Bohr Radius for Pluriharmonic Mappings in Separable Complex Hilbert Spaces

Let \(H_1\) $H_1$ be the Euclidean space \(\mathbb {C}^m\) ${\mathbb{C}}^m$ or \(\ell _2\) ${\ell }_2$ and let \(\mathbb {B}_{H_1}\) ${\mathbb{B}}_{H_1}$ be the unit ball of \(H_1\) $H_1$ . In this paper, we will give new generalizations of several results related to the Bohr radius for locally univalent harmonic functions on the unit disc \(\mathbb {U}\) $\mathbb{U}$ in \(\mathbb {C}\) $\mathbb{C}$ to pluriharmonic mappings on \(\mathbb {B}_{H_1}\) ${\mathbb{B}}_{H_1}$ with values in \(H_1\) $H_1$ which satisfy some assumptions. All of the results are sharp, and some of the results give improvements of the results on the unit disc.