On the second Bohr radius for vector valued pluriharmonic functions

Abstract: In this paper, we introduce the notion of the second Bohr radius for vector valued pluriharmonic functions on complete Reinhardt domains in $\mathbb{C}^n$. This investigation is motivated by the work of Lev Aizenberg [Proc. Amer. Math. Soc. 128 (2000), 1147-1155], where the corresponding problem was studied for complex valued holomorphic functions. We show that the second Bohr radius constant for pluriharmonic functions is strictly positive under suitable condition. In addition, we obtain its asymptotic behavior in the finite-dimensional settings using invariants from local Banach space theory. Asymptotic estimates for this constant are obtained on both convex and non-convex complete Reinhardt domains. Our results also apply to a broad class of Banach sequence spaces, including symmetric and convex Banach spaces. The framework developed here also includes the second Bohr radius problem for vector valued holomorphic functions. As an application of our results, we derive several consequences that extend known results in the scalar valued setting as well as existing results in the literature.