On the quasi-convex mappings on the unit polydisk in Cn

In this paper, firstly, we establish a sufficient condition for a quasi-convex mapping (including quasi-convex mapping of type A and quasi-convex mapping of type B) f(x) defined on the unit ball in a complex Banach space. Secondly, sharp estimations of all homogeneous expansions for f are given, where f(z)=(f1(z),f2(z),…,fn′(z)) is a normalized quasi-convex mapping (including quasi-convex mapping of type A and quasi-convex mapping of type B) defined on the open unit polydisk in Cn, and , k=1,2,…,n, , here , k,l=1,2,…,n, .