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

Article Type

Research Article

Publication Title

Forum Mathematicum

Abstract

We study a “p-powered” version Knp(F(R)) of the well-known Bohr radius problem for the family F(R) of holomorphic functions f : R → X satisfying ||f|| < ∞, where || . || is a norm in the function space F(R), R ⊂ Cn is a complete Reinhardt domain, and X is a complex Banach space. For all p > 0, we describe in full detail the asymptotic behavior of Knp(F(R)), where F(R) is: (a) the Hardy space of X-valued holomorphic functions defined in the open unit polydisk Dn; and (b) the space of bounded X-valued holomorphic or complex-valued pluriharmonic functions defined in the open unit ball B(lnt ) of the Minkowski space lnt . 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), no. 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), no. 1, Paper No. 109254] have been obtained as specific instances of our results.

First Page

765

Last Page

782

DOI

10.1515/forum-2023-0177

Publication Date

5-1-2024

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