An Extension Theorem and Subclasses of Univalent Mappings in Several Complex Variables
Titel:
An Extension Theorem and Subclasses of Univalent Mappings in Several Complex Variables
Auteur:
Graham, Ian Kohr, Gabriela
Verschenen in:
Complex variables and elliptic equations
Paginering:
Jaargang 47 (2002) nr. 1 pagina's 59-72
Jaar:
2002-01
Inhoud:
Let B be the unit ball in □n and let U be the unit disc in □. The aim of this work is to construct a family of operators Ψn,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 Ψn,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 Cn (for n ≤ 2.)
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