Bohr Phenomena for Holomorphic Mappings with Values in Several Complex Variables

In the first part of this paper, we study several Bohr radii for holomorphic mappings with values in the unit polydisc \(\mathbb {U}^N\) ${\mathbb{U}}^N$ in \(\mathbb {C}^{N}\) ${\mathbb{C}}^N$ . In particular, we obtain the new Bohr radius \(r_{k,m}^{***}\) $r_{k,m}^{\ast \ast \ast }$ for holomorphic mappings with lacunary series. Further, we show that when \(m\ge 1\) $m\ge 1$ , \(r_{k,m}^{***}\) $r_{k,m}^{\ast \ast \ast }$ is asymptotically sharp as \(N\rightarrow \infty \) $N\to \infty$ . Note that when \(m\ge 1\) $m\ge 1$ , \(r_{k,m}^{***}\) $r_{k,m}^{\ast \ast \ast }$ is completely different from the cases with values in the unit disc \(\mathbb {U}\) $\mathbb{U}$ and in the complex Hilbert balls with higher dimensions. In the second part of this paper, we obtain the Bohr type inequality for holomorphic mappings F with values in the unit ball of a JB \(^*\) $_{}^{\ast }$ -triple which is a generalization of that for holomorphic mappings F with values in the unit ball of a complex Banach space of the form \(F(z)=f(z)z\) $F(z)=f(z)z$ , where f is a \(\mathbb {C}\) $\mathbb{C}$ -valued holomorphic function.