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 convexity functional analysis. However, if we consider biholomorphisms between domains $${\mathbb {C}}^{n},$$ Cn, apart from of domains, various symmetries are also important. This follows Poincaré theorem showing Euclidean unit ball not biholomorphically equivalent to polydisc {C}}^{n},n>1$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">Cn,n>1 . From this reason second author 2003 considered some families locally biholomorphic mappings defined open using factorization and notion k -fold symmetry. The 2017 paper both authors contains results on absorption by family $$S(k),k\ge 2,$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">S(k),k≥2, above kind, starlike (convex) vice versa. In present there given new sufficient criterion for mapping f , {B}}^{n}$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">Bn into belong 2.$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">S(k),k≥2. result obtained an n -dimensional version Jack’s Lemma.