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

Abstract

Let $n\ge 2$. 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 $\Vert Df(z)-I_n\Vert <1$, $z\in {\mathbb{B}}^n$. Related results will be also obtained.

References in corpus (2)

• Resonances in Loewner equations
• Embedding Problems in Loewner Theory