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In this paper, we first obtain a refined version of the Bohr inequality ofnorm-type for holomorphic mappings with lacunary series on the polydisk in$\mathbb{C}^n$ under some restricted conditions. Next, we determine the refinedversion of the Bohr inequality for holomorphic functions defined on a balanceddomain $ 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 obtaina 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 areproved to be sharp.

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