Approximation properties of univalent mappings on the unit ball in \(\mathbb{C}^n\). (English) Zbl 1390.32004
Summary: 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 of mappings 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.
MSC:
| 32A30 | Other generalizations of function theory of one complex variable |
| 30C45 | Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.) |
Keywords:
automorphisms of \(\mathbb C^n\); convex, spiral-like and \(g\)-star-like mappings on the unit ball; approximationReferences:
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