Refined Bohr inequality for functions in ${\mathbb{C}}^n$ and in complex Banach spaces

Abstract

In 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.