Multidimensional analogues of Bohr’s theorem on power series

Abstract

Generalizing the classical result of Bohr, we show that if an n n -variable power series converges in n n -circular bounded complete domain D 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 ⋅ D r \cdot D , where r = 1 − 2 / 3 n r = 1- \sqrt [n]{2/3} . This constant is near to the best one for the domain D = { z : | z 1 | + … + | z n | D = \{z: |z_1 |+ \ldots + |z_n | > 1 } . > 1 \} .