q ‐pseudoconvex and q ‐holomorphically convex domains

In this article we prove a global result in the spirit of Basener's theorem regarding the relation between q ‐pseudoconvexity and q ‐holomorphic convexity: we prove that any open subset Ω ⊂ C nwith smooth boundary, strictly q ‐pseudoconvex, is ( q + 1 ) ‐holomorphically convex; moreover, assuming that Ω verifies an additional assumption, we prove that it is q ‐holomorphically convex. We also prove that any open subset of C n is n ‐holomorphically convex.