On an Application of Jack’s Lemma, Starlikeness and k-Fold Symmetry in $${\mathbb {C}}^{n}$$

Abstract

AbstractIt is known that the starlikeness plays a central role in complex analysis, similarly as the convexity in functional analysis. However, if we consider the biholomorphisms between domains in$${\mathbb {C}}^{n},$$Cn,apart from starlikeness of domains, various symmetries are also important. This follows from the Poincaré theorem showing that the Euclidean unit ball is not biholomorphically equivalent to a polydisc in$${\mathbb {C}}^{n},n>1$$Cn,n>1. From this reason the second author in 2003 considered some families of locally biholomorphic mappings defined in the Euclidean open unit ball using starlikeness factorization and a notion ofk-fold symmetry. The 2017 paper of both authors contains some results on the absorption by a family$$S(k),k\ge 2,$$S(k),k≥2,of the above kind, the families of mappings biholomorphic starlike (convex) and vice versa. In the present paper there is given a new sufficient criterion for a locally biholomorphic mappingf,  from the Euclidean ball$${\mathbb {B}}^{n}$$Bninto$${\mathbb {C}}^{n},$$Cn,to belong to the family$$S(k),k\ge 2.$$S(k),k≥2.The result is obtained using ann-dimensional version of Jack’s Lemma.