An abstract approach to Bohr’s phenomenon

Aizenberg, L.; Aytuna, A.; Djakov, P.

doi:10.1090/S0002-9939-00-05270-9

An abstract approach to Bohr’s phenomenon
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by L. Aizenberg, A. Aytuna and P. Djakov PDF

Proc. Amer. Math. Soc. 128 (2000), 2611-2619 Request permission

Abstract:

In 1914 Bohr discovered that there exists $r \in (0,1)$ such that if a power series converges in the unit disk and its sum has modulus less than $1$, then for $|z| < r$ the sum of absolute values of its terms is again less than $1$. Recently analogous results were obtained for functions of several variables. Our aim here is to present an abstract approach to the problem and show that Bohr’s phenomenon occurs under very general conditions.

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