## Bulletin of the Belgian Mathematical Society - Simon Stevin

- Bull. Belg. Math. Soc. Simon Stevin
- Volume 14, Number 4 (2007), 653-667.

### The asymptotical case of certain quasiconformal extension results for holomorphic mappings in $\mathbb{C}^n$

#### Abstract

Let $f(z,t)$ be a non-normalized subordination chain and assume that $f(\cdot,t)$ is $K$-quasiregular on $B^n$ for $t\in [0,\alpha]$. In this paper we obtain a sufficient condition for $f(\cdot,0)$ to be extended to a quasiconformal homeomorphism of $\overline{\mathbb{R}}^{2n}$ onto $\overline{\mathbb{R}}^{2n}$. Finally we obtain certain applications of this result. One of these applications can be considered the asymptotical case of the $n$-dimensional version of the well known quasiconformal extension result due to Ahlfors and Becker.

#### Article information

**Source**

Bull. Belg. Math. Soc. Simon Stevin, Volume 14, Number 4 (2007), 653-667.

**Dates**

First available in Project Euclid: 15 November 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.bbms/1195157134

**Digital Object Identifier**

doi:10.36045/bbms/1195157134

**Mathematical Reviews number (MathSciNet)**

MR2384461

**Zentralblatt MATH identifier**

1135.32017

**Subjects**

Primary: 32H 30C45: Special classes of univalent and multivalent functions (starlike, convex, bounded rotation, etc.)

**Keywords**

biholomorphic mapping Loewner differential equation Loewner chain subordination chain subordination quasiregular mapping quasiconformal mapping quasiconformal extension

#### Citation

Curt, Paula; Kohr, Gabriela. The asymptotical case of certain quasiconformal extension results for holomorphic mappings in $\mathbb{C}^n$. Bull. Belg. Math. Soc. Simon Stevin 14 (2007), no. 4, 653--667. doi:10.36045/bbms/1195157134. https://projecteuclid.org/euclid.bbms/1195157134