On the Rogosinski radius for holomorphic mappings and some of its applications

Lev Aizenberg ¹   ; Mark Elin ²   ; David Shoikhet ²

¹ Department of Mathematics Bar-Ilan University 52900 Ramat-Gan, Israel
² Department of Mathematics ORT Braude College 21982 Karmiel, Israel

Studia Mathematica, Tome 168 (2005) no. 2, pp. 147-158

Résumé

The well known theorem of Rogosinski asserts that if the modulus of the sum of a power series is less than $1$ in the open unit disk: $\vert \sum_{n=0}^{\infty }a_{n}z^{n}\vert 1,$ $|z|1$, then all its partial sums are less than $1$ in the disk of radius $1/2$: $$ \Big\vert \sum_{n=0}^{k}a_{n}z^{n}\Big\vert 1,\ \quad |z|\frac{1}{2}, $$ and this radius is sharp.We present a generalization of this theorem to holomorphic mappings of the open unit ball into an arbitrary convex domain. Other multidimensional analogs of Rogosinski's theorem as well as some applications to dynamical systems are considered.

Détail

DOI : 10.4064/sm168-2-5

Keywords: known theorem rogosinski asserts modulus sum power series unit disk vert sum infty vert its partial sums disk radius vert sum vert quad frac radius sharp present generalization theorem holomorphic mappings unit ball arbitrary convex domain other multidimensional analogs rogosinskis theorem applications dynamical systems considered