A class of Loewner chain preserving extension operators

We 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)]β ˆz), [f (z1)]β ˆz), ˆz = (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. © 2007 Elsevier Inc. All rights reserved.