PLURICOMPLEX GREEN AND LEMPERT FUNCTIONS FOR EQUALLY WEIGHTED POLES

PASCAL J. THOMAS NGUYEN VAN TRAO

1. Introduction

The pluricomplex Green function is an important tool of several variable complex analysis; in particular it provides a fundamental solution for the complex Monge-Ampère equation and information about the complex geometry of domains [10] (see [7] for an exposition of pluricomplex potential theory). For n ≥ 2, the complex Monge-Ampère equation is nonlinear, so studying the several-pole analogue of the Green function (introduced in [8]) is no easy task, see [4], and [1] or [3] for some of the few cases where it can be explicitly computed.

Let Ω be a domain in ℂ^{n}, and poles and weights denoted by

S = {(a_{1}, ν_{1}); ...; (a_{N}, ν_{N})} ⊂ Ω × ℝ_{+},

where ℝ_{+} = [0, +∞). The pluricomplex Green function is defined by

G_{S}(z) := sup {u(z) : u ∈ P SH_{−}(Ω), u(x) ≤ ν_{j} log ∥x − a_{j}∥ + C_{j} when x → a_{j}, j = 1, ..., N} .

Note that if N = 1 we might as well take ν_{1} = 1, G_{S} is the pluricomplex Green function with one pole.

We also recall the definition of Coman’s Lempert function [4]:

(1.1) ℓ_{S}(z) := inf{∑_{j=1}^{N} ν_{j} log |ζ_{j}| : φ(0) = z, φ(ζ_{j}) = a_{j}, j = 1, ..., N

for some φ ∈ O(𝔻, Ω)},

where 𝔻 is the unit disc in ℂ.

It is easy to see that ℓ_{S}(z) ≥ G_{S}(z) for all z ∈ Ω.

A remarkable theorem of Lempert [10] says that equality holds in the case where Ω is convex and N = 1. Later Coman [4] proved with considerable effort that this assertion also holds when Ω is the unit ball, N = 2, and the weights are equal. At the same time he conjectured that the equality might hold for any number of points and any convex domain in ℂ^{n}. Recently, Carlehed and Wiegerinck [2], [3] proved that Coman’s conjecture fails for the bidisc, with two poles lying on a coordinate axis and distinct weights. The main goal of this paper is to prove that Coman’s conjecture does not even hold in the case when all weights are equal.

Weights on the Green function are analogous to multiplicities for zeros. Since the work of Carlehed and Wiegerinck [2], [3] uses weights greater than 1, we focus on the behavior of Coman’s Lempert function with many poles when some group of poles tend to the same pole. Eventually, a counterexample is obtained Section 5 of this paper with the domain equal to the bidisc, and four poles at (a, 0), (b, 0), (b, ε), (a, ε), with ε small enough (Theorem 5.1). As is [3], one can deduce from this that the Coman conjecture fails for strictly convex smoothly bounded domains which are close enough to the bidisc.

Along the way, we need to introduce more general notions of Lempert functions, coming from generalizations of the Green function. The reason is as follows : when we consider the pluricomplex Green function as a fundamental solution for the complex Monge-Ampère operator in several variables, the quantity which we expect to see being preserved under convergence of poles to a single point is the total Monge-Ampère mass of the function, which is equal to ∑_{j} ν_{j}^{n} for a Green function with weights ν_{j}. Thus, when a group of N simple poles, with N ≠ a^{n} for any integer a, clusters to a single point, we cannot hope to have a usual weighted Green function arise as limit value for the sequence of Green functions for the separate poles. Simple examples yield explicit non-isotropic functions, i.e. which are not equivalent to constant multiples of the logarithm of a norm (see Lemma 2.6). From this point of view, weighted Green functions (with integer weights) are a special case among the possible limit values of unweighted Green functions.

Lelong and Rashkovskii [9] introduced a generalized pluricomplex Green function with several poles (see the definition in Section 2), which allows for non-isotropic singularities. We then study the problem of producing an analogous generalization of the case ν_{j} = 1 for Coman’s Lempert function (which may differ from Coman’s definition for the case ν_{j} ∈ ℤ_{+}, ν_{j} > 1). A motivation is that we know that Coman’s Lempert function is continuous with respect to z and to its poles when they stay away from each other (see [13] for the case of the ball), and we would like to extend such results to singular situations arising from ”collisions” of poles.

1

Unfortunately, this was not fully successful, since our best candidate (see Definition 3.6) is not in general the limit of the Lempert functions for the natural systems of points which tend to the given ”multiple poles” (see [14, Theorem 6.3]). However, we gather enough information to prove that in the four-point cases mentioned above, equality does not hold between the Lempert and Green functions.

Along the way to our counterexample, we give partial answers. There is equality between Lelong and Rashkovskii’s Green function and our first generalization of Coman’s Lempert function in the case of one pole, in the polydisc, with a simple enough singularity (Lemma 2.6; some hypothesis about integer multiplicities is of course necessary). We also prove equality between Lempert and Green functions in the case of the bidisc in ℂ^{2}, when all poles are on the first coordinate disc and all multiplicities equal to one ; also, our first generalization of the Lempert function provides a natural limit when poles collide along the first coordinate disc, producing ”horizontal” non-isotropic singularities, and this is still equal to the appropriate generalized Green function (this is made precise in Theorem 4.1).

The organization of the present paper is as follows : in Section 2, we give notations and definitions, introduce our generalization of the Lempert function and give Lemma 2.6 as a first motivation of this particular definition. In Section 3, we generalize to this new Lempert functions some of the results of [17]; the proofs we give are restricted to the particular cases which do occur in the examples below. Section 4 includes Theorem 4.1 and provides a few negative examples in the bidisc, the latter motivating Definition 3.6, which amends our first generalization of Coman’s Lempert function. Finally, the counterexample is proven in Section 5.

A longer version of this paper, with the proofs of some additional facts about our generalization of the Lempert function, is available as a preprint [14] and forms part of the second-named author’s Ph. D. dissertation (Ha Noi, Viet Nam, november 2002). Acknowledgements. The results of this paper were obtained in part during a stay of the second-named author at the Paul Sabatier university. He would like to thank Professor Do Duc Thai for stimulating discussions regarding this paper, the program FORMATH Vietnam (and in particular Professors Nguyen Thanh Van and Frédédric Pham) for the invitation and financial support, and the Emile Picard laboratory for hospitality. The first-named author thanks Stéphanie Nivoche and Evgueny Poletsky for interesting discussions concerning the topic, and also the latter for his hospitality in Syracuse. We thank the referee for his careful reading of the paper and his helpful suggestions.

2. Definitions

Definition 2.1. [9] We will say that Ψ ∈ P SH_{−}(𝔻^{n}) is an indicator (centered at 0) if and only if

Ψ(z_{1}, …, z_{n}) = g(log |z_{1}|, …, log |z_{n}|),

where g is a convex continuous nonpositive valued function defined on (ℝ_{−})^{n}, increasing with respect to each single variable, and positively homogeneous of degree 1: g(λx_{1}, …, λx_{n}) = λg(x_{1}, …, x_{n}), for any λ > 0.

This can be introduced in a less ad hoc way, see [9].

By [9], if Ψ is an indicator, it is a multiple of a fundamental solution to the complex Monge-Ampère equation, that is, there exists τ ≥ 0 such that

(dd^{c}Ψ(· − a))^{n} = τδ(a),

where δ(a) stands for the unit mass at the point a ∈ ℂ^{n}.

Let us fix the system S := {(a_{j}, Ψ_{j})}, 1 ≤ j ≤ N, where a_{j} ∈ Ω, 1 ≤ j ≤ N and Ψ_{j} are indicators.

Definition 2.2. The generalized Green function [9] is given by

G_{S}(z) := sup{u(z) : u ∈ P SH_{−}(Ω), u(x) ≤ Ψ_{j}(x − a_{j}) + C_{j}, 1 ≤ j ≤ N},

where the inequalities are required only for x belonging to a neighborhood of each a_{j}.

Remark 2.3. If Ω is a hyperconvex domain in ℂ^{n}, then Lelong and Rashkovskii [9] also showed that the Green function is the unique solution of the following Dirichlet problem (for short we write G instead of G_{S})

(a) G ∈ P SH_{−}(Ω) ∩ C(\overline{Ω});

(b) G(z) → 0 as z → ∂Ω;

(c) Ψ_{j}(z) = lim_{R→∞} R^{−1}G[a_{j} + (exp(u_{k} + iθ_{k} + R log |z_{k}|))_{1≤k≤n}], 1 ≤ j ≤ N, where the limit exists almost everywhere for x = (u_{k} + iθ_{k})_{1≤k≤n} and doesn’t depend on x;

(d) (dd^{c}G)^{n} = ∑_{j=1}^{N} τ_{j}δ(a_{j}).

We now introduce a new generalization of the Lempert function.

Definition 2.4.

L_{S}(z) := inf{∑_{j=1}^{N} τ_{j} log |ζ_{j}| : ∃φ ∈ O(𝔻, Ω), φ(0) = z,

∃U_{j} a neighborhood of ζ_{j} : Ψ_{j}(φ(ζ) − a_{j}) ≤ τ_{j} log |ζ − ζ_{j}| + C_{j}, ∀ζ ∈ U_{j}, 1 ≤ j ≤ N}.

Note that for the non-trivial case where τ_{j} ≠ 0, the conditions imposed on the map φ force φ(ζ_{j}) = a_{j}. In the basic case where Ψ_{j}(z) = log |z| for each j, we will have τ_{j} = 1 for each j, and we simply find the usual Lempert function ℓ_{S} with simple poles (ν_{j} = 1 for each j), since analytic maps are locally Lipschitz. But for N > 1, Ψ_{j}(z) = ν_{j} log |z| with some ν_{j} > 1 (and τ_{j} = ν_{j}^{n}), this is not a priori the same as the ℓ_{S} given in the Introduction (although we do not know of any example to exhibit this phenomenon, and do not know of any general inequality between the two functions).

Lemma 2.5. G_{S}(z) ≤ L_{S}(z), for any z ∈ Ω.

Proof. If φ : 𝔻 → Ω is an analytic disc in Ω, with φ(0) = z, φ(ζ_{j}) = a_{j}, 1 ≤ j ≤ N and Ψ_{j} ∘ φ(ζ) ≤ τ_{j} log |ζ − ζ_{j}| + C_{j}, 1 ≤ j ≤ N, then G_{S} ∘ φ is a subharmonic function on 𝔻, G_{S} ∘ φ is negative and

G_{S} ∘ φ(ζ) ≤ C_{j} + Ψ_{j} ∘ φ(ζ) ≤ C′_{j} + τ_{j} log |ζ − ζ_{j}|, 1 ≤ j ≤ N.

Thus G_{S} ∘ φ is a member in the defining family for the Green function on 𝔻 with poles ζ_{j} and weights τ_{j}, and hence,

G_{S} ∘ φ(ζ) ≤ ∑_{j=1}^{N} τ_{j} log \frac{|ζ_{j} − ζ|}{|1 − ζ\overline{ζ_{j}}|}.

It implies that

G_{S}(z) = G_{S} ∘ φ(0) ≤ ∑_{j=1}^{N} τ_{j} log |ζ_{j}|.

Thus G_{S}(z) ≤ L_{S}(z), for all z ∈ Ω. □

Recall (see e.g. [6], [12]) that the involutive Möbius map of 𝔻 which exchanges ξ ∈ 𝔻 and 0 is given by the following formula:

(2.1)  ϕ_{ξ}(ζ) := \frac{ξ − ζ}{1 − \overline{ξ}ζ} .

Therefore it is no loss of generality, in the case of a single pole a, to reduce ourselves to a = 0.

Lemma 2.6. Let Ω be the polydisc 𝔻^{n} in ℂ^{n}. If S has only one pole at (0, 0), and the indicator Ψ is of the following simple kind

Ψ(z) = max_{1≤j≤n} c_{j} log |z_{j}|,

where the numbers c_{j} are positive integers, then L_{S}(z) = G_{S}(z) = Ψ(z), for any z ∈ 𝔻^{n}.

Proof. By verifying the Dirichlet problem given by Lelong and Rashkovskii [9], we have

G_{S}(z) = max_{1≤j≤n} c_{j} log |z_{j}|.

We may assume that max_{1≤j≤n} c_{j} log |z_{j}| = c_{j_{0}} log |z_{j_{0}}| for some 1 ≤ j_{0} ≤ n. With this assumption we have G_{S}(z) = c_{j_{0}} log |z_{j_{0}}|. To prove the Lemma, it suffices to show that there exists a mapping φ ∈ O(𝔻, 𝔻^{n}) and ζ_{0} ∈ 𝔻 such that

(1) φ(0) = z,

(2) φ(ζ_{0}) = 0,

(3) Ψ ∘ φ(ζ) ≤ m log |ζ − ζ_{0}| + C, ∀ζ ∈ 𝔻, where m := ∏_{j=1}^{n} c_{j} = is the total mass of (dd^{c}Ψ)^{n},

(4) m log |ζ_{0}| = c_{j_{0}} log |z_{j_{0}}|.

The condition (3) can be rewritten as follows

(3’) φ_{j}^{(k)}(ζ_{0}) = 0, 1 ≤ k ≤ m_{j} − 1, 1 ≤ j ≤ n, where m_{j} := m/c_{j}.

We fulfill condition (4) by picking ζ_{0} ∈ 𝔻 such that

|ζ_{0}|^{m_{j_{0}}} = |z_{j_{0}}|

and put

φ_{j}(ζ) := [ϕ_{ζ_{0}}(ζ)]^{m_{j}}h_{j}(ϕ_{ζ_{0}}(ζ)), ∀ζ ∈ 𝔻, 1 ≤ j ≤ n

where h_{j} : 𝔻 → \overline{𝔻} is such that h_{j}(ζ_{0}) = \frac{z_{j}}{ζ_{0}^{m_{j}}} , 1 ≤ j ≤ n.

Then the function φ = (φ_{1}, ⋯, φ_{n}) and ζ_{0} satisfy all properties (1), (2), (3’) and (4). □

3. Existence of extremal discs

We now extend to this new Lempert function some known properties of its usual counterpart. The following generalizes [17, Theorem 2.4, p. 1054], or in the case of the unit ball [13, Proposition 3, p. 338] (see also [16, Papers V and VI]).

Proposition 3.1. Let Ω be a convex domain and S := {(a_{j}, Ψ_{j}) : 1 ≤ j ≤ N} and S′ := {(a_{j}, Ψ_{j}) : 1 ≤ j ≤ N − 1} where a_{j} ∈ Ω and Ψ_{j} are indicators centered at a_{j}. Then

L_{S}(z) ≤ L_{S′}(z), for all z ∈ Ω.

The proof of this proposition can be found in [14, Section 4]. Since that full proof is elementary and rather tedious, the one given below restricts itself to the special case where n = 2 and the indicators are of the type used in Lemma 2.6, with 1 ≤ c_{j} ≤ 2.

Proof of Proposition 3.1 This proof adapts the ideas of [17], [13, Proposition 3], and [15, Theorem 2.7]. Given any δ > 0, there exists a holomorphic map φ from the disc to Ω and points

ζ_{j}^{0} ∈ 𝔻, 1 ≤ j ≤ N − 1, such that φ(z) = 0,

L_{S′}(z) ≤ ∑_{j=1}^{N−1} τ_{j} log |ζ_{j}^{0}| ≤ L_{S′}(z) + δ,

and Ψ_{j} ∘ φ(ζ) ≤ τ_{j} log |ζ − ζ_{j}^{0}| + C_{j}, 1 ≤ j ≤ N − 1. Let r < 1, to be specified later.

We set φ^{r}(ζ) := φ(rζ). If r > max |ζ_{j}^{0}|, 1 ≤ j ≤ N − 1, we have \frac{ζ_{j}^{0}}{r} ∈ 𝔻 and

φ^{r}(\frac{ζ_{j}^{0}}{r}) = a_{j}, 1 ≤ j ≤ N − 1,

and more generally

(3.1) Ψ_{j} ∘ φ^{r}(ζ) ≤ τ_{j} log |r(ζ − \frac{ζ_{j}^{0}}{r})| + C_{j} ≤ τ_{j} log |(ζ − \frac{ζ_{j}^{0}}{r})| + C_{j}, 1 ≤ j ≤ N − 1.

We will introduce a correcting term to ensure that the same property hold for j = N, without destroying it for j ≤ N − 1.

Let K denote the convex hull of φ^{r}(\overline{𝔻}) ∪ \{a_{N}\}. Since φ^{r}(\overline{𝔻}) ∪ \{(a, 0)\} ⊂⊂ Ω, we can find an ε > 0 such that the distance between K and ∂Ω is at least εM_{1} where M_{1} := sup_{r\overline{𝔻}} |a_{N} − φ|.

Lemma 3.2. Given any m ∈ ℕ^{∗}, there exists h a holomorphic function on 𝔻 and some ζ^{∗} ∈ 𝔻 satisfying

• h(𝔻) ⊂ U_{ε} := ∪_{x∈[0,1]}D(x, ε),

• h(0) = 0,

• h(\frac{ζ_{j}^{0}}{r}) = h′(\frac{ζ_{j}^{0}}{r}) = 0, 1 ≤ j ≤ N − 1,

• h(ζ^{∗}) = 1, and h′(ζ^{∗}) = 0.

Accepting this lemma temporarily, define

φ̃(ζ) = φ^{r}(ζ) + h(ζ)(a_{N} − φ^{r}(ζ)).

The definition of ε and the first condition above show that φ̃(𝔻) ⊂ Ω. Clearly, φ̃(0) = z.

We have h(ζ) = O((ζ − \frac{ζ_{j}^{0}}{r})^{2}) for 1 ≤ j ≤ N − 1, so that the conditions (3.1), which reduce under our restrictive hypotheses to the vanishing of the derivatives of certain coordinate functions of φ^{r}, still hold for φ̃.

Finally, one also checks that φ̃(ζ) = a_{N} +(h(ζ)−1)(a_{N} −φ^{r}(ζ)) = a_{N} +O((ζ −ζ^{∗})^{2}), which will imply Ψ_{N} ∘φ̃(ζ) ≤ τ_{N} log |ζ −ζ^{∗}|+C_{N}. For the mapping φ̃, the logarithmic sum of the preimages yields

∑_{j=1}^{N−1} log |\frac{ζ_{j}^{0}}{r} | + log |ζ^{∗}| ≤ ∑_{j=1}^{N−1} log |ζ_{j}^{0}| + (N − 1) log \frac{1}{r} ≤ L_{S′}(z) + δ + (N − 1) log \frac{1}{r}.

Since this construction can be carried out for any r arbitrarily close to 1, we have L_{S}(z) ≤ L_{S′}(z). □

Proof of Lemma 3.2. Let ρ be a Riemann map from 𝔻 to U_{ε} so that ρ(0) = 0. We look for h under the form h = ρ ∘ h_{1}, where h_{1} is a holomorphic map from 𝔻 to itself such that

• h_{1}(0) = 0,

• h_{1}(\frac{ζ_{j}^{0}}{r}) = h′_{1}(\frac{ζ_{j}^{0}}{r}) = 0, 1 ≤ j ≤ N − 1, and

• there exists ζ^{∗} ∈ 𝔻 such that h_{1}(ζ^{∗}) = ρ^{−1}(1) and h′_{1}(ζ^{∗}) = 0. Let B_{0} be the finite Blaschke product with a single zero at the origin and double

zeroes at the points \frac{ζ_{j}^{0}}{r}, 1 ≤ j ≤ N − 1, and look for h_{1} under the form h_{1} = B_{0}g, where g is holomorphic and bounded by 1 in modulus on the unit disc. For ease of notation, write γ := ρ^{−1}(1) ∈ 𝔻.

The function h_{1} will fulfill the above conditions if and only if

g(ζ^{∗}) = \frac{γ}{B_{0}(ζ^{∗})}, g′(ζ^{∗}) = −g(ζ^{∗})\frac{B′_{0}(ζ^{∗})}{B_{0}(ζ^{∗})} = −γ \frac{B′_{0}(ζ^{∗})}{B_{0}^{2}(ζ^{∗})}.

By the Schwarz-Pick lemma, such a function can be found if and only if |g′(ζ^{∗})| ≤ (1 − |g(ζ^{∗})|^{2})/(1 − |ζ^{∗}|^{2}), i.e.

(1 − |ζ^{∗}|^{2})\frac{|B′_{0}(ζ^{∗})|}{|B_{0}^{2}(ζ^{∗})|} ≤ \frac{1 − |γ|^{2}|B_{0}(ζ^{∗})|^{−2}}{|γ|} .

Since |γ| is fixed and lim_{ζ→1} |B_{0}(ζ)| = 1, lim_{ζ→1} |B′_{0}(ζ)| = |B′_{0}(1)| < ∞, this is achieved for ζ^{∗} close enough to 1. □

We will use the shorthand S′ ⊂ S to mean that the sets of poles are included as noted, and that the indicators remain the same for all points of the smaller set, as in Proposition 3.1.

Proposition 3.3. Let Ω be a bounded taut domain, and S = {(a_{j}, Ψ_{j})}_{j=1,..,N}, N ≥ 2. If L_{S}(z) is not attained by any analytic disc, then

L_{S}(z) ≥ min_{S′⊊S} L_{S′}(z).

In particular, if Ω is convex and bounded, the conclusion becomes

L_{S}(z) = min_{S′⊊S} L_{S′}(z).

Proof. The proof of this Proposition is adapted from that of [17, Theorem 2.2, p. 1053].

Take a sequence of analytic discs φ^{k}, where

φ^{k}(0) = z and Ψ_{j} ∘ φ^{k}(ζ) ≤ τ_{j} log |ζ − ζ_{j}^{k}| + C_{j}^{k}, ∀ζ ∈ 𝔻, k ≥ 1, 1 ≤ j ≤ N

such that ∑_{j=1}^{N} τ_{j} log |ζ_{j}^{k}| converges to L_{S}(z), as k tends to 0.

By passing to a subsequence, using that Ω is taut, we may assume that φ^{k} converges locally uniformly to some φ ∈ O(𝔻, Ω). Also (if necessary, by passing to a subsequence again), we may assume that ζ_{j}^{k} → ζ_{j} ∈ \overline{𝔻}, for each 1 ≤ j ≤ N, as k → ∞.

We need to see that for each ζ_{j} ∈ 𝔻,

(3.2)  Ψ_{j} ∘ φ(ζ) ≤ τ_{j} log |ζ − ζ_{j}| + C_{j}, for ζ in a neighborhood of ζ_{j}.

Recall (from [9]) that Ψ being an indicator (centered at 0) means that

Ψ(z_{1}, …, z_{n}) = g(log |z_{1}|, …, log |z_{n}|),

where g is a convex continuous nonpositive valued function defined on (ℝ_{−})^{n}, increasing with respect to each single variable, and positively homogeneous of degree 1: g(λx_{1}, …, λx_{n}) = λg(x_{1}, …, x_{n}), for any λ > 0.

We study the situation for a fixed pole a_{j}. We must have for each k ≥ 0,

φ^{k}(ζ_{j}^{k} + h) = (φ_{l}^{k}(ζ_{j}^{k} + h), 1 ≤ l ≤ n) = (α_{k,l}h^{m_{k,l}} + O(|h|^{m_{k,l}+1}), 1 ≤ l ≤ n).

From the above expression,

Ψ_{j}(φ^{k}(ζ_{j}^{k} + h)) = g (−m_{k,l} + \frac{log |α_{k,l}| + O(h)}{| log |h||}) log |h|,

so the conditions on φ^{k} imply that

(3.3)  g(−m^{k}) ≤ τ_{j}, where m^{k} := (m_{k,1}, …, m_{k,n}).

Passing to a subsequence if needed, we may assume that m^{k} → m =: (m_{1}, …, m_{n}) ∈ (ℕ ∪ {∞})^{n}. The uniform convergence on compacta of the sequence φ^{k} implies that of all derivatives, and that in the limit φ_{l}^{(q)}(ζ_{j}) = 0 for q ≤ m_{l} − 1. This, together with (3.3), proves (3.2).

If no ζ_{j} ∈ ∂𝔻, φ is an analytic disc attaining the infimum in the definition of L_{S}(z). That is excluded by our hypothesis. Otherwise, assume after renumbering the coordinates that ζ_{j} ∈ 𝔻, 1 ≤ j ≤ M and ζ_{j} ∈ ∂𝔻 for M + 1 ≤ j ≤ N. (Note that not every ζ_{j} can be in ∂𝔻, as this would imply that L_{S}(z) = 0.) Then φ is a member in the defining family for L_{S′}, where S′ := {(a_{j}, Ψ_{j})}_{j=1,..,M}, and thus L_{S}(z) ≥ L_{S′}(z). □

Corollary 3.4. Let Ω be a bounded taut domain in ℂ^{n}, and let S be as above. Then for every z ∈ Ω there exists an analytic disc φ, such that φ(0) = z, passing through a (non empty) S_{0} ⊂ S such that φ attains the infimum in the definition of L_{S_{0}}(z), and L_{S_{0}}(z) = min_{∅≠S′⊂S} L_{S′}(z).

Proof. If S is a singleton, a normal family argument close to the one used in the previous proof will show that the corollary is true for this case.

Otherwise, by the previous proposition, either there is an analytic attaining the infimum, or min_{∅≠S′⊂S} L_{S′}(z) = L_{S_{0}}(z) for some proper subset S_{0} ⊂ S, and L_{S_{0}}(z) is attained by an analytic disc passing though z and the points in S_{0} (otherwise one could pass to a still smaller subset). □

As the consequence of Corollary 3.4 and Proposition 3.1 we have the following.

Theorem 3.5. Let Ω be a bounded convex domain, then the infimum in the definition of the function L_{S} is attained by an extremal disc that passes through a (non-empty) subset S′ ⊂ S (possibly the whole system S).

However, it would be natural to consider as well the more general case of the relationship between the Lempert functions of two systems S := {(a_{j}, Ψ_{j}) : 1 ≤ j ≤ N} and S′ := {(a_{j}, Ψ′_{j}) : 1 ≤ j ≤ N}, where Ψ_{j} ≤ Ψ′_{j}, for any 1 ≤ j ≤ N (S′ ⊂ S corresponds to the case where the Ψ′_{j} have τ_{j} = 0 for a_{j} outside the pole set of S′). Unfortunately, our generalized Lempert function is not in general monotone when we compare two such generalized pole sets, see a counter-example below (Proposition 4.2). We therefore introduce a corrected Lempert function L̃.

Definition 3.6. Let S := {(a_{j}, Ψ_{j}) : 1 ≤ j ≤ N} and S_{1} := {(a_{j}, Ψ_{j}^{1}) : 1 ≤ j ≤ N} where a_{j} ∈ Ω and Ψ_{j}, Ψ_{j}^{1} are indicators. We define

L̃_{S}(z) := inf{L_{S^{1}}(z) : Ψ_{j}^{1} ≥ Ψ_{j} + C_{j}, 1 ≤ j ≤ N}.

Lemma 3.7. G_{S}(z) ≤ L̃_{S}(z) ≤ L_{S}(z).

Proof. The fact that L̃_{S}(z) ≤ L_{S}(z) follows from the definition. For any S_{1} as in the definition, L_{S^{1}}(z) ≥ G_{S^{1}}(z) ≥ G_{S}(z), as follows from Lemma 2.5 and the definition of the pluricomplex Green function. □

In the situation related to the example in Proposition 4.2 where two fixed poles a_{1}, a_{2} lie on a coordinate axis, a_{3} lies on a line orthogonal to this axis at a_{1}, and a_{3} tends to a_{1}, then the limit of the ordinary Lempert functions is given by an L̃_{S}, and not by the corresponding L_{S} (the limit of the corresponding Green functions is not known in this case). A precise statement and a proof can be found in [14, Theorem 5.5]. However, there are other examples where L̃ also fails to be the limit of the Lempert functions for single poles [14, Theorem 6.3].

4. Examples in the bidisc

First, we would like to give one case where the Green function with several poles and indicator singularities is equal to its generalized Lempert counterpart. This is analogous in spirit to the result of Carlehed and Wiegerinck about the Green function with several poles in the bidisc [1], [3] (but easier).

Theorem 4.1. Let Ψ_{m}(z) = max {m log |z_{1}|; log |z_{2}|}, for any m ∈ ℕ^{∗}. Let a_{1}, a_{2}, …, a_{N} ∈ 𝔻, and

S := {((a_{1}, 0); Ψ_{m_{1}}); …, ((a_{N}, 0); Ψ_{m_{N}})}.

Then for any z ∈ 𝔻^{2},

L_{S}(z) = G_{S}(z) = max{∑_{j=1}^{N} m_{j} log |ϕ_{a_{j}}(z_{1})|; log |z_{2}|}.

As a consequence, if a_{j,i}^{(k)} ∈ 𝔻, 1 ≤ j ≤ N, 1 ≤ i ≤ m_{j}, are distinct points which verify

lim_{k→∞} a_{j,i}^{(k)} = a_{j}, , 1 ≤ i ≤ m_{j},

and S^{(k)} the pole system made up of all the a_{j,i}^{(k)} with equal weight 1, then lim_{k→∞} L_{S^{(k)}}(z) = L_{S}(z) and lim_{k→∞} G_{S^{(k)}}(z) = G_{S}(z), for any z ∈ 𝔻^{2}.

Proof. First of all, the Green function has the formula given above. To prove this assertion it suffices to show that the function defined by the right hand side verifies the Dirichlet problem in Remark 2.3. Indeed the conditions (a), (b) and (c) are trivially fulfilled. The last condition follows from the following theorem of Zeriahi [18], [19].

Theorem. For i = 1, 2, let Ω_{i} be an open set in ℂ^{n_{i}}, and u_{i} a locally bounded plurisubharmonic function in Ω_{i}, such that (dd^{c}u_{i})^{n_{i}} = 0 in Ω_{i}. Define v(z_{1}, z_{2}) = max{u_{1}(z_{1}), u_{2}(z_{2})}, n = n_{1} + n_{2}. Then (dd^{c}v)^{n} = 0 in Ω_{1} × Ω_{2}.

By our definition,

L_{S}(z) = inf { ∑_{j=1}^{N} m_{j} log |ζ_{j}| : ∃φ ∈ O(𝔻, 𝔻^{2}),

φ(0) = z, φ_{1}(ζ_{j}) = a_{j}, φ_{2}^{(k)}(ζ_{j}) = 0, 0 ≤ k ≤ m_{j} − 1, 1 ≤ j ≤ N}.

If z_{1} ∈ {a_{1}, …, a_{n}}, say z_{1} = a_{1}, then picking ζ_{1}^{m_{1}} = z_{2} and φ(ζ) = (a_{1}, ζ^{m_{1}}), we see by Proposition 3.1 that

log |z_{2}| = m_{1} log |ζ_{1}| ≥ L_{((a_{1},0),Ψ_{m_{1}})}(z) ≥ L_{S}(z) ≥ G_{S}(z) = log |z_{2}|,

so there is equality throughout.

If z_{1} ∉ {a_{1}, …, a_{n}}, we may reduce ourselves to z = (0, γ) and |a_{1}| ≥ |a_{2}| ≥ ... ≥ |a_{N}| > 0. Then

G_{S}(z) = max{log |a_{1}^{m_{1}} · a_{2}^{m_{2}} ⋯ a_{N}^{m_{N}} |; log |γ|}.

We will use induction on N. When N = 1 the equality follows from Lemma 2.6. Suppose that N > 1 and the theorem is proved for N − 1. We consider three cases.

Case 1. |γ| ≤ |a_{1}^{m_{1}} · a_{2}^{m_{2}} ⋯ a_{N}^{m_{N}} |.

Then G_{S}(z) = log |a_{1}^{m_{1}} · a_{2}^{m_{2}} ⋯ a_{N}^{m_{N}} |. The map

ζ ↦ (ζ, \frac{γ}{a_{1}^{m_{1}} · a_{2}^{m_{2}} ⋯ a_{N}^{m_{N}}} ∏_{j=1}^{N} (\frac{a_{j} − ζ}{1 − \overline{a_{j}}ζ} )^{m_{j}})

verifies all the requirements with ζ_{j} = a_{j}. This implies that G(z) = L(z). Case 2. |γ| ≥ |a_{2}^{m_{2}} ⋯ a_{N}^{m_{N}} |. Then G(z) = log |γ|. Moreover, G(z) is also equal to the Green function G_{1}(z) for

the system with N − 1 poles

S_{1} := {((a_{2}, 0); Ψ_{m_{2}}); …, ((a_{N}, 0); Ψ_{m_{N}})}.

By induction, G_{1} = L_{1}, where L_{1} is the generalized Lempert function with respect to S_{1}. On the other hand, we always have L_{S}(z) ≤ L_{1}(z) by Proposition 3.1. Hence G_{S}(z) = L_{S}(z).

Case 3. |a_{1}^{m_{1}} · a_{2}^{m_{2}} ⋯ a_{N}^{m_{N}} | < |γ| < |a_{2}^{m_{2}} ⋯ a_{N}^{m_{N}} |.

We now show that the G_{S}(z) = log |γ| is also equal to the new Lempert function, and the infimum in the definition of the new Lempert function is attained by an extremal disc φ passing through all poles (a_{1}, 0); (a_{2}, 0); ...; (a_{N}, 0) and z.

Set M := ∑_{j=1}^{N} m_{j} and define r ∈ (0, 1) by

r = \sqrt[M]{\frac{|a_{1}^{m_{1}} · a_{2}^{m_{2}} ⋯ a_{N}^{m_{N}} |}{|γ|}}.

We have, for any 1 ≤ j ≤ N,

|a_{j}|^{M} < |a_{j}|^{m_{1}} ≤ |a_{1}|^{m_{1}} < r^{M}

by the hypothesis on γ. So a_{j}/r ∈ 𝔻. We introduce the map φ : 𝔻 ↦ 𝔻^{2} given by

φ(ζ) = (rζ, e^{iθ} ∏_{j=1}^{N} (\frac{ζ_{j} − ζ}{1 − \overline{ζ_{j}}ζ} )^{m_{j}}),

where ζ_{j} = \frac{a_{j}}{r}, 1 ≤ j ≤ N, and θ is chosen such that

e^{iθ} (\frac{a_{1}}{r} )^{m_{1}} (\frac{a_{2}}{r} )^{m_{2}} ⋯ (\frac{a_{N}}{r} )^{m_{N}} = γ.

It is easy to verify that φ verifies the conditions in the definition of L_{S} and that |ζ_{1}^{m_{1}} · ζ_{2}^{m_{2}} ⋯ ζ_{N}^{m_{N}} | = |γ|. Hence, φ is an extremal disc for the new Lempert function, and G_{S}(z) = L_{S}(z) in this case. □

We will now give some negative results, mainly that the generalized Green function can be different from the generalized Lempert function as given in Definition 2.4.

We shall need some notation, to be used in this section and the next one.

For z ∈ 𝔻^{2}, we will use the following indicators:

(4.1) Ψ_{0}(z) := max(log |z_{1}|, log |z_{2}|), Ψ_{H}(z) := max(2 log |z_{1}|, log |z_{2}|), Ψ_{V} (z) := max(log |z_{1}|, 2 log |z_{2}|).

Here H stands for ”horizontal” and V for ”vertical”, for the obvious reasons : for a ∈ 𝔻^{2}, Ψ_{j}(φ(ζ) − a) ≤ τ_{j} log |ζ − ζ_{0}| + C translates to (τ_{0} = 1, τ_{H} = τ_{V} = 2):

φ(ζ_{0}) = a, when j = 0,
φ(ζ_{0}) = a, φ′_{2}(ζ_{0}) = 0 when j = H,
φ(ζ_{0}) = a, φ′_{1}(ζ_{0}) = 0 when j = V.

For a, b ∈ 𝔻, let

S_{a0} := {((a, 0), Ψ_{0})} = {(a, 0)}
S_{a0b0} := {((a, 0), Ψ_{0}); ((b, 0), Ψ_{0})} = {(a, 0); (b, 0)}
S_{aV} := {((a, 0), Ψ_{V})}
S_{bV} := {((a, 0), Ψ_{V})}
S_{a0bV} := {((a, 0), Ψ_{0}); ((b, 0), Ψ_{V})}
S_{aV bV} := {((a, 0), Ψ_{V}); ((b, 0), Ψ_{V})}.

We will denote with the corresponding subscripts the pertinent Green and Lempert functions, e.g. G_{a0bV}, L_{a0bV}, L̃_{a0bV}, etc. A special case of Theorem 4.1 is that L_{aHb0} = G_{aHb0} for any a and b in the disc, for instance.

We start by giving an example of a situation where L̃_{S}(z) < L_{S}(z), with S = S_{a0bV}.

Proposition 4.2. For z_{1} ∈ 𝔻, L_{a0bV} (z_{1}, 0) > L_{a0b0}(z_{1}, 0), and therefore L_{a0bV} (z_{1}, 0) > L̃_{a0bV} (z_{1}, 0) ≥ G_{a0bV} (z_{1}, 0).

Proof. From the above, L_{a0b0}(z_{1}, 0) = G_{a0b0}(z_{1}, 0) = log |ϕ_{a}(z_{1})| + log |ϕ_{b}(z_{1})|, where ϕ_{a} and ϕ_{b} are as in (2.1). We have

L_{a0bV} (z_{1}, 0) = inf{log |ζ_{1}| + 2 log |ζ_{2}| : ∃φ ∈ O(𝔻, 𝔻^{2}), φ(0) = (z_{1}, 0), φ(ζ_{1}) = (a, 0), φ(ζ_{2}) = (b, 0) and φ′_{1}(ζ_{2}) = 0},

L_{a0}(z_{1}, 0) = inf{log |ζ_{1}| : ∃φ ∈ O(𝔻, 𝔻^{2}), φ(0) = (z_{1}, 0) and φ(ζ_{1}) = (a, 0)},

L_{bV} (z_{1}, 0) = inf{2 log |ζ_{2}| : ∃φ ∈ O(𝔻, 𝔻^{2}), φ(0) = (z_{1}, 0), φ(ζ_{2}) = (b, 0) and φ′_{1}(ζ_{2}) = 0}.

So L_{a0bV} (z_{1}, 0) ≥ L_{a0}(z_{1}, 0) + L_{bV} (z_{1}, 0), since each of the infima on the right hand side is taken over a family of maps φ which is wider than the one used in the definition of L_{a0bV}.

By Lemma 2.6, L_{a0}(z_{1}, 0) = log |ϕ_{a}(z_{1})|, L_{bV} (z_{1}, 0) = log |ϕ_{b}(z_{1})|.

Now suppose that L_{a0bV} (z_{1}, 0) ≤ L_{a0b0}(z_{1}, 0). This means

L_{a0bV} (z_{1}, 0) ≤ G_{a0b0}(z_{1}, 0) = L_{a0}(z_{1}, 0) + L_{bV} (z_{1}, 0),

so there is equality throughout. Since L_{a0bV} (z_{1}, 0) < min(L_{a0}(z_{1}, 0), L_{bV} (z_{1}, 0)), Proposition 3.3 shows that the infimum in the definition of L_{a0bV} is attained by a map φ. It follows from the Schwarz Lemma applied to a and z_{1} that its first coordinate φ_{1} is a Möbius map of the disc. But we also had to have φ′_{1}(ζ_{2}) = 0. This is a contradiction. □

The following example is similar, and will be useful in the final construction.

Proposition 4.3. If a ≠ b ∈ 𝔻 and |γ|^{2} < |ab|, then

G_{aV bV} (0, γ) < L_{aV bV} (0, γ).

Proof. First of all we can rewrite the generalized Lempert function as follows

L_{aV bV} (z) = inf{2 log |ζ_{1}| + 2 log |ζ_{2}| : ∃φ ∈ O(𝔻, 𝔻^{2}), φ(0) = z, φ(ζ_{1}) = (a, 0), φ(ζ_{2}) = (b, 0) and φ′_{1}(ζ_{1}) = 0, φ′_{1}(ζ_{2}) = 0}.

As in the proof of Proposition 4.2, by Lemma 2.6 we have

L_{aV} (z) = G_{aV} (z) = max{log |ϕ_{a}(z_{1})|; 2 log |z_{2}|}, ∀z ∈ 𝔻^{2},

and similarly for L_{bV} (z) = G_{bV} (z).

By using the Dirichlet problem given by Lelong and Rashkovskii [9], we can verify that

G_{aV bV} (z) = max{log |ϕ_{a}(z_{1})| + log |ϕ_{b}(z_{1})|; 2 log |z_{2}|}.

Since |γ|^{2} < |ab|, G_{aV bV} (0, γ) = log |a| + log |b|.

From Lemma 2.5 we already know G_{aV bV} (z) ≤ L_{aV bV} (z), for any z ∈ 𝔻^{2}. Suppose equality holds at z_{0} := (0, γ). Then, by using Lemma 2.6 and the definition of L_{aV bV} we have

G_{aV bV} (z_{0}) = log |a| + log |b| = G_{aV} (z_{0}) + G_{bV} (z_{0}) = L_{aV} (z_{0}) + L_{bV} (z_{0}) ≤ L_{aV bV} (z_{0}) = G_{aV bV} (z_{0}).

Hence equality would hold throughout. Now, by Proposition 3.3, the infimum in the definition of L_{aV bV} is attained by an extremal disc φ that passes through both (a, 0) and (b, 0). It follows that φ must be extremal for L_{aV} and L_{bV}. We will prove that this is impossible.

First of all we characterize all extremal discs for L_{aV}. Let φ = (φ_{1}, φ_{2}) be such a disc. By the definition there exists ζ_{1} ∈ 𝔻 such that φ(0) = (0, γ), φ(ζ_{1}) = (a, 0), φ′_{1}(ζ_{1}) = 0, |ζ_{1}|^{2} = |a|.

Setting g := ϕ_{a} ∘ φ_{1} ∘ ϕ_{ζ_{1}}, we have

g(0) = 0, g′(0) = 0, g(ζ_{1}) = a, |ζ_{1}|^{2} = |a|.

The Schwarz Lemma now gives g(ζ) = e^{iθ}ζ^{2}, where θ ∈ ℝ. It implies that

φ_{1}(ζ) = ϕ_{a}(e^{iθ}(ϕ_{ζ_{1}}(ζ))^{2}), ∀ζ ∈ 𝔻.

If the function φ is an extremal disc for L_{bV}, then there is ζ_{2} ∈ 𝔻 such that

φ_{1}(0) = 0, φ_{1}(ζ_{2}) = b, φ′_{1}(ζ_{2}) = 0, |ζ_{2}|^{2} = |b|.

Clearly ζ_{1} ≠ ζ_{2} since a ≠ b. Since φ_{1} only has one critical point, the condition φ′_{1}(ζ_{2}) = 0 is not verified, so we have a contradiction. □

Proposition 4.4. If a ≠ b ∈ 𝔻, |γ| < |a|, and |γ|^{2} < |ab|, then

G_{aV bV} (0, γ) < L_{a0bV} (0, γ).

Proof. The arguments are similar to those in the proof of the above proposition, so we only indicate the differences. As in the proof of Proposition 4.2, L_{a0bV} (z) ≥ L_{a0}(z) + L_{bV} (z) = G_{a0}(z) + G_{bV} (z) by Lemma 2.6 ; because of the value of |γ|, this is equal to G_{aV bV (z)}. So if the conclusion was not true, equality would have to hold throughout, but the extremal disc φ in the definition of L_{a0}(0, γ) would have to have a Möbius map for its first coordinate φ_{1}, and since this has no critical point, it could not be extremal for L_{bV} (0, γ). □

5. The main counterexample

Theorem 5.1. Coman’s question admits a negative answer in the bidisc for equal weights. More precisely, consider, for ε ∈ ℂ,

S^{ε} := {(a, 0); (b, 0); (b, ε); (a, ε)} with b = −a,

where the weights are all equal to 1. Denote by G^{ε} and L^{ε} the corresponding Green and generalized Lempert functions. Let z = (0, γ) with |a|^{3/2} < |γ| < |a|. Then, lim inf_{ε→0}L^{ε}(z) > G_{aV bV} (z) and therefore, for |ε| small enough,

G^{ε}(z) < L^{ε}(z).

Proof. Using the result of Edigarian about the product property of the Green function, [5], we have

G^{ε}(0, γ) = max \{ log |a| + log |b|; log |γ| + log |\frac{ε − γ}{1 − \overline{ε}γ} |\}.

Thus

G_{aV bV} (0, γ) = lim_{ε→0}G^{ε}(0, γ) = log |a| + log |b| = log |a|^{2}.

By Propositions 4.3 and 4.4, and since L_{a0b0}(z) = log |γ| > log |a|^{2} = G_{aV bV} (z), with z = (0, γ), we have

(5.1) lim_{ε→0}G^{ε} = G_{aV bV} (z) < L̃_{aV bV} (z) = min{L_{a0b0}(z), L_{aV b0}(z), L_{a0bV} (z), L_{aV bV} (z)}.

We consider I := lim inf_{ε→0}L^{ε}(z). We want to prove that I > G_{aV bV} (z). In many

cases, this will follow from I ≥ L̃_{aV bV} (z).

Recall that L^{ε} is a Lempert function with simple poles, and thus the usual definition (the ℓ_{S} in the introduction) coincides here with our generalization given in Definition 2.4. For each ε, pick an analytic disc φ^{ε} ∈ O(𝔻, 𝔻^{2}) such that

φ^{ε}(0) = z, φ^{ε}(ζ_{1}^{ε}) = (a, 0), φ^{ε}(ζ_{2}^{ε}) = (b, 0), φ^{ε}(ζ_{3}^{ε}) = (b, ε), φ^{ε}(ζ_{4}^{ε}) = (a, ε)

and such that ∑_{j=1}^{4} log |ζ_{j}^{ε}| converges to I as ε → 0.

By passing to a subsequence, we may assume that φ^{ε} converges locally uniformly to some φ ∈ O(𝔻, 𝔻^{2}). Also (if necessary, by passing to a subsequence again), we may assume that ζ_{j}^{ε} → ζ_{j} ∈ \overline{𝔻}, for each j, as ε → 0.

Denote K = {k ∈ {1, 2, 3, 4} : ζ_{k} ∈ 𝔻}. It is easy to see that 𝔻∩{ζ_{1}, ζ_{4}}∩{ζ_{2}, ζ_{3}} = ∅.

If K = ∅ then I = 0, and hence we have I ≥ L̃_{aV bV} (z) > G_{aV bV} (z), by (5.1). So now we only consider the cases where K ≠ ∅.

If ζ_{j} ≠ ζ_{k}, ∀j ≠ k ∈ K, then I = ∑_{k∈K} log |ζ_{k}|, φ_{2} ∈ O(𝔻, 𝔻), φ_{2}(0) = γ and φ_{2}(ζ_{k}) = 0, k ∈ K. It implies that

φ_{2}(ζ) = ∏_{k∈K} (\frac{ζ_{k} − ζ}{1 − ζ\overline{ζ_{k}}} )h(ζ),

where h ∈ O(𝔻, \overline{𝔻}) and h(0) = \frac{γ}{∏_{k∈K} ζ_{k}} . Thus we have

L_{a0b0}(z) = log |γ| ≤ ∑_{k∈K} log |ζ_{k}| = I,

and hence, I ≥ L̃_{aV bV} (z) > G_{aV bV} (z).

If K = {2, 3} and ζ_{2} = ζ_{3}, then, since ζ_{2}^{ε} → ζ_{2}, ζ_{3}^{ε} → ζ_{2} and |ζ_{3}^{ε} − ζ_{2}^{ε}| ≥ |ε|,

φ′_{1}(ζ_{2}) = lim_{ε→0} \frac{0}{ζ_{3}^{ε} − ζ_{2}^{ε}} = 0.

Thus I ≥ L_{bV} (z) ≥ L_{a0bV} (z) by Proposition 3.1. So that I ≥ L̃_{aV bV} (z) > G_{aV bV} (z).

Similarly, if K = {1, 4} and ζ_{1} = ζ_{4}, then φ′_{1}(ζ_{1}) = 0. Thus I ≥ L_{aV} (z) ≥ L_{aV b0}(z) ≥ L̃_{aV bV} (z) > G_{aV bV} (z).

If K = {1, 2, 3}, ζ_{2} = ζ_{3}, then φ′_{1}(ζ_{2}) = 0. Thus I = log |ζ_{1}|+2 log |ζ_{2}| ≥ L_{a0bV} (z) ≥ L̃_{aV bV} (z) > G_{aV bV} (z). The same reasoning obtains if K = {4, 2, 3}, ζ_{2} = ζ_{3}.

Similarly, if either K = {1, 2, 4}, ζ_{1} = ζ_{4} or K = {1, 3, 4}, ζ_{1} = ζ_{4}, then φ′_{1}(ζ_{1}) = 0. This implies that I ≥ L_{aV b0}(z) ≥ L̃_{aV bV} (z) > G_{aV bV} (z).

If K = {1, 2, 3, 4} and ζ_{1} = ζ_{4}, ζ_{2} = ζ_{3}, then φ′_{1}(ζ_{1}) = φ′_{1}(ζ_{2}) = 0. It implies that I = 2 log |ζ_{1}| + 2 log |ζ_{2}| ≥ L_{aV bV} (z) ≥ L̃_{aV bV} (z) > G_{aV bV} (z).

Suppose now that K = {1, 2, 3, 4} and ζ_{1} ≠ ζ_{4}, ζ_{2} = ζ_{3}. This is the final and most delicate case (the proof of [14, Theorem 6.3] suggests that it does occur). Both previous types of argument now break down, because we only get

I < min(log |ζ_{1}|, log |ζ_{4}|) + 2 log |ζ_{2}| ≥ L_{a0bV} (z);

or, from the fact that φ_{2}(ζ_{1}) = φ_{2}(ζ_{4}) = φ_{2}(ζ_{2}) = 0 and φ_{2}(0) = γ,

I < log |ζ_{1}| + log |ζ_{4}| + log |ζ_{2}| ≥ log |γ| ≥ L_{a0b0}(z).

By using a rotation in the first coordinate we can assume that a > 0. We will prove that I > G_{aV bV} (z). If not, we would have

(5.2)  log |ζ_{1}| + log |ζ_{4}| + 2 log |ζ_{2}| = I = G_{aV bV} (z) = 2 log a.

Then the function φ_{1} has the following properties:

(5.3)  φ_{1}(0) = 0; φ_{1}(ζ_{1}) = φ_{1}(ζ_{4}) = a; φ_{1}(ζ_{2}) = −a; φ′_{1}(ζ_{2}) = 0.

Setting f := ϕ_{−a} ∘ φ_{1} ∘ ϕ_{ζ_{2}}, with ϕ_{ξ} defined as in (2.1), we have f(0) = 0, f′(0) = 0 and f(ζ_{2}) = −a. The Schwarz Lemma shows that |ζ_{2}|^{2} ≥ a, and hence

(5.4)  2 log |ζ_{2}| ≥ log a

Setting g := ϕ_{a} ∘ φ_{1}, we have g(ζ_{1}) = g(ζ_{4}) = 0 and g(0) = a. Thus the function g must have the following form

g(ζ) = ϕ_{ζ_{1}}(ζ)ϕ_{ζ_{4}}(ζ)h_{1}(ζ), ∀ζ ∈ 𝔻, where h_{1} ∈ O(𝔻, \overline{𝔻}) and h_{1}(0) = \frac{a}{ζ_{1}ζ_{4}} , hence

(5.5)  log |ζ_{1}| + log |ζ_{4}| ≥ log a.

The assumption (5.2) implies that all the inequalities in (5.4) and (5.5) become equalities. Now, since φ_{2}(0) = γ and φ_{2}(ζ_{1}) = φ_{2}(ζ_{2}) = φ_{2}(ζ_{4}) = 0,

φ_{2}(ζ) = ∏_{j=1,j≠3}^{4} (\frac{ζ_{j} − ζ}{1 − \overline{ζ_{j}}ζ} )h_{2}(ζ), where h_{2} ∈ O(𝔻, \overline{𝔻}) and h_{2}(0) = \frac{γ}{ζ_{1}ζ_{2}ζ_{4}} .

It implies that |γ| ≤ |ζ_{1}ζ_{2}ζ_{4}| = a^{3/2}. This contradicts the hypothesis |γ| > a^{3/2}, and the inequality I > G_{aV bV} (z) is proved.

If K = {1, 2, 3, 4} and ζ_{1} = ζ_{4}, ζ_{2} ≠ ζ_{3}, the proof is similar. □

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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

Nguyen Van Trao Department of Mathematics Dai Hoc Su Pham 1 (Pedagogical Institute of Hanoi) Cau Giay, Tu Liem Ha Noi

16 PASCAL J. THOMAS NGUYEN VAN TRAO

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ngvtrao@yahoo.com