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 quasiconvex 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 Dmfk(0)(zm) = zk( n l=1 aklmzm−1 l ), k = 1, 2, . . . , n, m = 2, 3, . . ., here aklm = ∂mfk (0) ∂zk∂zm−1 l , k, l = 1, 2, . . . , n, m = 2, 3, . . . . © 2007 Elsevier Inc. All rights reserved