# Wydawnictwa / Czasopisma IMPAN / Studia Mathematica / Wszystkie zeszyty

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

### Tom 168 / 2005

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

#### Streszczenie

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.

#### Autorzy

• Lev AizenbergDepartment of Mathematics
Bar-Ilan University
52900 Ramat-Gan, Israel
e-mail
• Mark ElinDepartment of Mathematics
ORT Braude College
21982 Karmiel, Israel
e-mail
• David ShoikhetDepartment of Mathematics
ORT Braude College
21982 Karmiel, Israel
e-mail

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