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

Abstract

Abstract We study a “p-powered” version K n p ⁢ ( F ⁢ ( R ) ) {K_{n}^{p}(F(R))} of the well-known Bohr radius problem for the family F ⁢ ( R ) {F(R)} of holomorphic functions f : R → X {f:R\to X} satisfying ∥ f ∥ < ∞ {\lVert f\rVert<\infty} , where ∥ ⋅ ∥ {\lVert\,\cdot\,\rVert} is a norm in the function space F ⁢ ( R ) {F(R)} , R ⊂ ℂ n {R\subset{\mathbb{C}}^{n}} is a complete Reinhardt domain, and X is a complex Banach space. For all p > 0 {p>0} , we describe in full detail the asymptotic behavior of K n p ⁢ ( F ⁢ ( R ) ) {K_{n}^{p}(F(R))} , where F ⁢ ( R ) {F(R)} is: (a) the Hardy space of X-valued holomorphic functions defined in the open unit polydisk 𝔻 n {{\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 ) {B(l_{t}^{n})} of the Minkowski space l t n {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 [C. Bénéteau, A. Dahlner and D. Khavinson, Remarks on the Bohr phenomenon, Comput. Methods Funct. Theory 4 2004, 1, 1–19] and [S. Chen and H. Hamada, Some sharp Schwarz–Pick type estimates and their applications of harmonic and pluriharmonic functions, J. Funct. Anal. 282 2022, 1, Paper No. 109254] have been obtained as specific instances of our results.