An Extension Theorem and Subclasses of Univalent Mappings in Several Complex Variables

Abstract

Let B be the unit ball in C n and let U be the unit disc in C. The aim of this work is to construct a family of operators Ψ n,α that provide a way to extend a locally univalent function ƒ ∈ H(U) to a locally univalent mapping Fα ∈ H(B), where α ∈ (0,1]. If ƒ is normalized univalent, then Fα can be imbedded in a Loewner chain. Also if ƒ ∈ S*, then Fα is starlike. We show that if ƒ belongs to a class of univalent functions which satisfy growth and distortion results, then the mapping Fα satisfies similar growth and distortion results. Also we study the concept of linear-invariant families as it relates to families generated by the operator Ψ n,0, and we obtain in this way another example of a L.I.F. that has minimum order (n + 1)/2 and is not a subset of the normalized convex mappings in the unit ball of C n (for n ≤ 2.)