H. BOHR’S THEOREM FOR BOUNDED SYMMETRIC DOMAINS

GUY ROOS

Abstract. A theorem of H. Bohr (1914) states that if f(z) = ∑_{k=0}^{∞} a_{k}z^{k} is a holomorphic map from the unit disc D ⊂ ℂ into itself, then ∑_{k=0}^{∞} |a_{k}z^{k}| ≤ 1 for |z| ≤ \frac{1}{3} ; the value \frac{1}{3} is optimal. This result has been extended by Liu Taishun and Wang Jianfei (2007) to the bounded symmetric domains of the four classical series, and to polydiscs. The result of Liu and Wang may be generalized to all bounded symmetric domains, with a proof which does not depend on classification.

Introduction

The following theorem was proved by Harald Bohr [Bohr 1914] for |z| < \frac{1}{6}, then soon improved to |z| < \frac{1}{3} from remarks by M. Riesz, I. Schur, F. Wiener^{1}:

Theorem 1. Let f : Δ → Δ be a holomorphic function from the unit disc Δ ⊂ ℂ into itself. If f(z) = ∑_{k=0}^{∞} a_{k}z^{k}, then

(0.1)  ∑_{k=0}^{∞} |a_{k}z^{k}| < 1

for |z| < \frac{1}{3}. The value \frac{1}{3} is optimal.

This result has been extended by Liu Taishun and Wang Jianfei [Liu–Wang 2007] to the bounded symmetric domains of the four classical series, and to polydiscs, using a case-by-case analysis, as follows:

Theorem 2. Let Ω be an irreducible bounded symmetric domain of classical type in the sense of Hua Luokeng [Hua 1963], or a polydisc. Denote by ∥ ∥_{Ω} the Minkowski norm associated to Ω. Let f : Ω → Ω be a holomorphic map and let

f(z) = ∑_{k=0}^{∞} f_{k}(z)

be its Taylor expansion in k-homogeneous polynomials f_{k}. Let ϕ ∈ Aut Ω such that ϕ(f(0)) = 0. Then

(0.2)  ∑_{k=0}^{∞} \begin{matrix}∥Dϕ(f(0)) · f_{k}(Z)∥ & Ω \\ ∥Dϕ(f(0))∥ & Ω \end{matrix} < 1

for all Z such that ∥Z∥_{Ω} < \frac{1}{3}.

Date: April 9, 2009. fused with Norbert Wiener.

2000 Mathematics Subject Classification. Primary 32A05. Secondary 30B10, 32M15, 17C40.

1See [Boas–Khavinson 2000] for biographical elements about Friedrich Wiener, not to be con-

For ∥Z∥_{Ω} > \frac{1}{3}, there exists a holomorphic map f : Ω → Ω such that (0.2) is not true.

Actually, the proof in [Liu–Wang 2007] depends on the following result, which is proved by the authors for classical domains of type I and IV, using ad hoc computations:

Theorem 3. Let Ω ⊂ V be a bounded circled symmetric domain. Let ∥ ∥_{Ω} be the associated spectral norm on V. Let u ∈ Ω and let ϕ ∈ Aut Ω such that ϕ(u) = 0. Then the operator norm of the derivative of ϕ at u is

∥d ϕ(u)∥_{Ω} = \frac{1}{1 − ∥u∥_{Ω}^{2}} .

In Section 2, we give a classification independent proof of this result and of Theorem 2, which is valid for any bounded circled symmetric domain.

In the above generalization of Bohr’s theorem, one considers the Taylor expansion

f(z) = ∑_{k=0}^{∞} f_{k}(z)

of a bounded holomorphic function or map, and one asks for which z the inequality (0.2) holds. One may ask the same type of question for other decompositions of f. For example, the following problem has been considered recently by several authors:

Problem 1. Let Ω be the unit ball of the ℓ^{p} norm:

Ω = {(z_{1}, ⋯, z_{n}) ∈ ℂ^{n} | ∑_{k=1}^{n} |z_{k}|^{p} < 1} .

Let f : Ω → Δ be a holomorphic function and consider the expansion of f in monomials

f(z) = ∑_{k_{1},...,k_{n}=0}^{∞} a_{k_{1}···k_{n}}z_{1}^{k_{1}} … z_{n}^{k_{n}}.

Determine the best constant K such that z ∈ KΩ ensures

∑_{k_{1},...,k_{n}=0}^{∞} |a_{k_{1}···k_{n}}z_{1}^{k_{1}} … z_{n}^{k_{n}} | < 1

for all f : Ω → Δ.

For results about this type of problem, see [Dineen–Timoney 1989], [Dineen–Timoney 1991], [Boas–Khavinson 1997], [Aizenberg 2000], [Boas 2000], [Defant et al. 2003]

1. H. Bohr’s theorem

1.1. We first recall Bohr’s theorem in dimension 1. We present here a rather elementary proof of this theorem, based on an inequality of Carathéodory or on an improvement of this inequality using a lemma of F. Wiener. This lemma has been generalized in [Liu–Wang 2007] to some classes of bounded symmetric domains. There are other proofs of Bohr’s theorem in the literature, for example by S. Sidon [Sidon 1927], following L. Fejér’s method of positive kernels [Fejér 1925]; the same proof was published later by M. Tomić [Tomić 1962], who added the case a_{0} = f(0) = 0 (see Section 1.2).

Theorem 4 ([Bohr 1914]). Let f : Δ → Δ be a holomorphic function from the unit disc Δ ⊂ ℂ into itself. If f(z) = ∑_{k=0}^{∞} a_{k}z^{k}, then

(1.1)  ∑_{k=0}^{∞} |a_{k}z^{k}| ≤ 1

for |z| ≤ \frac{1}{3}. The value \frac{1}{3} is optimal: for \frac{1}{3} < r < 1, there exists a holomorphic function f : D → D such that

∑_{k=0}^{∞} |a_{k}| r^{k} > 1.

For a holomorphic function f : Δ → Δ with Taylor expansion f(z) = ∑_{k=0}^{∞} a_{k}z^{k}, denote by 𝔐_{f}(r) the sum of absolute values:

𝔐_{f}(r) = ∑_{k=0}^{∞} |a_{k}| r^{k}.

Let us first prove that the bound \frac{1}{3} cannot be improved. For 0 < α < 1, let f be defined by

f(z) = \frac{α − z}{1 − αz} .

Then

f(z) = α + ∑_{k=1}^{∞} (α^{k+1} − α^{k−1}) z^{k},

so that f(z) = ∑_{k=0}^{∞} a_{k}z^{k}, with a_{0} = α, a_{k} = α^{k+1} − α^{k−1}. As a_{k} is negative for k > 0,

𝔐(r) = α − ∑_{k=1}^{∞} (α^{k+1} − α^{k−1}) r^{k} = 2α − f(r).

Then 𝔐(r) > 1 when r > \frac{1}{1+2α}; as \frac{1}{1+2α} → \frac{1}{3} + 0 when α → 1 − 0, there exists for each r > \frac{1}{3} a holomorphic function f : Δ → Δ such that 𝔐_{f}(r) > 1.

To prove the direct part of Theorem 4, one may use one of the following lemmas.

Lemma 1. Let f : Δ → Δ be a holomorphic function, with f(z) = ∑_{k=0}^{∞} a_{k}z^{k}. Then for all k > 0,

(1.2)  |a_{k}| ≤ 2(1 − |a_{0}|).

Lemma 2. Let f : Δ → Δ be a holomorphic function, with f(z) = ∑_{k=0}^{∞} a_{k}z^{k}. Then for all k > 0,

(1.3)  |a_{k}| ≤ 1 − |a_{0}|^{2} .

The inequalities (1.2) result directly from inequalities, due to C. Carathéodory, for holomorphic functions in Δ with positive real part; they can also be used to prove Bohr’s theorem. Note that (1.3) is slightly sharper than (1.2), as (1 − t^{2}) ≤ 2(1 − t) for all t ∈ ℝ. We will prove directly the second lemma.

First we prove the special case:

(1.4)  |a_{1}| ≤ 1 − |a_{0}|^{2} .

Consider the automorphism ϕ of Δ defined by

ϕ(z) = \frac{z − a_{0}}{1 − \overline{a_{0}}z};

we have

ϕ(a_{0}) = 0,  ϕ′(a_{0}) = \frac{1}{1 − a_{0}\overline{a_{0}}} ,
(ϕ ∘ f)′ (0) = ϕ′(a_{0})f′(0) = \frac{a_{1}}{1 − a_{0}\overline{a_{0}}} .

As ϕ∘f maps Δ into Δ and (ϕ∘f)(0) = 0, we have |(ϕ ∘ f)′ (0)| ≤ 1, which implies

|a_{1}| ≤ 1 − a_{0}\overline{a_{0}}.

From (1.4), we will now deduce (1.2) for all k > 0. For k > 0, define

g_{k}(z) = \frac{1}{k} ∑_{j=1}^{k} f(e^{2 i πj/k} z) = ∑_{m=0}^{∞} a_{km}z^{km}.

From the first equality, it results that g_{k} maps Δ into Δ, and so does also h_{k} defined by

h_{k}(z) = ∑_{m=0}^{∞} a_{km}z^{m}.

We have h_{k}(0) = a_{0} and h′_{k}(0) = a_{k} ; applying the inequality (1.4) to the function h_{k} yields

|a_{k}| ≤ 1 − a_{0}\overline{a_{0}}.

Proof of Bohr’s theorem. Using (1.3), we deduce

𝔐(r) = ∑_{k=0}^{∞} |a_{k}| r^{k} ≤ |a_{0}| + (1 − a_{0}\overline{a_{0}}) \frac{r}{1 − r}

and, for r ≤ \frac{1}{3},

𝔐(r) ≤ |a_{0}| + \frac{1}{2} (1 − a_{0}\overline{a_{0}}) = 1 − \frac{1}{2} (1 − |a_{0}|^{2}) < 1,

as |a_{0}| = |f(0)| < 1. □

1.2. Another Bohr type theorem holds in dimension 1 for the class of holomorphic functions f : Δ → Δ such that f(0) = 0. It was proved in [Tomić 1962] using an analogue of the argument in [Sidon 1927]. An elementary proof for this, which uses another result of L. Fejér [Fejér 1914], can also be found in [Landau 1925].

This type of problem was generalized and partially solved in [Ricci 1955] and [Bombieri 1962] by G. Ricci and E. Bombieri.

Definition 1. Consider the following classes of analytic functions:

F_{0} = {f : Δ → Δ} ,
F_{0,α} = {f : Δ → Δ | f(0) = α}  (0 ≤ α < 1),
F_{m} = {f : Δ → Δ | f(z) = ∑_{k=m}^{∞} a_{k}z^{k}}  (m ∈ ℕ),

F_{m,α} = {f : Δ → Δ | f(z) = αz^{m} + ∑_{k=m+1}^{∞} a_{k}z^{k}}  (m ∈ ℕ, 0 ≤ α < 1).

Define the corresponding Bohr numbers by

B_{0} = sup {r | 𝔐_{f}(r) < 1 for all f ∈ F_{0}} ,

B_{0,α} = sup {r | 𝔐_{f}(r) < 1 for all f ∈ F_{0,α}} ,
B_{m} = sup {r | 𝔐_{f}(r) < 1 for all f ∈ F_{m}} ,
B_{m,α} = sup {r | 𝔐_{f}(r) < 1 for all f ∈ F_{m,α}} .

The original Bohr theorem means that B_{0} = \frac{1}{3}.

The result in [Tomić 1962] (already proved in [Landau 1925]) says that B_{1} > \frac{1}{2}. In [Ricci 1955], the author proves, among other results, that \frac{3}{5} < B_{1} ≤ \frac{1}{\sqrt{2}}. The optimal result B_{1} = \frac{1}{\sqrt{2}} was obtained by [Bombieri 1962].

Let us prove that Ricci’s result B_{1} > \frac{3}{5} easily follows from Wiener’s lemma 2.

Proposition 1. Let f : Δ → Δ be a holomorphic function such that f(0) = 0, f(z) = ∑_{k=1}^{∞} a_{k}z^{k}. Then

∑_{k=0}^{∞} |a_{k}z^{k}| ≤ 1

for |z| ≤ \frac{3}{5}.

Proof. Consider the function

g(z) = \frac{f(z)}{z} = ∑_{k=0}^{∞} a_{k+1}z^{k}.

By the Schwarz lemma, g maps Δ into Δ. Lemma 2 applied to g gives

|a_{k}| ≤ 1 − |a_{1}|^{2} (k > 1).

Hence

𝔐_{f}(r) = ∑_{k=0}^{∞} |a_{k}| r^{k} ≤ |a_{1}| r + (1 − |a_{1}|^{2}) \frac{r^{2}}{1 − r} ,

𝔐_{f} (\frac{3}{5}) ≤ \frac{3}{5} (|a_{1}| + \frac{3}{2} (1 − |a_{1}|^{2})) = 1 − \frac{9}{10} (\frac{1}{3} − |a_{1}|)^{2} ≤ 1.
□

2. H. Bohr’s theorem for bounded symmetric domains

2.1. Let Ω ⊂ V be a bounded circled homogeneous domain in a finite dimensional complex vector space V. See the appendix for notations and general results. Denote by ∥ ∥_{Ω} the spectral norm associated to Ω.

The main result in [Liu–Wang 2007] is the following theorem, which is proved there for domains of type I (rectangular matrices) and for polydiscs.

Theorem 5. Let Ω be a bounded circled symmetric domain. Denote by ∥ ∥_{Ω} the spectral norm associated to Ω. Let f : Ω → Ω be a holomorphic map and let

f = ∑_{k=0}^{∞} f_{k}

be its Taylor expansion in k-homogeneous polynomials f_{k}. Let ϕ ∈ Aut Ω such that ϕ(f(0)) = 0. Then

(2.1)  ∑_{k=0}^{∞} \begin{matrix}∥d ϕ(f(0)) · f_{k}(z)∥ & Ω \\ ∥d ϕ(f(0))∥ & Ω \end{matrix} < 1

for all z such that ∥z∥_{Ω} ≤ \frac{1}{3}.

The bound \frac{1}{3} is optimal: for each z ∈ Ω with ∥z∥_{Ω} > \frac{1}{3}, there exists a holomorphic map f = ∑_{k=0}^{∞} f_{k} : Ω → Ω such that

∑_{k=0}^{∞} \begin{matrix}∥d ϕ(f(0)) · f_{k}(z)∥ & Ω \\ ∥d ϕ(f(0))∥ & Ω \end{matrix} > 1.

This theorem will be proved below for all bounded symmetric domains.

2.2. The following properties generalize to all bounded circled symmetric domains results in [Liu–Wang 2007] for the differential at u ∈ Ω of an automorphism ϕ ∈ Aut Ω such that ϕ(u) = 0, and may be of independent interest.

For u ∈ V, denote by τ_{u} the translation z ↦ z + u and by τ̃_{u} the rational map

τ̃_{u}(z) = z^{u},

where z^{u} is the quasi-inverse

z^{u} = B(z, u)^{−1} (z − Q(z)u) ,

defined in the open set of points z such that B(z, u) = id_{V} −D(z, u) + Q(x)Q(u) is invertible.

Recall that for u ∈ Ω, the operator B(u, u) is positive (with respect to the Hermitian scalar product on V: (x | y) = tr D(x, y)), so that B(u, u)^{t} is well defined for t ∈ ℝ. Let u ∈ Ω and consider a spectral decomposition

u = λ_{1}e_{1} + ⋯ + λ_{r}e_{r},
1 > λ_{1} ≥ ⋯ ≥ λ_{r} ≥ 0,

where e = (e_{1}, …, e_{r}) is a frame. Then, by (A11),

B(u, u) = ∑_{0≤i≤j≤r} (1 − λ_{i}^{2}) (1 − λ_{j}^{2}) p_{ij}

and B(u, u)^{t} is given by

(2.2)  B(u, u)^{t} = ∑_{0≤i≤j≤r} (1 − λ_{i}^{2})^{t} (1 − λ_{j}^{2})^{t} p_{ij},

where (p_{ij}) is the family of orthogonal projectors onto the subspaces of the simultaneous Peirce decomposition with respect to the frame e = (e_{1}, …, e_{r}).

The following result is well known.

Lemma 3. For u ∈ Ω, the map

(2.3)  ϕ_{u} = τ̃_{u} ∘ B(u, u)^{−1/2} ∘ τ_{−u}

is an automorphism of Ω which sends u to 0. The derivative of ϕ_{u} at u is

(2.4)  d ϕ_{u}(u) = B(u, u)^{−1/2}.

Proof. Let

ψ_{u} = τ_{u} ∘ B(u, u)^{1/2} ∘ τ̃_{−u}.

Then, by [Loos 1977, Proposition 9.8 (1)], ψ_{u} is an automorphism of Ω. As τ̃_{−u}(0) = 0 and B(u, u)^{1/2} is linear, we have ψ_{u}(0) = u. The inverse of ψ_{u} is

ϕ_{u} = τ̃_{u} ∘ B(u, u)^{−1/2} ∘ τ_{−u}

and ϕ_{u}(u) = 0. The derivative of τ̃_{u} is

d τ̃_{u}(z) = B(z, u)^{−1}

(see [Roos 2000, Proposition III.4.1 (ii)]). As B(0, u) = id_{V}, it follows that

d ϕ_{u}(u) = B(u, u)^{−1/2}.
□

Denote by ∥ ∥ the Hermitian norm on V (∥z∥^{2} = tr D(z, z)) and by ∥ ∥ the associated operator norm for linear endomorphisms of V. The spectral norm ∥ ∥_{Ω} on V is given by

∥z∥_{Ω}^{2} = \frac{1}{2} ∥D(z, z)∥ = ∥Q(z)∥;

the unit ball of this norm is Ω (see the appendix). We denote also by ∥ ∥_{Ω} the associated operator norm for linear endomorphisms of V.

Lemma 4. For u ∈ Ω, t ∈ ℝ,

(2.5)  ∥B(u, u)^{t}∥_{Ω} = ∥B(u, u)^{t}∥ .

That is, the operator B(u, u)^{t} has the same operator norm with respect to the spectral norm or to the Hermitian norm.

Proof. Let u ∈ Ω and consider a spectral decomposition

u = λ_{1}e_{1} + ⋯ + λ_{r}e_{r},
1 > λ_{1} ≥ ⋯ ≥ λ_{r} ≥ 0,

where e = (e_{1}, …, e_{r}) is a frame. Then, by (A11),

B(u, u) = ∑_{0≤i≤j≤r} (1 − λ_{i}^{2}) (1 − λ_{j}^{2}) p_{ij}

and

B(u, u)^{t} = ∑_{0≤i≤j≤r} (1 − λ_{i}^{2})^{t} (1 − λ_{j}^{2})^{t} p_{ij},

where (p_{ij}) is the family of orthogonal projectors onto the subspaces of the simultaneous Peirce decomposition with respect to the frame e = (e_{1}, …, e_{r}).

For t ≥ 0, let μ_{j} ≥ 0 be defined by

(1 − λ_{i}^{2})^{t} = 1 − μ_{j}^{2}

and define

w = μ_{1}e_{1} + ⋯ + μ_{r}e_{r}.

Then, again by (A11),

B(u, u)^{t} = B(w, w).

Using (A5), we get for t ≥ 0 and all z ∈ V

(2.6)  Q(B(u, u)^{t}z) = B(u, u)^{t}Q(z)B(u, u)^{t}.

This implies

(2.7)  Q(B(u, u)^{−t}z) = B(u, u)^{−t}Q(z)B(u, u)^{−t},

so that (2.6) is finally valid for all t ∈ ℝ.

Now, by (A12), for t ∈ ℝ, z ∈ V,

∥B(u, u)^{t}z∥_{Ω}^{2} = ∥Q(B(u, u)^{t}z)∥ ≤ ∥B(u, u)^{t}∥^{2} ∥Q(z)∥ = ∥B(u, u)^{t}∥^{2} ∥z∥_{Ω}^{2} .

This implies

∥B(u, u)^{t}∥_{Ω} ≤ ∥B(u, u)^{t}∥ .

Recall that

B(u, u)^{t} = ∑_{0≤i≤j≤r} (1 − λ_{i}^{2})^{t} (1 − λ_{j}^{2})^{t} p_{ij},

where the p_{ij}’s are orthogonal projections with respect to the Hermitian metric. Then ∥B(u, u)^{t}∥ is the greatest eigenvalue of B(u, u)^{t}, and ∥B(u, u)^{t}∥_{Ω} ≥ ∥B(u, u)^{t}∥ . □

For u ∈ Ω with spectral decomposition u = λ_{1}e_{1} + ⋯ + λ_{r}e_{r}, 1 > λ_{1} ≥ ⋯ ≥ λ_{r} ≥ 0, denote by β(u) the greatest eigenvalue of B(u, u).

Lemma 5. Let Ω be a bounded irreducible symmetric domain with invariants a, b, r. For u ∈ Ω with spectral decomposition u = λ_{1}e_{1} + ⋯ + λ_{r}e_{r}, 1 > λ_{1} ≥ ⋯ ≥ λ_{r} ≥ 0, the value of β(u) is

(1) if Ω is the unit disc of ℂ (b = 0, r = 1),

β(u) = (1 − ∥u∥_{Ω}^{2})^{2} ;

(2) if Ω is the Hermitian ball of dimension n > 1 (b = n − 1, r = 1),

β(u) = 1 − ∥u∥_{Ω}^{2} ;

(3) if Ω is of tube type (b = 0),

β(u) = (1 − λ_{r}^{2})^{2} ;

(4) if Ω is of non tube type (b > 0),

β(u) = 1 − λ_{r}^{2}.

Proof. The lemma follows from the fact that the eigenvalues of B(u, u) are: (1−λ_{i}^{2})^{2} (1 ≤ i ≤ r), (1 − λ_{i}^{2})(1 − λ_{j}^{2}) (1 ≤ i < j ≤ r) and only in the non tube case 1 − λ_{i}^{2} (1 ≤ i ≤ r). □

Note that λ_{1} = λ_{r} occurs if and only if u is a scalar multiple of a maximal tripotent (an element of the Shilov boundary of Ω). In this case, one has β(u) =

(1 − ∥u∥_{Ω}^{2})^{2} or β(u) = 1 − ∥u∥_{Ω}^{2}, depending on whether the domain Ω is of tube type or not.

Proposition 2. Let Ω be a bounded circled symmetric domain. For u ∈ Ω we have

(2.8)  ∥B(u, u)^{−1/2}∥_{Ω} = \frac{1}{1 − ∥u∥_{Ω}^{2}} ,
(2.9)  ∥B(u, u)^{−1/2}u∥_{Ω} = \frac{∥u∥_{Ω}}{1 − ∥u∥_{Ω}^{2}} ,
(2.10)  ∥B(u, u)^{1/2}∥_{Ω} = β(u)^{1/2}.

For any automorphism ϕ ∈ Aut Ω such that ϕ(u) = 0,

(2.11)  ∥d ϕ(u)∥_{Ω} = \frac{1}{1 − ∥u∥_{Ω}^{2}} ,
(2.12)  ∥d ϕ(u)u∥_{Ω} = \frac{∥u∥_{Ω}}{1 − ∥u∥_{Ω}^{2}}

(2.13)  ∥(d ϕ(u))^{−1}∥_{Ω} = β(u)^{1/2}.

Proof. Let u = λ_{1}e_{1}+⋯+λ_{r}e_{r}, 1 > λ_{1} ≥ ⋯ ≥ λ_{r} ≥ 0 be a spectral decomposition of u. The operator norm ∥B(u, u)^{−1/2}∥ with respect to the Hermitian product is

its greatest eigenvalue (1 − λ_{1}^{2})^{−1} = (1 − ∥u∥_{Ω}^{2})^{−1}; then Lemma 4 implies (2.8). In the same way, Lemma 4 implies (2.10). We have

B(u, u)^{−1/2}u = ∑_{i=1}^{r} \frac{λ_{i}}{(1 − λ_{i}^{2})^{1/2}} e_{i},

hence

∥B(u, u)^{−1/2}u∥_{Ω} = \frac{λ_{1}}{(1 − λ_{1}^{2})^{1/2}}

and (2.9).

Let ϕ be any automorphism ϕ ∈ Aut Ω such that ϕ(u) = 0. Then ϕ = k ∘ ϕ_{u}, where k is a linear automorphism of Ω; the spectral norm is invariant by k and

∥d ϕ(u)∥_{Ω} = ∥k ∘ B(u, u)^{−1/2}∥_{Ω} = ∥B(u, u)^{−1/2}∥_{Ω} ,
∥d ϕ(u)u∥_{Ω} = ∥k ∘ B(u, u)^{−1/2}u∥_{Ω} = ∥B(u, u)^{−1/2}u∥_{Ω} ,
∥(d ϕ(u))^{−1}∥_{Ω} = ∥B(u, u)^{1/2}∥_{Ω} .

□

2.3. Let Ω be a bounded circled symmetric domain. If f : Ω → Ω is a holomorphic map, denote by

f(z) = ∑_{k=0}^{∞} f_{k}(z)

its Taylor expansion in k-homogeneous polynomials f_{k}. The following lemma generalizes Lemma 2 to bounded symmetric domains.

Lemma 6. Let u = f(0) and let ϕ ∈ Aut Ω such that ϕ(u) = 0. Then

(2.14)  ∥d ϕ(u) · f_{k}(z)∥_{Ω} ≤ ∥z∥_{Ω}^{k} .

Proof. For k > 1, consider g_{k} : Ω → Ω defined by

g_{k}(z) = \frac{1}{k} ∑_{j=1}^{k} f (e^{2 i πj/k} z) = ∑_{m=0}^{∞} f_{mk}(z).

Let h_{k} = ϕ ∘ g_{k}. Then

h_{k}(z) = ϕ ( ∑_{m=0}^{∞} f_{mk}(z))

= ϕ(u) + d ϕ(u) · ( ∑_{m=1}^{∞} f_{mk}(z)) + ⋯ = d ϕ(u) · f_{k}(z) + ⋯

and, for z ∈ Ω,

d ϕ(u) · f_{k}(z) = ∫_{0}^{1} ϕ ∘ g_{k} (e^{2 i πt} z) e^{−2 i πkt} d t.

This implies

∥d ϕ(u) · f_{k}(z)∥_{Ω} ≤ 1

for z ∈ Ω, and by homogeneity of f_{k},

∥d ϕ(u) · f_{k}(z)∥_{Ω} ≤ ∥z∥_{Ω}^{k}

for z ∈ V. □

Proposition 3. Let Ω be a bounded circled symmetric domain. Let u ∈ Ω and let ϕ ∈ Aut Ω such that ϕ(u) = 0.

Then

(2.15)  ∑_{k=0}^{∞} \begin{matrix}∥d ϕ(u) · f_{k}(z)∥ & Ω \\ ∥d ϕ(u)∥ & Ω \end{matrix} < 1

for all f : Ω → Ω such that f(0) = u and for all z such that ∥z∥_{Ω} ≤ \frac{1}{2+∥u∥_{Ω}} .

Proof. The proof goes along the same lines as in [Liu–Wang 2007]. For z ∈ V we have by Lemma 6

∑_{k=0}^{∞} \begin{matrix}∥d ϕ(f(0)) · f_{k}(z)∥ & Ω \\ ∥d ϕ(f(0))∥ & Ω \end{matrix} ≤ ∥u∥_{Ω} + \frac{1}{∥d ϕ(u)∥_{Ω}} ∑_{k=1}^{∞} ∥z∥_{Ω}^{k} .

Using ∥d ϕ(u)∥_{Ω} = \frac{1}{1−∥u∥_{Ω}^{2}} from Proposition 2, we get for ∥z∥_{Ω} ≤ r < 1

∑_{k=0}^{∞} \begin{matrix}∥d ϕ(f(0)) · f_{k}(z)∥ & Ω \\ ∥d ϕ(f(0))∥ & Ω \end{matrix} ≤ ∥u∥_{Ω} + (1 − ∥u∥_{Ω}^{2}) \frac{r}{1 − r} .

For fixed u, the right hand side is less than 1 if and only if r < \frac{1}{2+∥u∥_{Ω}} . □

Proposition 4. Let Ω be a bounded circled symmetric domain. Let u ∈ Ω and let ϕ ∈ Aut Ω such that ϕ(u) = 0. Assume ∥u∥_{Ω} > \frac{1}{3} and \frac{1}{1+2∥u∥_{Ω}} < a < 1.

Then there exist a holomorphic map f : Ω → Ω with f(0) = u and z ∈ Ω with ∥z∥_{Ω} = a, such that

∑_{k=0}^{∞} \begin{matrix}∥d ϕ(u) · f_{k}(z)∥ & Ω \\ ∥d ϕ(u)∥ & Ω \end{matrix} > 1.

Proof. Consider u ∈ Ω with spectral decomposition

u = λ_{1}e_{1} + ⋯ + λ_{r}e_{r},
1 > ∥u∥_{Ω} = λ_{1} ≥ ⋯ ≥ λ_{r} ≥ 0.

Define f : Ω → Ω by

f(z) = ∑_{i=1}^{r} \frac{λ_{i} − (z | e_{i})}{1 − λ_{i} (z | e_{i})}e_{i}.

This is a well defined holomorphic map, as |(z | e_{i})| < 1 for all z ∈ Ω, due to the convexity of Ω. From

\frac{λ − ζ}{1 − λζ} = λ + ∑_{k=1}^{∞} (λ^{k+1} − λ^{k−1}) ζ^{k},

H. BOHR’S THEOREM 11

we get the Taylor expansion of f:

f = ∑_{k=0}^{∞} f_{k},
u = f(0) = f_{0}(z),
f_{k}(z) = ∑_{i=1}^{r} (λ_{i}^{k+1} − λ_{i}^{k−1}) (z | e_{i})^{k} e_{i}  (k ≥ 1).

In particular, taking z_{0} = ae_{1}, we have

f_{k}(z_{0}) = (λ_{1}^{k+1} − λ_{1}^{k−1}) a^{k}e_{1} (k ≥ 1).

Take ϕ_{u} as in Lemma 3. Then

d ϕ_{u}(u) = B(u, u)^{−1/2},
∥d ϕ_{u}(u)∥_{Ω} = \frac{1}{1 − ∥u∥_{Ω}^{2}} .

From (2.2) we get

B(u, u)^{−1/2}e_{1} = (1 − ∥u∥_{Ω}^{2})^{−1}e_{1},
∥d ϕ_{u}(u) · u∥ = (1 − ∥u∥_{Ω}^{2})^{−1} ∥u∥_{Ω} ,
d ϕ_{u}(u) · f_{k}(z_{0}) = −a^{k}λ_{1}^{k−1}e_{1},
∥d ϕ_{u}(u) · f_{k}(z_{0})∥_{Ω} = a^{k}λ_{1}^{k−1} = a^{k} ∥u∥_{Ω}^{k−1} .

Hence

∑_{k=0}^{∞} \begin{matrix}∥d ϕ(u) · f_{k}(z_{0})∥ & Ω \\ ∥d ϕ(u)∥ & Ω \end{matrix} = ∥u∥_{Ω} + (1 − ∥u∥_{Ω}^{2}) ∑_{k=1}^{∞} a^{k} ∥u∥_{Ω}^{k−1}

= ∥u∥_{Ω} + (1 − ∥u∥_{Ω}^{2})\frac{a}{1 − a ∥u∥_{Ω}} .

An elementary computation shows that this is greater than 1 if and only if

a > \frac{1}{1 + 2 ∥u∥_{Ω}} .

□

Proof of Theorem 5. Proposition 3 shows that (2.15) is satisfied for all maps f : Ω → Ω and all z such that ∥z∥_{Ω} < \frac{1}{3}. Moreover, for ∥z∥_{Ω} = \frac{1}{3},we have

∑_{k=0}^{∞} \begin{matrix}∥d ϕ(f(0)) · f_{k}(z)∥ & Ω \\ ∥d ϕ(f(0))∥ & Ω \end{matrix} ≤ ∥f(0)∥_{Ω} + (1 − ∥f(0)∥_{Ω}^{2}) ∑_{k=1}^{∞} \frac{1}{3^{k}}

= ∥f(0)∥_{Ω} + \frac{1 − ∥f(0)∥_{Ω}^{2}}{2} = 1 − \frac{(1 − ∥f(0)∥_{Ω})^{2}}{2} < 1.

As \frac{1}{1+2∥u∥_{Ω}} → \frac{1}{3} as ∥u∥_{Ω} → 1 − 0, proposition 4 implies that \frac{1}{3} is the optimal bound. □

2.4. Open questions.

Problem 2. Let f : Ω → Ω be a holomorphic map.

(1) With the assumption f(0) = 0, Theorem 5 gives

(2.16)  ∑_{k=0}^{∞} ∥f_{k}(z)∥_{Ω} < 1

for all z such that ∥z∥_{Ω} < \frac{1}{2}. Is the optimal bound equal to \frac{1}{\sqrt{2}}, as proved by E. Bombieri [Bombieri 1962] in the one dimensional case?

(2) What is the same optimal bound for all maps f satisfying f(0) = u, with u ∈ Ω fixed? Propositions 3 and 4 show that this optimal bound belongs to [\frac{1}{2+∥u∥_{Ω}} , \frac{1}{1+2∥u∥_{Ω}} ]. In the one dimensional case, this is Ricci’s estimate (see [Ricci 1955]). But Bombieri’s results for the one dimensional case show that this estimate may be sharpened: in particular, the optimal bound is \frac{1}{\sqrt{2}} when u = 0, \frac{1}{1+2∥u∥_{Ω}} when \frac{1}{2} < ∥u∥_{Ω} < 1.

For Ω a bounded circled symmetric domain, a natural problem generalizing H. Bohr’s problem would be the following.

Problem 3. Let Ω ⊂ V be a bounded circled symmetric domain of rank r. Let f : Ω → Ω be a holomorphic map and consider the Schmidt decomposition of its Taylor expansion

f(z) = ∑_{k_{1}≥...k_{r}≥0} f_{k_{1}···k_{r}}(z)

(where the f_{k_{1}···k_{r}}’s are polynomials in the irreducible K-modules for the linear subgroup K of Aut Ω). Determine the best constant C such that z ∈ CΩ ensures

∑_{k_{1}≥...k_{r}≥0} ∥f_{k_{1}···k_{r}}(z)∥_{Ω} < 1

for all holomorphic maps f : Ω → Ω.

Appendix

We recall here some notations and results about complex bounded symmetric domains and their associated Jordan triple structure (see [Loos 1977], [Roos 2000]).

Bounded symmetric domains and Jordan triples. Let Ω be an irreducible bounded circled homogeneous domain in a complex vector space V. This circled realization is unique up to a linear isomorphism. Let K be the identity component of the (compact) Lie group of (linear) automorphisms of Ω leaving 0 fixed. Let ω be a volume form on V, invariant by K and by translations. Let K be the Bergman kernel of Ω with respect to ω, that is, the reproducing kernel of the Hilbert space H^{2}(Ω, ω) = Hol(Ω) ∩ L^{2}(Ω, ω). The Bergman metric at z ∈ Ω is defined by

h_{z}(u, v) = ∂_{u}\overline{∂}_{v} log K(z).

The Jordan triple product on V is characterized by

h_{0}(\{uvw\}, t) = ∂_{u}\overline{∂}_{v}∂_{w}\overline{∂}_{t} log K(z) |_{z=0} .

The triple product (x, y, z) ↦ {xyz} is complex bilinear and symmetric with respect to (x, z), complex antilinear with respect to y. It satisfies the Jordan identity

(J)  {xy{uvw}} − {uv{xyw}} = {{xyu}vw} − {u{vxy}w}.

The space V endowed with the triple product {xyz} is called a (Hermitian) Jordan triple system. For x, y, z ∈ V, denote by D(x, y) and Q(x, z) the operators defined by

(A1)  {xyz} = D(x, y)z = Q(x, z)y.

The Bergman metric at 0 is related to D by

(A2)  h_{0}(u, v) = tr D(u, v).

A Jordan triple system is called Hermitian positive if (u|v) = tr D(u, v) is positive definite. As the Bergman metric of a bounded domain is always definite positive, the Jordan triple system associated to a bounded symmetric domain is Hermitian positive.

The quadratic representation

Q : V ⟶ End_{ℝ}(V)

is defined by Q(x)y = \frac{1}{2}\{xyx\}. The Bergman operator B is defined by

(A3)  B(x, y) = I − D(x, y) + Q(x)Q(y),

where I denotes the identity operator in V. The quadratic representation and the Bergman operator satisfy to many identities, the most important of which are

Q(Q(x)y) = Q(x)Q(y)Q(x), (A4)

Q(B(x, y)z) = B(x, y)Q(z)B(y, x) (A5)

(see [Loos 1975], [Roos 2000]).

The Bergman operator gets its name from the following property:

h_{z} (B(z, z)u, v) = h_{0}(u, v) (z ∈ Ω; u, v ∈ V).

The Bergman kernel of Ω is then given by

K(z) = \frac{1}{vol Ω} \frac{1}{det B(z, z)}.

Spectral theory. Let V be an Hermitian positive Jordan triple system. An element c ∈ V is called tripotent if c ≠ 0 and {ccc} = 2c.

Two tripotents c_{1} and c_{2} are called orthogonal if D(c_{1}, c_{2}) = 0. If c_{1} and c_{2} are orthogonal tripotents, then D(c_{1}, c_{1)} and D(c_{2}, c_{2}) commute and c_{1} + c_{2} is also a tripotent.

A tripotent c is called primitive (or minimal) if it is not the sum of two orthogonal tripotents. A tripotent c is maximal if there is no tripotent orthogonal to c.

A frame of V is a maximal sequence (c_{1}, …, c_{r}) of pairwise orthogonal primitive tripotents. Then there exist frames for V. All frames have the same number of elements, which is the rank r of V. The frames of V form a manifold F, which is called the Satake-Furstenberg boundary of Ω.

Let V be a simple Hermitian positive Jordan triple system. Then any x ∈ V can be written in a unique way

(A6)  x = λ_{1}c_{1} + λ_{2}c_{2} + ⋯ + λ_{p}c_{p},

where λ_{1} > λ_{2} > ⋯ > λ_{p} > 0 and c_{1}, c_{2} …, c_{p} are pairwise orthogonal tripotents. The element x is called regular iff p = r; then (c_{1}, c_{2}, …, c_{r}) is a frame of V. The decomposition (A6) is called the spectral decomposition of x.

Let c = (c_{1}, …, c_{r}) be a frame. For 0 ≤ i ≤ j ≤ r, let

(A7)  V_{ij}(c) = {x ∈ V | D(c_{k}, c_{k})x = (δ_{i}^{k} + δ_{j}^{k})x, 1 ≤ k ≤ r}.

The decomposition

(A8)  V = ⨁_{0≤i≤j≤r} V_{ij}(c)

is orthogonal with respect to the Hermitian product (A2) and is called the simultaneous Peirce decomposition with respect to the frame c.

Let (p_{ij}) be the family of orthogonal projectors onto the subspaces of the decomposition (A8). Then, for x = λ_{1}c_{1} + λ_{2}c_{2} + ⋯ + λ_{r}c_{r}, λ_{i} ∈ ℝ (1 ≤ i ≤ r) and λ_{0} = 0,

(A9)  D(x, x) = ∑_{0≤i≤j≤r} (λ_{i}^{2} + λ_{j}^{2}) p_{ij},

(A10)  Q(x)^{2} = ∑_{0≤i≤j≤r} λ_{i}^{2}λ_{j}^{2}p_{ij},

(A11)  B(x, x) = ∑_{0≤i≤j≤r} (1 − λ_{i}^{2}) (1 − λ_{j}^{2}) p_{ij}.

(See [Loos 1977, Corollary 3.15]).

The map x ↦ λ_{1}, where x = λ_{1}c_{1}+λ_{2}c_{2}+⋯+λ_{p}c_{p} is the spectral decomposition of x (λ_{1} > λ_{2} > ⋯ > λ_{p} > 0) is a norm on V, called the spectral norm. We will denote the spectral norm by ∥x∥_{Ω}. It satisfies

(A12)  ∥x∥_{Ω}^{2} = ∥Q(x)∥ = \frac{1}{2} ∥D(x, x)∥ ,

where ∥u∥ denotes the operator norm of an ℝ-linear operator u ∈ End_{ℝ} V with respect to the Hermitian norm ∥x∥^{2} = tr D(x, x).

If V has the Jordan triple structure associated to the bounded symmetric domain Ω ⊂ V, then Ω is the unit ball of V for the spectral norm. Conversely, if V is a positive Hermitian Jordan triple, the unit ball of the spectral norm is a bounded circled symmetric domain, whose associated Jordan triple is V.

References

[Aizenberg 2000] Aizenberg, Lev. Multidimensional analogues of Bohr’s theorem on power series.

Proc. Amer. Math. Soc. 128:4 (2000), 1147–1155. MR1636918 (2000i:32001). Zbl 0948.32001. [Aizenberg et al.2000] Aizenberg, Lev; Aytuna, Aydin; Djakov, Plamen. An abstract approach

to Bohr’s phenomenon. Proc. Amer. Math. Soc. 128:9 (2000), 2611–2619. MR1657738 (2000m:32002). Zbl 0958.46015.

[Boas 2000] Boas, Harold P. Majorant series. J. Korean Math. Soc. 37:2 (2000), 321–337.

MR1775963 (2001j:32001). Zbl 0965.32001. [Boas–Khavinson 1997] Boas H.P.; Khavinson D. Bohr’s power series theorem in several variables.

Proc. Amer. Math. Soc. 125:10 (1997), 2975–2979. Zbl 0888.32001. [Boas–Khavinson 2000] Boas, Harold P.; Khavinson, Dmitry. Vita: Friedrich Wilhelm Wiener.

Math. Intell. 22:2 (2000), 73–75. MR1764270 (2001g:01037). Zbl 1052.01516. [Bohr 1914] Bohr, Harald. A theorem concerning power series, Proc. London Math. Soc. (2) 13

(1914), 1–5. JFM 44.0289.01. [Bombieri 1962] Bombieri, Enrico. Sopra un teorema di H. Bohr e G. Ricci sulle funzioni maggio-

ranti delle serie di potenze. Boll. Unione Mat. Ital., III. Ser. 17 (1962), 276–282. MR0162918 (29 #222). Zbl 0109.04801.

[Bombieri–Bourgain 2004] Bombieri, E.; Bourgain, J. A remark on Bohr’s inequality. Int. Math. Res. Not. 80 (2004), 4307–4330. MR2126627 (2006a:30003). Zbl 1069.30001.

[Defant et al. 2003] Defant, Andreas; García, Domingo; Maestre, Manuel. Bohr’s power series theorem and local Banach space theory. J. Reine Angew. Math. 557 (2003), 173–197. MR1978407 (2004d:46012). Zbl 1031.46014.

[Dineen–Timoney 1989] Dineen, Seán; Timoney, Richard M. Absolute bases, tensor products and a theorem of Bohr. Studia Math., 94:3 (1989), 227–234. MR1019790 (91e:46006). Zbl 0713.46005.

[Dineen–Timoney 1991] Dineen, Seán; Timoney, Richard M. On a problem of H. Bohr. Bull. Soc.

Roy. Sci. Liège 68:6 (1991), 401–404. MR1162791 (93e:46050). Zbl 0795.46034. [Djakov–Ramanujan 2000] Djakov P.B.; Ramanujan M.S. A remark on Bohr’s theorem and its

generalizations. J. Anal. 8 (2000), 65–77. MR1806396 (2001k:30001). Zbl 0969.30001. [Fejér 1914] Fejér, L. Über gewisse durch die Fourier’sche und Laplace’sche Reihe definierten

Mittelkurven und Mittelflächen. Palermo Rend. (1) 38 (1914), 79-97. JFM 45.0412.03. [Fejér 1925] Fejér L. Über die Positivität von Summen, die nach trigonometrischen oder Legen-

dreschen Funktionen fortschreiten. I. Acta Szeged 2 (1925), 75–86. JFM 51.0219.01. [Hua 1963] Hua, L.K. (Hua, Luo-Keng). Harmonic analysis of functions of several complex vari-

ables in classical regions. Translated from the Russian by Leo Ebner and Adam Korányi. iv+168 p. Translations of Mathematical Monographs, 6. American Mathematical Society, Providence, RI, 1963. MR0171936 (30 #2162). Zbl 0112.07402.

[Hua 1979] Hua, L.K. (Hua, Luo-Keng). Harmonic analysis of functions of several complex variables in the classical domains. Transl. from the Russian, which was a translation of the Chinese orig., by Leo Ebner and Adam Koranyi. With a foreword by M.I. Graev. Reprint of the 1963 edition. iv+186 p. Translations of Mathematical Monographs, 6. American Mathematical Society, Providence, R.I. 1979. MR0598469 (82c:32032). Zbl 0507.32025.

[Kahane 1980] Kahane, Jean-Pierre. Sur les polynômes à coefficients unimodulaires. Bull. London

Math. Soc. 12:5 (1980), 321–342. MR0587702 (82a:30003). Zbl 0443.30005. [Landau 1925] Landau, Edmund. Über einen Fejérschen Satz. Nachrichten Göttingen 1925, 22

(1925). JFM 51.0219.02. [Landau 1929] Landau, E. (Landau, Edmund Georg Hermann). Darstellung und Begründung

einiger neuerer Ergebnisse der Funktionentheorie, 2. Aufl. 122S., 10 Fig. Julius Springer, Berlin, 1929. JFM 55.0171.03.

[Liu–Wang 2007] Liu, Taishun; Wang, Jianfei. An absolute estimate of the homogeneous expansions of holomorphic mappings. Pacific J. Math. 231:1 (2007), 155–166. MR2304626 (2008h:32001). Zbl pre05366236.

[Loos 1975] Loos, Ottmar. Jordan pairs, xvi+218 p. Lecture Notes in Mathematics, vol. 460,

Springer Verlag, Berlin–Heidelberg–New York, 1975. MR0444721 (56 #3071). Zbl 0301.17003 [Loos 1977] Loos, Ottmar. Bounded symmetric domains and Jordan pairs, Math. Lectures, Univ.

of California, Irvine, CA, 1977. [Paulsen et al. 2002] Paulsen V.I.; Popescu G.; Singh D. On Bohr’s inequality. Proc. London

Math. Soc. (3) 85:2 (2002), 493–512. MR1912059 (2003h:47025). Zbl 1033.47008. [Ricci 1955] Ricci, Giovanni. Complementi a un teorema di H. Bohr riguardante le serie di potenze.

Rev. Un. Mat. Argentina 17, 185–195 (1956). MR0081336 (18,385a). Zbl 0072.07301. [Roos 2000] Roos, Guy. Jordan triple systems. Faraut, Jacques et al., Analysis and geometry on

complex homogeneous domains, Prog. Math. 185, Birkhäuser, Boston, MA, 425–536 (2000). Zbl 1043.17017.

[Sidon 1927] Sidon S., Über einen Satz von Herrn Bohr, Math. Z. 26:1 (1927), 731–732.

MR1544888. JFM 53.0281.04. [Tomić 1962] Tomić M., Sur un théorème de H. Bohr, Math. Scand. 11 (1962), 103–106.

MR0176040 (31 #316). Zbl 0109.30202.

Nevski pr. 113/4-53, 191024 St Petersburg, Russian Federation

E-mail address: guy.roos@normalesup.org