Approximation properties of univalent mappings on the unit ball in $\mathbb{C}^n$

TL;DR

This paper studies how various classes of univalent holomorphic mappings on the unit ball in complex n-space can be approximated by automorphisms of the same space, extending approximation results to several geometric and parametric classes.

Contribution

It establishes new approximation properties of univalent mappings with specific geometric properties and parametric representations by automorphisms in complex n-space.

Findings

Approximation of spirallike, convex, and g-starlike mappings by automorphisms.

Approximation of mappings with A-parametric representation for nonresonant A.

Approximation of mappings close to the identity in the derivative norm.

Abstract

Let $n\geq 2$. In this paper, we obtain approximation properties of various families of normalized univalent mappings $f$ on the Euclidean unit ball $\mathbb{B}^n$ in $\mathbb{C}^n$ by automorphisms of $\mathbb{C}^n$ whose restrictions to $\mathbb{B}^n$ have the same geometric property of $f$. First, we obtain approximation properties of spirallike, convex and $g$-starlike mappings $f$ on $\mathbb{B}^n$ by automorphisms of $\mathbb{C}^n$ whose restrictions to $\mathbb{B}^n$ have the same geometric property of $f$, respectively. Next, for a nonresonant operator $A$ with $m(A)>0$, we obtain an approximation property ofmappings which have $A$-parametric representation by automorphisms of $\mathbb{C}^n$ whose restrictions to $\mathbb{B}^n$ have $A$-parametric representation. Certain questions will be also mentioned. Finally, we obtain an approximation property by automorphisms of $\mathbb{C}^n$ for a subset of $S_{I_n}^0(\mathbb{B}^n)$ consisting of mappings $f$ which satisfy the condition $\|Df(z)-I_n\|<1$, $z\in\mathbb{B}^n$. Related results will be also obtained.