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

If G : C n − 1 → C is a holomorphic function such that G ( 0 ) = 0 and D G ( 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 ⊆ C n given by Φ G ( f ) ( z ) = ( f ( z 1 ) + G ( f ′ ( z 1 ) z ˆ ) , f ′ ( z 1 ) z ˆ ) , z ∈ B , z ˆ = ( z 2 , … , z n ) . While for a given f, Φ G ( 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 Φ G , 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 Φ G ( f ) is a mapping of the same type on B .