On the Rogosinski radius for holomorphic mappings and some of its applications
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: |∞n=0 anz| < 1, |z| < 1, then all its partial sums are less than 1 in the disk of radius 1/2: ∣∣ k ∑ n=0 anz n ∣∣ < 1, |z| < 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. 1. Preliminaries. Let Ω be a domain in Cn. By Hol(Ω,C) we denote the space of holomorphic functions in Ω with the topology of uniform convergence on compact sets. Let {φk(z)}k=0 be a given basis in Hol(Ω,C). Definition 1.1. The Bohr radius B(Ω, {φk}) is the largest number r > 0 such that if f ∈ Hol(Ω) has the expansion ∞ ∑ k=0 ckφk(z) = f(z) and if |f(z)| < 1 in Ω, then ∞ ∑ k=0 |ckφk(z)| < 1 in Ωr = rΩ, homothety of Ω. Definition 1.2. The Rogosinski radius R(Ω, {φk}) is the largest number r > 0 such that, under the conditions of the previous definition, the 2000 Mathematics Subject Classification: Primary 32A05.