Some generalizations of Bohr's theorem

<jats:p>Let <jats:italic>X</jats:italic> be a complex Banach space and <jats:italic>Y</jats:italic> be a JB*‐triple. Let <jats:italic>G</jats:italic> be a bounded balanced domain in <jats:italic>X</jats:italic> and <jats:italic>B</jats:italic><jats:sub><jats:italic>Y</jats:italic></jats:sub> be the unit ball in <jats:italic>Y</jats:italic>. Let <jats:italic>f</jats:italic> : <jats:italic>G</jats:italic> → <jats:italic>B</jats:italic><jats:sub><jats:italic>Y</jats:italic></jats:sub> be a holomorphic mapping. In this paper, we obtain some generalization of Bohr's theorem that if <jats:italic>a</jats:italic> = <jats:italic>f</jats:italic>(0), then we have <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/mma2633-math-0001.gif" xlink:title="urn:x-wiley:01704214:media:mma2633:mma2633-math-0001" /> for <jats:italic>z</jats:italic> ∈ (1 / 3)<jats:italic>G</jats:italic>, where <jats:italic>φ</jats:italic><jats:sub><jats:italic>a</jats:italic></jats:sub> ∈ Aut(<jats:italic>B</jats:italic><jats:sub><jats:italic>Y</jats:italic></jats:sub>) such that <jats:italic>φ</jats:italic><jats:sub><jats:italic>a</jats:italic></jats:sub>(<jats:italic>a</jats:italic>) = 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 <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/mma2633-math-0002.gif" xlink:title="urn:x-wiley:01704214:media:mma2633:mma2633-math-0002" />. Copyright © 2012 John Wiley & Sons, Ltd.</jats:p>