AN EXAMPLE OF LIMIT OF LEMPERT FUNCTIONS

PASCAL J. THOMAS

1. Introduction

Let Ω be a domain in ℂ^{n}, and a_{j} ∈ Ω, j = 0, ..., N. Coman’s Lempert function is defined by [2], [5]:

(1.1) ℓ(z) := ℓ_{a_{0},...,a_{N}} (z) := inf { ∑_{j=0}^{N} log |ζ_{j}| : φ(0) = z,

φ(ζ_{j}) = a_{j}, j = 0, ..., N for some φ ∈ O(𝔻, Ω)},

where 𝔻 is the unit disc in ℂ.

For most of this paper, we will consider Ω = 𝔻^{2}, |z| := max(|z_{1}|, |z_{2}|), a_{0} = (0, 0), a_{1} = (ε_{1}, 0), and a_{2} = (0, ε_{2}), where ε_{j} ∈ 𝔻, j = 1, 2. We will write ℓ_{ε}(z) for the Lempert function with respect to the three poles a_{0}, a_{1}, a_{2} evaluated at the point z = (z_{1}, z_{2}) ∈ 𝔻. It is clear that the Lempert function approaches −∞ near each of its poles a_{j}. When ε_{j} → 0, all poles concentrate at the origin of ℂ^{2}, and the Lempert function may converge to some limit with its singularities concentrated at the origin.

Our goal is to understand in detail the aspect of this singularity. A first remark is that the Lempert function is always related to the corresponding Green function for the same poles,

g(z) := sup {u ∈ P SH(Ω, ℝ_{−}) : u(z) ≤ log |z − a_{j}| + C_{j}, j = 0, ..., N} ,

where P SH(Ω, ℝ_{−}) stands for the set of all negative plurisubharmonic functions in Ω. The inequality g(z) ≤ ℓ(z) always holds, and it is known that it can be strict [1], [7], [6]. If ℓ ever turns out to be plurisubharmonic itself, then it must be equal to g [2].

In the special case that we are studying, the Green function (denoted by g_{ε}) is not known either, nor do we know whether it always admits a limit as ε_{j} → 0, but it can be compared to the Green functions for the three following systems of points:

S_{1} := {a_{0}, a_{1}} , S_{2} := {a_{0}, a_{2}} , S_{3} := {a_{0}, a_{1}, a_{2}, (ε_{1}, ε_{2})} .

Those are all product sets, so their Green functions are explicitly known [3] as well as their limits when ε_{j} → 0, which are respectively

g_{1}(z) := max(2 log |z_{1}|, log |z_{2}|), g_{2}(z) := max(log |z_{1}|, 2 log |z_{2}|), g_{3}(z) := max(2 log |z_{1}|, 2 log |z_{2}|).

Nguyen Van Trao remarked that it follows from the definition of the Green function that

(1.2)  g_{3} ≤ g_{ε} ≤ min(g_{1}, g_{2})

throughout the bidisk, and therefore when |z_{2}| ≤ |z_{1}|^{2}, g_{ε}(z) = g_{3}(z) = g_{1}(z) = 2 log |z_{1}|, and when |z_{1}| ≤ |z_{2}|^{2}, g_{ε}(z) = g_{3}(z) = g_{2}(z) = 2 log |z_{2}|. Also, for any z in the bidisk,

lim inf_{ε_{1},ε_{2}→0} ℓ_{ε}(z) ≥ g_{ε}(z) ≥ g_{3}(z) = 2 log |z|.

We first give our result in a special case where the picture is more complete.

Theorem 1.1. Suppose ε_{1} = ε_{2} = ε. Then

(1) If z_{1} = 0 or z_{2} = 0 or z_{1} + z_{2} = 0, there exists a constant C > 0

such that

lim sup_{ε_{1},ε_{2}→0} ℓ_{ε}(z) ≤ 2 log |z| + C.

(2) For any c_{0} > 0, there exists a constant C = C(c_{0}) > 0 such that for any z = (z_{1}, z_{2}) verifying c_{0} ≤ |\frac{z_{1}}{z_{2}} | ≤ c_{0}^{−1}, and c_{0} ≤ |1 + \frac{z_{1}}{z_{2}} |, then

lim inf_{ε→0} ℓ_{ε}(z) ≥ \frac{3}{2} log |z| − C

and

lim inf_{ε→0} ℓ_{ε}(z) ≤ \frac{3}{2} log |z| + C.

Corollary 1.2. If g is some cluster point of the family {g_{ε}}, then for any z ∈ 𝔻^{2},

g(z) ≤ \frac{3}{2} log |z|,

and g(z_{1}, −z_{1}) = 2 log |z_{1}|.

This is an improvement over the inequalities (1.2) when |z_{2}|^{2} < |z_{1}| < |z_{2}|^{1/2}.

Proof. The function g must be plurisubharmonic, so subharmonic on any complex line going through the origin. It is also negative everywhere. The inequality that its restriction on a complex line satisfies near the origin implies that it is bounded above by the corresponding one-variable Green function, which gives us the required upper bound. The equality in the special case follows from (1.2). □

Theorem 1.1 will follow from the more detailed result below.

Theorem 1.3. (1) If z_{1} = 0 or z_{2} = 0 or if z_{1} ≠ 0, z_{2} ≠ 0 and lim_{ε_{1},ε_{2}→0} \frac{ε_{1}}{ε_{2}} = −\frac{z_{1}}{z_{2}} , there exists a constant C > 0 such that

lim sup_{ε_{1},ε_{2}→0} ℓ_{ε}(z) ≤ 2 log |z| + C.

(2) For any c_{0} > 0, there exists a constant C = C(c_{0}) > 0 such that for any z = (z_{1}, z_{2}) verifying c_{0} ≤ |\frac{z_{1}}{z_{2}} | ≤ c_{0}^{−1}, with the (ε_{1}, ε_{2}) involved in the limes (inferior and superior) below always verifying

(1.3)  |\frac{z_{1}}{|z_{1}|} \frac{|z_{2}|}{z_{2}} + \frac{ε_{1}}{|ε_{1}|} \frac{|ε_{2}|}{ε_{2}} | ≥ c_{0},

then

(1.4)  lim inf_{ε_{1},ε_{2}→0} ℓ_{ε}(z) ≥ \frac{3}{2} log |z| − C.

If, furthermore,

lim_{ε_{1},ε_{2}→0} \frac{ε_{1}^{2}}{ε_{2}} = lim_{ε_{1},ε_{2}→0} \frac{ε_{2}^{2}}{ε_{1}} = 0,

then

(1.5)  lim sup_{ε_{1},ε_{2}→0} ℓ_{ε}(z) ≤ \frac{3}{2} log |z| + C.

2. Upper estimates

To prove the upper estimates in the above theorem, we shall need to construct appropriate maps φ from the disk to the bidisk. It will be useful to relax a little the condition that φ(𝔻) ⊂ 𝔻^{2}.

Lemma 2.1. Suppose that a_{j}(ε) ∈ 𝔻^{2}, j = 0, ..., N depend on some parameter ε ∈ ℂ^{m}, with lim_{ε→0} a_{j}(ε) = a_{j} ∈ Ω, j = 0, ..., N, and that there exists a function γ(ε) such that lim_{ε→0} γ(ε) = 0 and λ ∈ ℝ with the following property: for any ε > 0, z ∈ 𝔻^{2}\{a_{1}, …, a_{N}}, there exists a holomorphic map φ : 𝔻 ⟶ D(0, 1 + γ(ε))^{2} and ζ_{0}, …, ζ_{N} such that

(2.1) φ(0) = z, φ(ζ_{j}) = a_{j}(ε), j = 0, ..., N, and ∑_{j=0}^{N} log |ζ_{j}| ≤ λ.

Then lim sup_{ε→0} ℓ_{a_{0}(ε),...,a_{N} (ε)}(z) ≤ λ.

Proof. First note that by applying an automorphism ϕ of the bidisk exchanging z and (0, 0), we have a map φ̃ := ϕ ∘ φ such that

φ̃(0) = (0, 0), φ(ζ_{j}) = ϕ(a_{j}(ε)), j = 0, ..., N, and φ̃ : 𝔻 ⟶ D(0, 1+γ(ε))^{2},

for another function γ with the same property as the original one.

Likewise, to estimate ℓ_{a_{1}(ε),...,a_{N} (ε)}(z) it is equivalent to look for maps ψ and points ζ′_{0}, …, ζ′_{N} such that ψ(𝔻) ⊂ 𝔻^{2}, and

ψ(0) = (0, 0), φ(ζ′_{j}) = ϕ(a_{j}(ε)), j = 0, ..., N, and lim sup_{ε→0} ∑_{j=0}^{N} log |ζ′_{j}| ≤ λ

Applying the usual Schwarz lemma to each coordinate of φ̃, we see that φ̃(D(0, (1 + γ(ε))^{−1}) ⊂ 𝔻^{2}, and therefore

ψ(ζ) := φ̃(\frac{ζ}{1 + γ(ε)}) and ζ′_{j} := (1 + γ(ε))ζ_{j}

will satisfy our requirements. □

Notation: we always write ϕ_{a}(ζ) := \frac{a−ζ}{1−ζā} for the involutive automorphism of the unit disk exchanging a and 0. Applying an automorphism of 𝔻 exchanging ζ_{0} and 0, the original problem of construction of maps as in (2.1) is equivalent to finding a holomorphic map φ : 𝔻 ⟶ D(0, 1 + γ(ε))^{2} and new points ζ_{0}, …, ζ_{N} such that (2.2)

φ(0) = (0, 0), φ(ζ_{j}) = a_{j}(ε), j = 1, ..., N, and |ζ_{0}|+∑_{j=1}^{N} log |ϕ_{ζ_{0}}(ζ_{j})| ≤ λ.

Proof of Theorem 1.3, part (1).

Assume z_{2} = 0. The case z_{1} = 0 would be treated in the same way. By the formulation (2.2) of our problem, we need to construct a map

φ, with an appropriate control of the image of 𝔻, such that

(2.3)  \{ \begin{matrix}φ(0) & = & (0, 0) \\ φ(ζ_{1}) & = & (ε_{1}, 0) \\ φ(ζ_{2}) & = & (0, ε_{2}) \\ φ(ζ_{0}) & = & (z_{1}, 0) \end{matrix}

First we choose ζ_{1} := ε_{1}, ζ_{0} := z_{1}, and ζ_{2} close to 1, to be specified later. An approximate solution of this interpolation problem, will be given by the following map from 𝔻 to 𝔻^{2}:

φ^{0}(ζ) := (ζϕ_{ζ_{2}}(ζ), ζ \frac{ϕ_{ε_{1}}(ζ)}{ϕ_{ε_{1}}(1)} \frac{ϕ_{z_{1}}(ζ)}{ϕ_{z_{1}}(1)}ε_{2}) .

The errors with respect to the requirements in (2.3) are now given by

(2.4) \{ \begin{matrix}φ^{0}(0) − (0, 0) & = & (0, 0) \\ φ^{0}(ε_{1}) − (ε_{1}, 0) & = & (ε_{1}(ϕ_{ζ_{2}}(ε_{1}) − 1), 0) & =: (E_{1}, 0) \\ φ^{0}(ζ_{2}) − (0, ε_{2}) & = & (0, ε_{2}(ζ_{2} \frac{ϕ_{ε_{1}}(ζ_{2})}{ϕ_{ε_{1}}(1)} \frac{ϕ_{z_{1}}(ζ_{2})}{ϕ_{z_{1}}(1)} − 1)) & =: (0, E_{2}) \\ φ^{0}(z_{1}) − (z_{1}, 0) & = & (z_{1}(ϕ_{ζ_{2}}(z_{1}) − 1), 0) & =: (E_{3}, 0) \end{matrix}

A computation shows that

|ϕ_{ζ_{2}}(ζ) − 1| = |\frac{(1 − ζ_{2}) + ζ(1 − ζ̄_{2})}{1 − ζζ̄_{2}} | ≤ (\frac{1 + |ζ|}{1 − |ζ|}) |1 − ζ_{2}|,

so that for |z_{1}| ≤ \frac{1}{2}, ε_{1} ≤ \frac{1}{2}|z_{1}|, which we may assume,

(2.5)  |E_{1}| ≤ \frac{5}{3}|1 − ζ_{2}||ε_{1}|, |E_{3}| ≤ 3|1 − ζ_{2}||z_{1}|.

In the same way, one sees that under the above assumption, |E_{2}| ⪯ |1 − ζ_{2}||ε_{2}|.

To get a map satisfying (2.3), we subtract from φ^{0} a correcting term φ^{1}(ζ) = (φ_{1}^{1}(ζ), φ_{2}^{1}(ζ)) obtained by Lagrange interpolation. More precisely, we choose

φ_{2}^{1}(ζ) = \frac{ζ}{ζ_{2}} \frac{ϕ_{ε_{1}}(ζ)}{ϕ_{ε_{1}}(ζ_{2})} \frac{ϕ_{z_{1}}(ζ)}{ϕ_{z_{1}}(ζ_{2})}E_{2},

and φ_{1}^{1}(ζ) = ζϕ_{ζ_{2}}(ζ)h(ζ), where h must satisfy

h(ε_{1}) = \frac{E_{1}}{ε_{1}ϕ_{ζ_{2}}(ε_{1})} =: E′_{1}, h(z_{1}) = \frac{E_{3}}{z_{1}ϕ_{ζ_{2}}(z_{1})} =: E′_{3}.

We can then set

h(ζ) := E′_{1} \frac{ζ − z_{1}}{ε_{1} − z_{1}} + E′_{3} \frac{ζ − ε_{1}}{z_{1} − ε_{1}} .

It is easy to see that |φ_{2}^{1}(ζ)| ⪯ |E_{2}| ≤ |ε| for ζ_{2} close enough to 1. It follows from (2.5) that |E′_{1}|, |E′_{3}| ⪯ |1 − ζ_{2}|, therefore |φ_{1}^{1}(ζ)| ⪯ |1−ζ_{2}||z_{1}|^{−1} ≤ |ε| for ζ_{2} close enough to 1. So the map φ^{0} −φ^{1} satisfies the hypotheses of Lemma 2.1 and since we have

|ζ_{0}|+∑_{j=1}^{2} log |ϕ_{ζ_{0}}(ζ_{j})| ≤ log |z_{1}|+log |ϕ_{z_{1}}(ε_{1})| ≤ log |z_{1}|+log(|z_{1}|+|ε_{1}|),

we may take λ = 2 log |z_{1}| + η, for any η > 0, which implies the conclusion of Theorem 1.3, part (1), with C = 0. □

Proof of Theorem 1.3, part (2), (1.5).

We will need a few notations. We set μ := z_{2}/z_{1}, σ := ε_{2}/ε_{1}. Exchanging coordinates if needed, we may assume |μ| ≤ 1, and therefore |z| = |z_{1}|. The hypothesis in the theorem is that

c_{0} ≤ |\frac{|μ|}{μ} + \frac{|σ|}{σ}| = |1 + (\frac{|μ|}{μ})^{−1} \frac{|σ|}{σ}| ,

so the complex number σ/μ lies outside of certain plane sector containing −1, and in particular |1 + (σ/μ)| ≥ c_{0}. We choose a complex number ν such that ν^{2} := (1 + (σ/μ))^{−1}; this remains bounded.

We choose a complex number ζ_{0} such that ζ_{0}^{2} = z_{1}. This means that z = (ζ_{0}^{2}, μζ_{0}^{2}). We also choose, for each ε_{1}, an ε′_{1} such that ε′_{1}^{2} = ε_{1}. Now we set (2.6)

ζ_{1} := νε′_{1} = (1 + \frac{ε_{2}}{ε_{1}μ})^{−1/2} ε_{1}^{1/2}, ζ_{2} := −\frac{σ}{μ}ζ_{1} = −\frac{ε_{2}}{ε_{1}^{1/2}} (1 + \frac{ε_{2}}{ε_{1}μ})^{−1/2} .

We will follow the pattern of the previous proof. In order to apply Lemma 2.1, we need to produce a map satisfying

(2.7)  \{ \begin{matrix}φ(0) & = & (0, 0) \\ φ(ζ_{1}) & = & (ε_{1}, 0) \\ φ(ζ_{2}) & = & (0, ε_{2}) \\ φ(ζ_{0}) & = & (z_{1}, z_{2}) \end{matrix}

We choose ζ_{0}, ζ_{1}, ζ_{2} as above and set

φ^{0}(ζ) = (ζ \frac{ζ − ζ_{2}}{1 − ζζ̄_{2}} , μζ \frac{ζ − ζ_{1}}{1 − ζζ̄_{1}} ) .

We remark that our ζ_{j} have been chosen so that

ζ_{1}(ζ_{1} − ζ_{2}) = ε_{1}, μζ_{2}(ζ_{2} − ζ_{1}) = ε_{2}, and |ζ_{1}ζ_{2}| = |ν|^{2}|μ|^{−1}|ε_{2}| ≤ c_{0}^{−2}|ε_{2}|.

Elementary computations yield (2.8)

\{ \begin{matrix}φ^{0}(0) − (0, 0) & = & (0, 0) \\ φ^{0}(ζ_{1}) − (ε_{1}, 0) & = & (ε_{1} \frac{ζ_{1}ζ̄_{2}}{1 − ζ_{1}ζ̄_{2}} , 0) & =: (E_{1}, 0) \\ φ^{0}(ζ_{2}) − (0, ε_{2}) & = & (0, ε_{2} \frac{ζ̄_{1}ζ_{2}}{1 − ζ̄_{1}ζ_{2}} ) & =: (0, E_{2}) \\ φ^{0}(ζ_{0}) − z & = & (ζ_{0} \frac{ζ_{0}^{2}ζ̄_{2} − ζ_{2}}{1 − ζ_{0}ζ̄_{2}} , μζ_{0} \frac{ζ_{0}^{2}ζ̄_{1} − ζ_{1}}{1 − ζ_{0}ζ̄_{1}} ) & =: (E_{3}, E_{4}) \end{matrix} .

We construct a correcting term φ^{1} by Lagrange interpolation:

\begin{aligned}φ_{1}^{1}(ζ) & = & E_{1} \frac{ζ(ζ−ζ_{2})(ζ−ζ_{0})}{ζ_{1}(ζ_{1}−ζ_{2})(ζ_{1}−ζ_{0})} + E_{3} \frac{ζ(ζ−ζ_{1})(ζ−ζ_{2})}{ζ_{0}(ζ_{0}−ζ_{1})(ζ_{0}−ζ_{2})}, \\ φ_{2}^{1}(ζ) & = & E_{2} \frac{ζ(ζ−ζ_{1})(ζ−ζ_{0})}{ζ_{2}(ζ_{2}−ζ_{1})(ζ_{2}−ζ_{0})} + E_{4} \frac{ζ(ζ−ζ_{1})(ζ−ζ_{2})}{ζ_{0}(ζ_{0}−ζ_{1})(ζ_{0}−ζ_{2})}. \end{aligned}

The map φ^{0} − φ^{1} will assume the correct values, now we need to see that it sends 𝔻 to a neighborhood of the bidisk by estimating the size of φ^{1}.

First note that for |ε| small enough, |E_{3}| ≤ 3|z_{1}|^{1/2}|ζ_{2}| ≤ 3c_{0}^{−3/2}|z_{1}|^{1/2}|ε_{1}|^{−1/2}|ε_{2}|, and |E_{4}| ≤ 3|z_{1}|^{1/2}|ζ_{1}| ≤ 3c_{0}^{−1/2}|z_{1}|^{1/2}|ε_{1}|^{1/2}, therefore, using the last hypothesis of the theorem, for |ε| small enough (depending on |z_{1}|), the second terms in φ_{1}^{1}(ζ) and φ_{2}^{1}(ζ) can be made arbitrarily small.

On the other hand, for |ε| small enough,

|\frac{E_{1}}{ζ_{1}(ζ_{1} − ζ_{2})}| = |\frac{ζ_{1}ζ̄_{2}}{1 − ζ_{1}ζ̄_{2}} | ≤ 2c_{0}^{−2}|ε_{2}|,

|\frac{E_{2}}{ζ_{2}(ζ_{2} − ζ_{1})}| = |\frac{μζ̄_{1}ζ_{2}}{1 − ζ̄_{1}ζ_{2}} | ≤ 2c_{0}^{−3}|ε_{2}|.

So the first terms in φ_{1}^{1}(ζ) and φ_{2}^{1}(ζ) are bounded above by 2c_{0}^{−3}|ε_{2}||z_{1}|^{−1/2}, which once again can be made arbitrarily small for |ε| small enough.

Finally, the relevant sum

log |ζ_{0}|+log |ϕ_{ζ_{0}}(ζ_{1})|+log |ϕ_{ζ_{0}}(ζ_{2})| ≤ log |ζ_{0}|+log(|ζ_{0}|+|ζ_{1}|)+log(|ζ_{0}|+|ζ_{2}|),

so that λ = 3 log |ζ_{0}| + η = \frac{3}{2} log |z_{1}| + η ≤ \frac{3}{2} log |z| + η may be used to apply Lemma 2.1, for any η > 0. □

3. Lower Estimates

Proof of Theorem 1.3, part (2), (1.4).

We will assume that the conclusion fails, i.e. that for any C_{H} > 0 there exist arbitrarily small values of ε = (ε_{1}, ε_{2}) such that

(3.1)  ℓ_{ε}(z) ≤ \frac{3}{2} log |z| − C_{H},

which means (we change the value of the constant slightly while keeping the same notation) that there exists a holomorphic map φ from 𝔻 to 𝔻^{2} and points ζ_{j} ∈ 𝔻 satisfying the conditions in (2.7) with

(3.2)  log |ζ_{0}| + log |ϕ_{ζ_{0}}(ζ_{1})| + log |ϕ_{ζ_{0}}(ζ_{2})| ≤ \frac{3}{2} log |z| − C_{H}.

The interpolation conditions in (2.7) are equivalent to the existence of two holomorphic functions h_{1}, h_{2} from 𝔻 to itself such that

φ(ζ) = (ζϕ_{ζ_{2}}(ζ)h_{1}(ζ), ζϕ_{ζ_{1}}(ζ)h_{2}(ζ)) ,

such that furthermore

(3.3)  h_{1}(ζ_{1}) = \frac{ε_{1}}{ζ_{1}ϕ_{ζ_{2}}(ζ_{1})} =: w_{1},
(3.4)  h_{1}(ζ_{0}) = \frac{z_{1}}{ζ_{0}ϕ_{ζ_{2}}(ζ_{0})} =: w_{2},
(3.5)  h_{2}(ζ_{2}) = \frac{ε_{2}}{ζ_{2}ϕ_{ζ_{1}}(ζ_{2})} =: w_{4},
(3.6)  h_{2}(ζ_{0}) = \frac{z_{2}}{ζ_{0}ϕ_{ζ_{1}}(ζ_{0})} =: w_{3}.

For convenience, we will use the invariant (pseudohyperbolic) distance between points of the unit disk given by

d_{G}(a, b) := |ϕ_{a}(b)| = |ϕ_{b}(a)| .

By the invariant Schwarz Lemma, the existence of a holomorphic function h_{1} mapping 𝔻 to itself and satisfying (3.3) and (3.4) is equivalent to

(3.7) |w_{1}| < 1, |w_{2}| < 1, and d_{G} (w_{1}, w_{2}) < d_{G} (ζ_{1}, ζ_{0}) = |ϕ_{ζ_{1}}(ζ_{0})| .

In the same way, the existence of h_{2} is equivalent to

(3.8) |w_{3}| < 1, |w_{4}| < 1, and d_{G} (w_{3}, w_{4}) < d_{G} (ζ_{2}, ζ_{0}) = |ϕ_{ζ_{2}}(ζ_{0})| .

The proof will proceed as follows: one main elementary tool is the fact that if the invariant distance of two points in the unit disk is small with respect to one of their moduli, then the difference of their arguments (or rather, the distance between their projections to the unit circle) must be small (Lemma 3.1).

Then, using (3.1) and the fact that |w_{2}| and |w_{3}| are in the unit disk, we will show that |ϕ_{ζ_{1}}(ζ_{0})| and |ϕ_{ζ_{2}}(ζ_{0})| must both be relatively small. This will imply that |w_{2}| and |w_{3}| are also relatively big, and because of (3.7), (3.8), so will be |w_{1}| and |w_{4}|, and because ε is small, this will force |ϕ_{ζ_{2}}(ζ_{1})| to be small too, and therefore | arg(ζ_{1}/ζ_{2})|. This will allow us to show that arg(w_{1}/w_{4}) is close to arg(−ε_{1}/ε_{2}), because ϕ_{ζ_{2}}(ζ_{1}) is almost opposite to ϕ_{ζ_{1}}(ζ_{2}). On the other hand, use of the triangle inequality on the unit circle will show that arg(w_{1}/w_{4}) = arg[(w_{1}/w_{2})(w_{2}/w_{3})(w_{3}/w_{4})] is close to arg(z_{1}/z_{2}). Hypothesis (1.3) will lead to a contradiction when C_{H} is big enough.

Lemma 3.1. Suppose that a, b ∈ 𝔻, and that d_{G}(a, b) = δ < |a|, δ ≤ \frac{1}{2}. Then

|arg (\frac{a}{b})| ≤ arcsin (\frac{δ(1 − |a|^{2})}{|a|(1 − δ^{2})}) ≤ \frac{3δ}{|a|}.

Note that in this result, and the computations that follow, we implicitly only consider arguments in the range [−\frac{π}{2}, \frac{π}{2}].

Proof. The set of all points b ∈ 𝔻 such that d_{G}(a, b) ≤ δ is a disk of center γ := \frac{a(1−δ^{2})}{1−|a|^{2}δ^{2}} and of radius ρ := \frac{δ(1−|a|^{2})}{1−|a|^{2}δ^{2}} [4, Chapter I, Section 1, p. 3]. Elementary geometry shows that the absolute value of the sine of the angle between a and b is bounded by ρ/|γ|. □

Our starting remark is that (3.2) can be rewritten as

(3.9) − log |w_{2}| − log |w_{3}| = log |\frac{ζ_{0}ϕ_{ζ_{1}}(ζ_{0})}{z_{2}} | + log |\frac{ζ_{0}ϕ_{ζ_{0}}(ζ_{2})}{z_{1}} |

≤ log |ζ_{0}| + \frac{3}{2} log |z| − log |z_{1}| − log |z_{2}| − C_{H}

≤ log |ζ_{0}| − \frac{1}{2} log |z| − log c_{0} − C_{H}.

Since both terms on the left hand side of the first inequality sign are positive by (3.7), (3.8), they are each bounded by the right hand side, and as a consequence, writing C′_{H} := exp(C_{H}),

(3.10)  |ϕ_{ζ_{1}}(ζ_{0})| < \frac{|z|^{3/2}}{|z_{1}|C′_{H}} ≤ \frac{|z|^{1/2}}{c_{0}C′_{H}} ,
(3.11)  |ϕ_{ζ_{2}}(ζ_{0})| < \frac{|z|^{3/2}}{|z_{2}|C′_{H}} ≤ \frac{|z|^{1/2}}{c_{0}C′_{H}} .

Furthermore, since the right hand side must also be positive, we have

(3.12)  |ζ_{0}| > c_{0}C′_{H}|z|^{1/2} ≥ (c_{0}C′_{H})^{2} max(|ϕ_{ζ_{1}}(ζ_{0})| , |ϕ_{ζ_{2}}(ζ_{0})|).

Lower bounds for |w_{2}| and |w_{3}|, and |w_{1}| and |w_{4}|.

By (3.11),

(3.13)  |w_{2}| = \frac{|z_{1}|}{|ζ_{0}||ϕ_{ζ_{2}}(ζ_{0})|} > \frac{c_{0}C′_{H}|z|^{1/2}}{|ζ_{0}|} ,

and since d_{G}(w_{2}, w_{1}) < |ϕ_{ζ_{1}}(ζ_{0})|, classical facts about the invariant distance [4, Lemma 1.4, p. 4] imply

(3.14) |w_{1}| ≥ \frac{|w_{2}| − |ϕ_{ζ_{1}}(ζ_{0})|}{1 − |w_{2}||ϕ_{ζ_{1}}(ζ_{0})|} ≥ \frac{c_{0}C′_{H}|z|^{1/2}}{|ζ_{0}|} − \frac{|z|^{1/2}}{c_{0}C′_{H}} ≥ \frac{c_{0}C′_{H}|z|^{1/2}}{2|ζ_{0}|}

when C_{H} is big enough (depending on c_{0}). The same computations go through for w_{3} and w_{4} respectively.

Upper bound for d_{G}(ζ_{1}, ζ_{2}) = |ϕ_{ζ_{2}}(ζ_{1})|.

By definition of w_{1}, we have ϕ_{ζ_{2}}(ζ_{1}) = ε_{1}/(ζ_{1}w_{1}). We estimate the modulus of |ζ_{1}| from below by noting that ζ_{1} is close to ζ_{0}, by (3.10) and using (3.12):

(3.15)  |ζ_{1}| ≥ \frac{|ζ_{0}| − |ϕ_{ζ_{1}}(ζ_{0})|}{1 − |ζ_{0}||ϕ_{ζ_{1}}(ζ_{0})|} ≥ \frac{1}{2}|ζ_{0}|,

when C_{H} is big enough (depending on c_{0}). So by (3.14)

(3.16)  |ϕ_{ζ_{2}}(ζ_{1})| ≤ 4\frac{|ε_{1}|}{c_{0}C′_{H}|z|^{1/2}} .

Since |ε| will tend to 0 as z remains fixed, this estimate is much stronger than the previous (3.10) and (3.11). From this, (3.15) and Lemma 3.1, we deduce

(3.17)  |arg (\frac{ζ_{1}}{ζ_{2}} )| ≤ 24\frac{|ε_{1}|}{(c_{0}C′_{H})^{2}|z|}.

Argument estimates for w_{1}/w_{4}.

First compute

\frac{w_{1}}{w_{4}} = \frac{ε_{1}}{ε_{2}} \frac{ζ_{2}}{ζ_{1}} \frac{ϕ_{ζ_{1}}(ζ_{2})}{ϕ_{ζ_{2}}(ζ_{1})} = −\frac{ε_{1}}{ε_{2}} \frac{ζ_{2}}{ζ_{1}} \frac{1 − ζ̄_{2}ζ_{1}}{1 − ζ̄_{1}ζ_{2}} .

We need to bound arg(1 − ζ̄_{2}ζ_{1}). We use the fact that ζ_{1} and ζ_{2} are close to each other :

1 − ζ̄_{2}ζ_{1} = (1 − |ζ_{1}|^{2}) (1 − \frac{(ζ̄_{2} − ζ̄_{1})}{1 − |ζ_{1}|^{2}}ζ_{1}) ,

and we know that \frac{a−b}{1−|a|^{2}} = O(d_{G}(a, b)), so that (3.16) implies that

|arg(1 − ζ̄_{2}ζ_{1})| ≤ c_{1} \frac{|ε_{1}|}{c_{0}C′_{H}|z|^{1/2}} ,

where c_{1} is some absolute constant. To avoid problems of definition of arguments, use the notation w^{∗} := w/|w| for any nonzero complex number. We have proved that

(3.18)  |(\frac{w_{1}}{w_{4}} )^{∗} − (−\frac{ε_{1}}{ε_{2}} )^{∗}| ≤ c_{1} \frac{|ε_{1}|}{c_{0}C′_{H}|z|^{1/2}} ,

where c_{1} is some absolute constant (not the same as above).

Argument estimates for w_{2}/w_{3}.

First note that

\frac{w_{2}}{w_{3}} = \frac{z_{1}}{z_{2}} \frac{ϕ_{ζ_{1}}(ζ_{0})}{ϕ_{ζ_{2}}(ζ_{0})} = \frac{z_{1}}{z_{2}} \frac{ϕ_{ζ_{1}}(ζ_{0})}{ϕ_{ζ_{0}}(ζ_{1})} \frac{ϕ_{ζ_{0}}(ζ_{1})}{ϕ_{ζ_{0}}(ζ_{2})} \frac{ϕ_{ζ_{0}}(ζ_{2})}{ϕ_{ζ_{2}}(ζ_{0})}.

Since ϕ_{ζ_{0}} is an automorphism of the unit disk, it preserves the invariant distances, so d_{G}(ϕ_{ζ_{0}}(ζ_{1}), ϕ_{ζ_{0}}(ζ_{2})) = d_{G}(ζ_{1}, ζ_{2}). To apply Lemma 3.1, we now need a lower bound for |ϕ_{ζ_{0}}(ζ_{1})|, say. But the fact that |w_{3}| < 1 (see (3.8)) already implies that |ϕ_{ζ_{0}}(ζ_{1})| > |z_{2}| > c_{0}^{−1}|z|. Finally, using the Lemma with (3.16),

(3.19)  |arg (\frac{ϕ_{ζ_{0}}(ζ_{1})}{ϕ_{ζ_{0}}(ζ_{2})})| ≤ 12\frac{|ε_{1}|}{C′_{H}|z|^{3/2}} ,

provided that |ε| is small enough.

Now computations as in the previous paragraph yield

\frac{ϕ_{ζ_{1}}(ζ_{0})}{ϕ_{ζ_{0}}(ζ_{1})} \frac{ϕ_{ζ_{0}}(ζ_{2})}{ϕ_{ζ_{2}}(ζ_{0})} = \frac{1 − ζ̄_{0}ζ_{1}}{1 − ζ̄_{1}ζ_{0}} \frac{1 − ζ̄_{2}ζ_{0}}{1 − ζ̄_{0}ζ_{2}} .

Since

(1−ζ̄_{0}ζ_{1})(1−ζ̄_{2}ζ_{0}) = (1−ζ̄_{0}ζ_{1})(1−(ζ̄_{1}+ζ̄_{2}−ζ̄_{1})ζ_{0}) = |1−ζ̄_{0}ζ_{1}|^{2} (1 + \frac{(ζ̄_{2} − ζ̄_{1})ζ_{0}}{1 − ζ̄_{1}ζ_{0}} ) ,

we can estimate arg (\frac{1−ζ̄_{0}ζ_{1}}{1−ζ̄_{1}ζ_{0}} \frac{1−ζ̄_{2}ζ_{0}}{1−ζ̄_{0}ζ_{2}} ) by a constant multiple of d_{G}(ζ_{1}, ζ_{2}), and as in the previous paragraph,

|arg (\frac{ϕ_{ζ_{1}}(ζ_{0})}{ϕ_{ζ_{0}}(ζ_{1})} \frac{ϕ_{ζ_{0}}(ζ_{2})}{ϕ_{ζ_{2}}(ζ_{0})})| ≤ c_{1} \frac{|ε_{1}|}{c_{0}C′_{H}|z|^{1/2}} ,

changing the value of c_{1} as needed.

Using this and (3.19), the final result of this paragraph is then that (3.20)

|(\frac{w_{2}}{w_{3}} )^{∗} − (\frac{z_{1}}{z_{2}} )^{∗}| ≤ c_{1} \frac{|ε_{1}|}{c_{0}C′_{H}|z|^{1/2}} + 12\frac{|ε_{1}|}{C′_{H}|z|^{3/2}} ≤ c_{2} \frac{|ε_{1}|}{c_{0}C′_{H}|z|^{3/2}} .

Triangle inequality, and contradiction.

The conditions about invariant distances in (3.7) and (3.8) express the fact that w_{1} is close to w_{2}, and that w_{4} is close to w_{3}. In fact, applying Lemma 3.1 and the estimates (3.13) and (3.10), we obtain

|arg (\frac{w_{1}}{w_{2}} )| ≤ arcsin ( \begin{matrix}\frac{|z|^{1/2}}{c_{0}C′_{H}} & (1 − \frac{(c_{0}C′_{H})^{2}|z|}{|ζ_{0}|^{2}}) \\ \frac{c_{0}C′_{H}|z|^{1/2}}{|ζ_{0}|} & (1 − \frac{|z|}{(c_{0}C′_{H})^{2}} ) \end{matrix}) ≤ \frac{3}{(c_{0}C′_{H})^{2}} ,

provided that C′_{H} is large enough (depending on c_{0}). As before, the same reasoning applies to w_{4} and w_{3}, and we finally have

|(\frac{w_{1}}{w_{2}} )^{∗} − 1| ≤ \frac{3}{(c_{0}C′_{H})^{2}} , |(\frac{w_{3}}{w_{4}} )^{∗} − 1| ≤ \frac{3}{(c_{0}C′_{H})^{2}} .

Now we combine this with (3.20) (and the fact that rotations are isometries) to obtain

|(\frac{w_{1}}{w_{4}} )^{∗} − (\frac{z_{1}}{z_{2}} )^{∗}| = |(\frac{w_{1}}{w_{2}} )^{∗} (\frac{w_{3}}{w_{4}} )^{∗} (\frac{w_{2}}{w_{3}} )^{∗} − (\frac{z_{1}}{z_{2}} )^{∗}|

≤ |(\frac{w_{1}}{w_{2}} )^{∗} − 1| + |(\frac{w_{3}}{w_{4}} )^{∗} − 1| + |(\frac{w_{2}}{w_{3}} )^{∗} − (\frac{z_{1}}{z_{2}} )^{∗}|

≤ \frac{6}{(c_{0}C′_{H})^{2}} + c_{2} \frac{|ε_{1}|}{c_{0}C′_{H}|z|^{3/2}} .

Combining this with (3.18), we see that, for |ε| small enough (depending on |z|),

|(\frac{z_{1}}{z_{2}} )^{∗} − (−\frac{ε_{1}}{ε_{2}} )^{∗}| ≤ \frac{6}{(c_{0}C′_{H})^{2}} + c_{1} \frac{|ε_{1}|}{c_{0}C′_{H}|z|^{1/2}} + c_{2} \frac{|ε_{1}|}{c_{0}C′_{H}|z|^{3/2}} .

However, our hypothesis (1.3) precisely says that the left hand side of this must be greater than c_{0}. Taking C′_{H} large enough (depending on c_{0} only), we may assume \frac{6}{(c_{0}C′_{H})^{2}} < \frac{c_{0}}{2} , so that by taking |ε| small enough (depending on |z|), we obtain a contradiction, q.e.d.

□

4. The remaining case

Proof of Theorem 1.3, part (1), when lim_{ε_{1},ε_{2}→0} \frac{ε_{1}}{ε_{2}} = −\frac{z_{1}}{z_{2}} .

We shall reuse the notation μ = z_{1}/z_{2}, and reduce ourselves to the case |μ| ≤ 1. The hypothesis implies that ε_{2} = −(μ − γ)ε_{1}, where γ = γ(ε) tends to 0 as ε tends to 0.

This time, instead of constructing an explicit map satisfying an approximate version of our interpolation problem, we will provide values of ζ_{0}, ζ_{1} and ζ_{2} such that the conditions (3.7) and (3.8) are satisfied. From now on we take ζ_{0} = 1/2, ζ_{1} = ζ_{0} + ξ, ζ_{2} = ζ_{1} + ξ′, where

ξ = C_{1}z_{1}, ξ′ = \frac{ε_{1}}{z_{1}} \frac{ζ_{0}}{ζ_{1}} \frac{1 − |ζ_{1}|^{2}}{1 − ζ_{0}ζ̄_{1}} ξ,

and C_{1} = 40.

Standing assumptions.

Throughout, we will assume that |z_{1}| is small enough so that |ξ| ≤ 1/4, so that 1/4 ≤ Reζ_{1} ≤ |ζ_{1}| ≤ 3/4. This implies that 1/2 ≤ |1 − ζ_{0}ζ̄_{1}| ≤ 1, and therefore

(4.1)  |ξ′| ≤ 4|ε_{1}|\frac{|ξ|}{|z_{1}|} ≤ \frac{1}{2}|ξ|,

for ε small enough (depending on z_{1}). Also note that |ξ′| ≤ |ε_{1}|/10. The estimate (4.1) implies that |ξ + ξ′| ≥ \frac{1}{2}|ξ|, and also that

\frac{1}{2} ≤ |1 − ζ_{0}ζ̄_{2}| ≤ 1, \frac{1}{4}|1 − ζ_{2}ζ̄_{1}| ≤ 1.

Now we compute the invariant distances between the ζ_{j}’s. We have

(4.2) ϕ_{ζ_{1}}(ζ_{0}) = \frac{ξ}{1 − ζ_{0}ζ̄_{1}} , ϕ_{ζ_{2}}(ζ_{0}) = \frac{ξ + ξ′}{1 − ζ_{0}ζ̄_{2}} ,

ϕ_{ζ_{2}}(ζ_{1}) = \frac{ξ′}{1 − ζ_{1}ζ̄_{2}} , ϕ_{ζ_{1}}(ζ_{2}) = \frac{−ξ′}{1 − ζ_{2}ζ̄_{1}} .

This implies in particular that

(4.3)  |ϕ_{ζ_{1}}(ζ_{0})| ≥ |ξ| = C_{1}|z_{1}|, |ϕ_{ζ_{2}}(ζ_{0})| ≥ |ξ′| ≥ \frac{1}{2}|ξ| = \frac{C_{1}}{2} |z_{1}|.

Then

log |ζ_{0}| + log |ϕ_{ζ_{0}}(ζ_{1})| + log |ϕ_{ζ_{0}}(ζ_{2})|

= − log 2 + log 2|ξ| + log 4|ξ + ξ′| ≤ 2 log |z_{1}| + C = 2 log |z| + C.

To prove that the above choice of ζ_{j}’s allows for a map passing through the required points, we will show that, for small enough |ε|, the conditions (3.7) and (3.8) are satisfied. To see this, we need to compute the quantities w_{j}, using (4.2).

(4.4) w_{1} = \frac{ε_{1}}{ζ_{1}ϕ_{ζ_{2}}(ζ_{1})} = \frac{ε_{1}}{ζ_{1}} \frac{1 − ζ_{1}ζ̄_{2}}{ξ′} = \frac{z_{1}}{ζ_{0}} \frac{1 − ζ_{1}ζ̄_{2}}{1 − |ζ_{1}|^{2}} \frac{1 − ζ_{0}ζ̄_{1}}{ξ} ,
(4.5) w_{2} = \frac{z_{1}}{ζ_{0}ϕ_{ζ_{2}}(ζ_{0})} = \frac{z_{1}}{ζ_{0}} \frac{1 − ζ_{0}ζ̄_{2}}{ξ + ξ′} ,
(4.6) w_{3} = \frac{z_{2}}{ζ_{0}ϕ_{ζ_{1}}(ζ_{0})} = μ\frac{z_{1}}{ζ_{0}} \frac{1 − ζ_{0}ζ̄_{1}}{ξ} ,
(4.7) w_{4} = \frac{ε_{2}}{ζ_{2}ϕ_{ζ_{1}}(ζ_{2})} = \frac{(μ − γ)ε_{1}}{ζ_{2}} \frac{1 − ζ_{2}ζ̄_{1}}{ξ′}
= (μ − γ)\frac{z_{1}}{ζ_{0}} \frac{ζ_{1}}{ζ_{2}} \frac{1 − ζ_{2}ζ̄_{1}}{1 − |ζ_{1}|^{2}} \frac{1 − ζ_{0}ζ̄_{1}}{ξ} .

We need to see that all those w_{j} lie in the unit disk. It follows from the Standing assumptions above that

(4.8)  |w_{1}| ≤ |\frac{z_{1}}{ξ} | \frac{1}{(1 − |ζ_{1}|^{2})ζ_{0}} ≤ \frac{5}{C_{1}} = \frac{1}{8},
(4.9)  |w_{2}| ≤ |\frac{z_{1}}{ξ} | \frac{2}{ζ_{0}} ≤ \frac{4}{C_{1}} = \frac{1}{10},
(4.10)  |w_{3}| ≤ |\frac{z_{1}}{ξ} | \frac{1}{ζ_{0}} ≤ \frac{2}{C_{1}} = \frac{1}{20},
(4.11)  |w_{4}| ≤ |μ − γ| |\frac{z_{1}}{ξ} | \frac{2}{(1 − |ζ_{1}|^{2})ζ_{0}} ≤ \frac{20}{C_{1}} = \frac{1}{2},

where the last inequality holds when we assume ε small enough so that |γ| ≤ 1. It also follows from the above estimates that |1−w_{1}w̄_{2}| ≥ 1/2, |1 − w_{3}w̄_{4}| ≥ 1/2, so that d_{G}(w_{1}, w_{2}) ≤ 2|w_{1} − w_{2}| and d_{G}(w_{3}, w_{4}) ≤ 2|w_{3} − w_{4}|. We proceed to the computation of those Euclidean distances, which we will compare to the estimates (4.3).

w_{2} − w_{1} = \frac{z_{1}}{ζ_{0}} [(1 − ζ_{0}ζ̄_{2}) (\frac{1}{ξ + ξ′} − \frac{1}{ξ}) +

\frac{1}{ξ} ((1 − ζ_{0}ζ̄_{2}) − (1 − ζ_{0}ζ̄_{1})) + \frac{1 − ζ_{0}ζ̄_{1}}{ξ} (1 − \frac{1 − ζ_{1}ζ̄_{2}}{1 − |ζ_{1}|^{2}} )] .

Each term is estimated by

|\frac{1}{ξ + ξ′} − \frac{1}{ξ}| = |\frac{ξ′}{ξ(ξ + ξ′)}| ≤ \frac{8|ε_{1}|}{C_{1}|z_{1}|^{2}} ,
|(1 − ζ_{0}ζ̄_{2}) − (1 − ζ_{0}ζ̄_{1})| = |ξ′ζ_{0}| ≤ 2|ε_{1}|C_{1},
|1 − \frac{1 − ζ_{1}ζ̄_{2}}{1 − |ζ_{1}|^{2}} | = \frac{|ξ′ζ_{1}|}{1 − |ζ_{1}|^{2}} ≤ 2|ξ′| ≤ 8|ε_{1}|C_{1},

so |w_{2} − w_{1}| ≤ 40|ε_{1}|C_{1}/|z_{1}|.

Similarly,

w_{3} − w_{4} = μ\frac{z_{1}}{ζ_{0}} \frac{1 − ζ_{0}ζ̄_{1}}{ξ} [1 − \frac{ζ_{1}}{ζ_{2}} + \frac{ζ_{1}}{ζ_{2}} (1 − \frac{1 − ζ_{1}ζ̄_{2}}{1 − |ζ_{1}|^{2}} )]

+ γ \frac{z_{1}}{ζ_{0}} \frac{ζ_{1}}{ζ_{2}} \frac{1 − ζ_{1}ζ̄_{2}}{1 − |ζ_{1}|^{2}} \frac{1 − ζ_{0}ζ̄_{1}}{ξ} .

Using the fact that |1 − \frac{ζ_{1}}{ζ_{2}} | ≤ 8|ξ′|, we find

|w_{3} − w_{4}| ≤ 96|ε_{1}| + 12|γ|/C_{1}.

It is then easy to see, using (4.3), that (3.7) and (3.8) can be verified whenever we choose ε_{1} small enough (and therefore |γ| small enough), depending on |z_{1}|. □

References

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2. D. Coman, The pluricomplex Green function with two poles of the unit ball of ℂ^{n}, Pacific J. Math. 194, no 2, 257–283 (2000).

3. A. Edigarian, Remarks on the pluricomplex Green function, Univ. Iagel. Acta Math. 37 (1999), 159–164.

4. J. B. Garnett, Bounded Analytic Functions, Academic Press, Inc. New York-London, 1981.

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Pascal J. Thomas Laboratoire Emile Picard, UMR CNRS 5580 Université Paul Sabatier 118 Route de Narbonne F-31062 TOULOUSE CEDEX France pthomas@cict.fr