Mixed Bohr radius in several variables

الأصل · EN

Let K(Bℓₚₙ,Bℓqₙ) be the n-dimensional (p,q)-Bohr radius for holomorphic functions on Cⁿ. That is, K(Bℓₚₙ,Bℓqₙ) denotes the greatest constant r≥ 0 such that for every entire function f(z)=∑α cα zα in n-complex variables, we have the following (mixed) Bohr-type inequality ∈ ᵣ · Bℓqₙ ∑α | cα zα | ≤ ∈ Bℓₚₙ | f(z) |, where Bℓᵣₙ denotes the closed unit ball of the n-dimensional sequence space ℓᵣⁿ. For every 1 ≤ p, q ≤ ∞, we exhibit the exact asymptotic growth of the (p,q)-Bohr radius as n (the number of variables) goes to infinity.