The $p$-Bohr radius for vector-valued holomorphic and pluriharmonic functions

Abstract

We study a "$p$-powered" version $K_n^p(F(R))$ of the well-known Bohr radius problem for the family $F(R)$ of holomorphic functions $f: R\to X$ satisfying $\|f\|<\infty$, where $\|.\|$ is a norm in the function space $F(R)$, $R\subset\mathbb{C}^n$ is a complete Reinhardt domain and $X$ is a complex Banach space. For all $p>0$, we describe in full details the asymptotic behaviour of $K_n^p(F(R))$, where $F(R)$ is (a) the Hardy space of $X$-valued holomorphic functions defined in the open unit polydisk $\mathbb{D}^n$, and (b) the space of bounded $X$-valued holomorphic or complex-valued pluriharmonic functions defined in the open unit ball $B(l_t^n)$ of the Minkowski space $l_t^n$. We give an alternative definition of the optimal cotype for a complex Banach space $X$ in the light of these results. In addition, the best possible versions of two theorems from [B\'en\'eteau et. al., Comput. Methods Funct. Theory, 4 (2004), no. 1, 1-19] and [Chen & Hamada, J. Funct. Anal., 282 (2022), no. 1, Paper No. 109254, 42 pp] have been obtained as specific instances of our results.