General Univalence Criteria and Quasiconformal Extensions Starting From Loewner Chains Theory

Abstract

The aim of this paper is to obtain general  univalence conditions  and quasiconformal extensions to $\mathcal C^n$ of holomorphic mappings defined on the Euclidian unit ball $B$. The asymptotical case of the quasiconformal extension results is also presented. We extend several results obtained by Hamada and Kohr (2011) in [15]to a more general case. In particular our results improve certain univalence criteria and quasiconformal extension results previously obtained by Pfaltzgraff [21], [22], Curt and Pascu [8], Curt [5], Hamada and Kohr [16], Curt and Kohr [6], [7] and Rău aducanu [24]. As  applications we present general forms of the n-dimensional version of the well-known univalence criterion due to Lewandowski [19] and its quasiconformal extension.