Invariant Distances and Metrics in Complex Analysis

Hyperbolicity and Vitali properties of unbounded domains in Banach spaces

“…A less obvious property is provided by the next result whose proof is the same as the one given in the finite dimensional case (see Proposition 3.1.13 in [6]). …”

“…For λ ∈ ∆ we let p(λ) := 1 2 log 1 + |λ| 1 − |λ| , be the Poincare distance from 0 to λ. As in the finite dimensional case (see [6], p.73), the Lempert function can now be used to define the Kobayashi pseudo-distance on Ω as follows. For z, w ∈ Ω we let…”

Comparison of invariant functions and metrics

“…[9]) and the fact that the sublevels {z ∈ D i : c D (z i , w) = α} have empty interior for any w ∈ D i and α > 0 (which clearly follows from the fact that non-constants holomorphic functions are open).…”

A Julia's Lemma for the symmetrized bidisc 𝔾2

“…After putting f a ðs, pÞ ¼ 2apÀs 2Àas , it is clear that f a is holomorphic in (C\{2/a}) Â C and that actually for any a 2 D the function f a maps G 2 in D. Moreover, the following important relation (see, e.g. [2]) can be found:…”

“…where k à G 2 is the Lempert function in G 2 and m D is the pseudo-metric in D. In particular (see again [2]), the Kobayashi pseudo-distance 2 k G 2 can be obtained as follows:…”