# Computing Bohr Radii

The Bohr radius $$R$$ for $$\mathcal{H}(\mathbb{D})$$ is defined as $$R = \sup\limits_{0 where $$|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}

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 $$\geq.20741$$.