The Bohr radius $R$ for $\mathcal{H}(\mathbb{D})$ is defined as $$R = \sup\limits_{0<r<1} \Bigl\{ r\ \Big|\ \sum\limits_{k=0}^{\infty}|a_k|r^k \leq |f|_\mathbb{D} \text{ for all }f(z)=\sum\limits_{k=0}^{\infty} a_{k}z^{k}\in\mathcal{H}(\mathbb{D}) \Bigr\},$$ where $|f|_{\mathbb{D}}=\sup\limits_{z\in\mathbb{D}} |f(z)|$, $\mathcal{H}(\mathbb{D})$ is the space of functions analytic in $\mathbb{D}$, and $\sum\limits_{k=0}^{\infty}|a_k|r^k$ is known as the majorization of $f$. Harald Bohr and later M. Riesz, I. Shur, and F. Wiener showed that $R=\frac{1}{3}$, and it is known the only extremal functions satisfying $\sum\limits_{k=0}^\infty |a_k|(\frac{1}{3})^k=|f|_{\mathbb{D}}$ are constant. There are several extensions of this result (for instance one can exactly compute the radius in terms of entries in an associated Toeplitz matrix when $\mathcal{H}(\mathbb{D})$ is replaced with $\mathcal{P}_{n}$, polynomials of degree of at most $n$, another interesting example being functions on the bi-disk $f\in\mathcal{H}(\mathbb{D}^2)$, where the radius is unknown).

One interesting question is by what means could one effectively numerically estimate the Bohr radius for a space of functions (in the case the space is polynomials in $n$ variables of degree at most $m$, say).

Question: How can one numerically (or better yet exactly) compute a sequence of polynomials $p_{k}(z_1,\dots,z_n)$ (for small degree and number of variables even!) each of degree at most $m$, and the corresponding values of $r_{k}$ so that the majorization of $p_{k}(z_1,\dots,z_n)$, $p^{*}_{k}$ attains the maximum value of $|p_{k}|$ on the poly-disk $\mathbb{D}^n$ at $p^{*}_{k}(r,r,...,r)$ for value $r=r_k$, and $r_k\rightarrow R^{n}_{m}$, the corresponding Bohr radius of this space, as $k\rightarrow \infty$ where $$R^{n}_{m}=\sup {r}\ \hspace{3 mm} \text{s.t.}\hspace{3 mm} \ \Bigl|\sum\limits_{|\alpha|\leq m} c_{\alpha}z^{\alpha}\Bigr|<1\ \text{for }(z_1,\dots,z_{n})\ \text{ s.t. } \|z\|_{\infty}=\max\limits_{1\leq j\leq n} |z_{j}|<1 \implies \sum\limits_{|\alpha|\leq m} |c_{\alpha}z^{\alpha}|<1\ \text{for } \|z\|_{\infty}<r$$

Edit: For anyone interested, some lower bounds for the Bohr radius in multiple variables are easily derived following a Cauchy-Schwarz argument as in H. P. Boas and D. Khavinson's paper (which gave first lower and upper bounds for the Bohr radius on the poly-disk). For two variable polynomials of degree $2,3,$ and $4$ respectively the bounds one gets are $\geq.26654,\geq.25261,$ and $\geq.24936$. For three variables one gets for degree $2,3,$ and $4$ the lower bounds $\geq.22014,\geq.20967,$ and $\geq.20741$.


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