Exact value of the n-dimensional Bohr radius Kn

Determine the exact value of the n-dimensional Bohr radius Kn for holomorphic functions on the unit polydisk D^n when n > 1. Specifically, Kn is defined as the largest number r in (0,1) such that, for every holomorphic function f(z) = sum_{α} a_{α} z^{α} on D^n with |f(z)| < 1 on D^n, the inequality sum_{α} |a_{α} z^{α}| ≤ 1 holds whenever max_{1≤j≤n} |z_j| ≤ r.

Background

The paper reviews the multidimensional generalizations of Bohr’s theorem, defining the n-dimensional Bohr radius Kn as the largest radius ensuring the Bohr-type coefficient sum bound on the unit polydisk. Foundational bounds were established by Boas and Khavinson, with subsequent improvements to lower bounds by Defant and Frerick and further results via hypercontractivity techniques.

Bayart, Pellegrino, and Seoane-Sepúlveda later determined the asymptotic behavior of Kn as n grows, but the exact value of Kn for each fixed n > 1 remains unknown. The authors highlight this as an ongoing open problem in the field.