Necessary conditions for the existence of higher order extensions of univalent mappings from the disk to the ball

Abstract

If G:Cn−1→C is a holomorphic function such that G(0)=0 and DG(0)=0 and f is a normalized univalent mapping of the unit disk D⊆C, we consider the normalized extension of f to the Euclidean unit ball B⊆Cn given by ΦG(f)(z)=(f(z1)+G(f′(z1)zˆ),f′(z1)zˆ), z∈B, zˆ=(z2,…,zn). While for a given f, ΦG(f) will maintain certain geometric properties of f, such as convexity or starlikeness, if G is a polynomial of degree 2 of sufficiently small norm, these properties may be lost whenever G contains a nonzero term of higher degree. By establishing separate necessary and sufficient conditions for the extension of Loewner chains from D to B through ΦG, we are able to completely classify those starlike and convex mappings f on D for which there exists a G with nonzero higher degree terms such that ΦG(f) is a mapping of the same type on B.