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

### Volume 168 / 2005

Studia Mathematica 168 (2005), 147-158
MSC: Primary 32A05.
DOI: 10.4064/sm168-2-5

#### Abstract

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.