Ω) must be constant.

(2) Let z 6= w in Ω be two arbitrary points. If H^∞(Ω) separates points in Ω, there exists a function g ∈ H^∞(Ω) such that g(z) 6= g(w) and by a rescaling also a function f : Ω → D such that