A class of Loewner chain preserving extension operators. (English) Zbl 1133.32007
Summary: We consider operators that extend locally univalent mappings of the unit disk \(\Delta\) in \(\mathbb C\) to locally biholomorphic mappings of the Euclidean unit ball B of \(\mathbb C^n\). For such an operator \(\Phi\), we seek conditions under which \(e^t\Phi(e^{-t}f(\cdot,t))\), \(t\geq 0\), is a Loewner chain on \(B\) whenever \(f(\cdot,t)\), \(t\geq 0\), is a Loewner chain on \(\Delta\). We primarily study operators of the form
\[
[\Phi_{G,\beta}(f)](z)=(f(z_1)+G([f'(z_1)]^\beta\widehat z),[f'(z_1)]^\beta\widehat z),\quad \widehat z=(z_2,\dots,z_n),
\]
where \(\beta\in[0,1/2]\) and \(G:\mathbb C^{n-1}\to\mathbb C\) is holomorphic, finding that, for \(\Phi G,\beta\) to preserve Loewner chains, the maximum degree of terms appearing in the expansion of \(G\) is a function of \(\beta\). Further applications involving Bloch mappings and the radius of starlikeness are given, as are elementary results concerning extreme points and support points.
where \(\beta\in[0,1/2]\) and \(G:\mathbb C^{n-1}\to\mathbb C\) is holomorphic, finding that, for \(\Phi G,\beta\) to preserve Loewner chains, the maximum degree of terms appearing in the expansion of \(G\) is a function of \(\beta\). Further applications involving Bloch mappings and the radius of starlikeness are given, as are elementary results concerning extreme points and support points.
MSC:
| 32H02 | Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables |
| 30C45 | Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.) |
| 32A30 | Other generalizations of function theory of one complex variable |
Keywords:
biholomorphic mapping; starlike mapping; convex mapping; Loewner chain; parametric representationReferences:
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