Relations among starlikeness, convexity and k-fold symmetry of locally biholomorphic mappings in Cn

The subject of the paper concerns the existence of different inclusions between families St,Sc of Cn - biholomorphic starlike or convex mappings and a family S(k),k≥2, of Cn - locally biholomorphic mappings. In the definition of S(k) we used a combination of a Kikuchi-Matsuno-Suffridge's starlikeness condition and a property of (j,k)-symmetrical mappings in Cn. This definition generalizes a notion of planar Sakaguchi's functions onto n-dimensional case. A motivation for the family S(k) comes from the paper [11] of the second author and the paper [6] of Hamada and Kohr. The family S(k) is a superset of a family which is connected with a problem posed in the first paper and solved in the second one.