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Some generalizations of Bohr's theorem
Author(s) -
Hamada Hidetaka,
Honda Tatsuhiro
Publication year - 2012
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.2633
Subject(s) - bohr model , mathematics , holomorphic function , unit sphere , bounded function , banach space , unit (ring theory) , complex plane , pure mathematics , ball (mathematics) , domain (mathematical analysis) , generalization , combinatorics , discrete mathematics , mathematical analysis , quantum mechanics , physics , mathematics education
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.