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

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 ? ? H(U) to a locally univalent mapping F_α ? H(B), where α ? (0,1]. If ? is normalized univalent, then F_α can be imbedded in a Loewner chain. Also if ? ? S?, then F_α is starlike. We show that if ? belongs to a class of univalent functions which satisfy growth and distortion results, then the mapping F_α 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.)