Invariant distances and metrics in complex analysis by Jarnicki M., Pflug P.

By Jarnicki M., Pflug P.

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25 max We extend formally the definitions of dmin G (p, ·) and dG (p, ·) to the case where min max p : G −→ [0, +∞]; dG (p, ·) = dG (p, ·) :≡ 0 if there exists a z0 ∈ G with p(z0 ) = +∞. 3. 4 ([Jar-Jar-Pfl 2003]). f. f. (dG )G (with integer-valued weights) we have max dmin G (p, ·) ≤ dG (p, ·) ≤ dG (p, ·) 23 24 For h : A −→ [0, 1], we put Note that if sup p(f −1 (µ 0 )) a∈A h(a) := inf B⊂A #B<+∞ a∈B h(a). [mE (µ, f (z))]sup p(f = +∞ for a µ0 ∈ f (G), then µ∈f (G) 25 We will see (cf. 4) that in fact dmin G (A, ·) = mG (A, ·).

Consequently, h is a rational function of degree ≤ 1 and, therefore, h must be an automorphism of the unit disc. 12. 11, then for any g ∈ Aut(E), the mapping ψ := Hg ◦ ϕ satisfies the same assumptions. Proof. 6). Let g = τ ha for some τ ∈ ∂E, a ∈ E. Fix a λ and let ϕ(λ) = (S(λ), P (λ)) = π(z1 , z2 ). Then ψ(λ) = π(g(z1 ), g(z2 )) = (τ (ha (z1 ) + ha (z2 )), τ 2 h(z1 )h(z2 )) = τ (1 + |a|2 )(z1 + z2 ) − 2az1 z2 − 2a 2 z1 z2 − a(z1 + z2 ) + a2 ,τ . 1 − a(z1 + z2 ) + a2 z1 z2 1 − a(z1 + z2 ) + a2 z1 z2 Consequently, ψ= τ (1 + |a|2 )S − 2aP − 2a 2 P − aS + a2 ,τ 1 − aS + a2 P 1 − aS + a2 P (1 + |a|2 )S − 2aP − 2aP0 2 P − aS + a2 P0 ,τ .

Moreover, (b) need not be true if D is not convex — cf. 7. For the behavior of the pluricomplex Green function under coverings see [Azu 1995], [Azu 1996]. Proof. (a) We only need to show gD (q, F (z)) ≥ gG (q ◦ F, z), z ∈ G; cf. 1(e). Put S := {z ∈ G : det F (z) = 0}, Σ := F (S). It is well-known that F |G\F −1 (Σ) : G \ F −1 (Σ) −→ D \ Σ is a holomorphic covering. Let N denote its multiplicity. Let u : G −→ [0, 1) be a logarithmically plurisubharmonic function such that u(z) ≤ C(a) z − a q(F (a)) , a, z ∈ G.

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