Applications holomorphes propres d'un domaine quasi-disqué sur un domaine disqué

A result of F. Berteloot and G. Patrizio [F. Berteloot, G. Patrizio, A cartan theorem for proper holomorphic mappings of complete circular domains, Adv. Math. 153 (2000) 342–352] states that if f is a proper holomorphic map between two bounded complete circular domains Ω ₁ and Ω ₂ in C n + 1 ( n ⩾ 1 ), such that f − 1 { 0 } = { 0 } and such that the principal part f p of the Taylor expansions of f at the origine is nondegenerated i.e. f p − 1 { 0 } = { 0 } , then f = f p .

Here, we give a partial extension of this result to the case where f is a nondegenerated proper holomorphic map between a quasi-circular domain Ω ₁ and a complete circular domain Ω ₂ , which are pseudo-convex but not necessarily bounded.

We show that if f and its principal part f p are nondegenerated at the origine, then f p − 1 ( Ω ₂ ) = Ω ₁ .