Existence domains of holomorphic functions of restricted growth

Abstract

The paper presents a large class of domains G G of holomorphy in C n {{\mathbf {C}}^n} such that, for any N > 0 N > 0 , there exists a nonextendable holomorphic function f f on G G with | f | δ G N |f|\delta _G^N bounded where δ G ( z ) := min ( ( 1 + | z | 2 ) − 1 / 2 , dist ⁡ ( z , ∂ G ) ) {\delta _G}(z): = \min ({(1 + |z{|^2})^{ - 1 / 2}},\,\operatorname {dist} (z,\,\partial G)) . Any fat Reinhardt domain of holomorphy belongs to this class. On the other hand we characterize those Reinhardt domains of holomorphy which are existence domains of bounded holomorphic functions.