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

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Multidisciplinary Methods for Analysis Optimization and Control of Complex Systems

This e-book contains lecture notes of a summer season university named after the past due Jacques Louis Lions. The summer season college was once designed to alert either Academia and to the expanding function of multidisciplinary equipment and instruments for the layout of complicated items in numerous components of socio-economic curiosity.

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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.