Some generalizations of Bohr's theorem

Let 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  =  f (0), then we have∑ k = 0 ∞ ∥ D φ a( a ) [D k f ( 0 ) (z k) ] ∥ / ( k ! ∥ D φ a( a ) ∥ ) < 1 for z  ∈ (1 / 3) G , where φ a  ∈ Aut( B Y ) such that φ a ( a ) = 0. Moreover, we show that the constant 1 / 3 is best possible. This result generalizes Bohr's theorem for the open unit disc Δ in the complex plane C . Copyright © 2012 John Wiley & Sons, Ltd.