Multidimensional analogues of Bohr’s theorem on power series

Aizenberg, Lev

doi:10.1090/S0002-9939-99-05084-4

Multidimensional analogues of Bohr’s theorem on power series
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Proc. Amer. Math. Soc. 128 (2000), 1147-1155 Request permission

Abstract:

Generalizing the classical result of Bohr, we show that if an $n$-variable power series converges in $n$-circular bounded complete domain $D$ and its sum has modulus less than 1, then the sum of the maximum of the modulii of the terms is less than 1 in the homothetic domain $r \cdot D$, where $r = 1- \sqrt [n]{2/3}$. This constant is near to the best one for the domain $D = \{z: |z_1 |+ \ldots + |z_n |$$< 1 \} .$

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