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In this article, we study a generalized Bohr radius
$R_{p, q}(X), p, q\in [1, \infty )$
defined for a complex Banach space X. In particular, we determine the exact value of
$R_{p, q}(\mathbb {C})$
for the cases (i)
$p, q\in [1, 2]$
, (ii)
$p\in (2, \infty ), q\in [1, 2]$
, and (iii)
$p, q\in [2, \infty )$
. Moreover, we consider an n-variable version
$R_{p, q}^n(X)$
of the quantity
$R_{p, q}(X)$
and determine (i)
$R_{p, q}^n(\mathcal {H})$
for an infinite-dimensional complex Hilbert space
$\mathcal {H}$
and (ii) the precise asymptotic value of
$R_{p, q}^n(X)$
as
$n\to \infty $
for finite-dimensional X. We also study the multidimensional analog of a related concept called the p-Bohr radius. To be specific, we obtain the asymptotic value of the n-dimensional p-Bohr radius for bounded complex-valued functions, and in the vector-valued case, we provide a lower estimate for the same, which is independent of n.
