A class of Loewner chain preserving extension operators

AbstractWe consider operators that extend locally univalent mappings of the unit disk Δ in C to locally biholomorphic mappings of the Euclidean unit ball B of Cn. For such an operator Φ, we seek conditions under which etΦ(e−tf(⋅,t)), t⩾0, is a Loewner chain on B whenever f(⋅,t), t⩾0, is a Loewner chain on Δ. We primarily study operators of the form [ΦG,β(f)](z)=(f(z1)+G([f′(z1)]βzˆ),[f′(z1)]βzˆ), zˆ=(z2,…,zn), where β∈[0,1/2] and G:Cn−1→C is holomorphic, finding that, for ΦG,β to preserve Loewner chains, the maximum degree of terms appearing in the expansion of G is a function of β. Further applications involving Bloch mappings and radius of starlikeness are given, as are elementary results concerning extreme points and support points