The sharp estimate of the third homogeneous expansion for a class of starlike mappings of order α on the unit polydisk in ℂn

Abstract

In this article, first, a sufficient condition for a starlike mapping of order αf(x) defined on the unit ball in a complex Banach space is given. Second, the sharp estimate of the third homogeneous expansion for f is established as well, where f(z) = (f1(z), f2(z), …, fn(z))' is a starlike mapping of order α or a normalized biholomorphic starlike mapping defined on the unit polydisk in ℂn, and D2fk(0)(z2)2!=zk(∑i=1naklzl),k=1,2,…,n, here, akl=1∂2fk(0)2!∂zk∂zl,k,l=1,2,…,n. Our result states that the Bieberbach conjecture in several complex variables (the case of the third homogeneous expansion for starlike mappings of order α and biholomorphic starlike mappings) is partly proved.