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

Abstract

It 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}, 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. 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 of k-fold symmetry. The 2017 paper of both authors contains some results on the absorption by a family S(k),kge 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 mapping f, from the Euclidean ball {mathbb {B}}^{n} into {mathbb {C}}^{n}, to belong to the family S(k),kge 2. The result is obtained using an n-dimensional version of Jack’s Lemma.