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The generalized Bohr radius $R_{p, q}(X), p, q\in[1, \infty)$ for a complex Banach space $X$ was introduced by Blasco in 2010. In this article, 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 $\mathcal{H}$, (ii) the precise asymptotic value of $R_{p, q}^n(X)$ as $n\to\infty$ for finite dimensional $X$. We also study the multidimensional analogue of a related concept called the $p$-Bohr radius, introduced by Djakov and Ramanujan in 2000. In particular, 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 $n$. In a similar vein, we investigate in detail the multidimensional $p$-Bohr radius problem for functions with positive real part. Towards the end of this article, we pose one more generalization $R_{p, q}(Y, X)$ of $R_{p, q}(X)$-considering functions that map the open unit ball of another complex Banach space $Y$ inside the unit ball of $X$, and show that the existence of nonzero $R_{p, q}(Y, X)$ is governed by the geometry of $X$ alone.