Refined Bohr inequality for functions in $\mathbb{C}^n$ and in complex Banach spacesIn this paper, we first obtain a refined version of the Bohr inequality of
norm-type for holomorphic mappings with lacunary series on the polydisk in
$\mathbb{C}^n$ under some restricted conditions. Next, we determine the refined
version of the Bohr inequality for holomorphic functions defined on a balanced
domain $ G $ of a complex Banach space $ X $ and take values from the unit disk
$ \mathbb{D} $. Furthermore, as a consequence of one of this results, we obtain
a refined version of the Bohr-type inequalities for harmonic functions $
f=h+\bar{g} $ defined on a balanced domain $ G\subset X $. All the results are
proved to be sharp.
arxiv.org 
