Some generalizations of Bohr's theorem

Abstract

Communicated by T. QianLet X be a complex Banach space and Y be a JB*-triple. Let G be a bounded balanced domain in X and B Y be the unit ball in Y. Let f : G ! B Y be a holomorphic mapping. In this paper, we obtain some generalization of Bohr's theorem that if a D f .0/, then we have P 1 kD0 kD' a .a/OED k f .0/.z k /k=.kŠkD' a .a/k/ < 1 for z 2 .1=3/G, where ' a 2 Aut.B Y / such that ' a .a/ D 0. Moreover, we show that the constant 1=3 is best possible. This result generalizes Bohr's theorem for the open unit disc in the complex plane C.

On the Bohr Inequality

“…Mathematicians have studied various generalizations of Bohr theorem, for example, in the works of [12,14,24,25,40,41,48,49] and the references therein. One generalization uses power series representation of holomorphic functions defined on a complete Reinhardt domain, that is, a bounded complete n-circular domain in C n .…”

Improved Bohr inequality for harmonic mappings

“…As mentioned above, there exist Bohr's theorems to more general domains or higher dimensional spaces, holomorphic functions defined on bounded complete Reinhardt domain in $${\mathbb {C}}^n$$, and operator‐theoretic Bohr radius. See for example, [3–5, 15, 26].…”