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 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 in the unit disk . Furthermore, as a consequence of one of these results, we obtain a refined version of the Bohr-type inequality for harmonic functions defined on a balanced domain . All the results are proved to be sharp.
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