Multi-dimensional Bohr radii of Banach space valued holomorphic functions

In this article, we study the multi-dimensional Bohr radii of holomorphic functions defined on the Banach sequence spaces with values in the Banach spaces. For the case of finite dimensional Banach spaces, we exhibit the exact asymptotic growth of the Bohr radius. To achieve our goal in the finite case, we use ${\ell }_{p^{\prime }}$-summability of certain coefficients of a given polynomial in terms of its uniform norm on ${\ell }_p^n$. The infinite case is handled using the techniques developed in recent years from the work of Defant, Maestre and Schwarting. We crucially use several properties of the symmetric M-linear mapping associated with a homogeneous polynomial of degree M in our analysis. Furthermore, we study the bounds of the arithmetic Bohr radius of Banach space-valued holomorphic functions defined on the Banach sequence spaces, which generalizes the work of Defant, Maestre, and Prengel in this direction.