Asymptotic Bohr radius for the polynomials in one complex variable (Q2827659)

Let \(\mathbb{D}\) be the open unit disk in the complex plane \(\mathbb{C}\). Let \(H^{\infty}\) be the Banach space of bounded analytic functions on \(\mathbb{D}\) with the norm \(\|f\|_{\infty}=\sup_{z\in \mathbb{D}}|f(z)|\). Let \(\mathcal{P}_n\) denote the subspace of \(H^{\infty}\) consisting of all complex polynomials of degree at most \(n\). The Bohr-type radius \(R_n\) for the class \(\mathcal{P}_n\) is defined by NEWLINE\[NEWLINER_n= \sup\bigg\{r\in (0, 1): \sum_{k=0}^{\infty}|a_k|r^k\leq \|p\|_{\infty}, \;\;\text{for all} \;\;p(z)=\sum_{k=0}^{n}a_k z^k \in \mathcal{P}_n\bigg\}.NEWLINE\]NEWLINE Based on numerical evidence, \textit{R. Fournier} [J. Math. Anal. Appl. 338, No. 2, 1100--1107 (2008; Zbl 1155.30002)] conjectured that NEWLINE\[NEWLINER_n= \frac{1}{3} + \frac{\pi^2}{3n^2}+ o\Big(\frac{1}{n^2}\Big).NEWLINE\]NEWLINE In the paper under review, the author proves that this conjecture is true. In fact, the author proves that NEWLINE\[NEWLINE\lim_{n\rightarrow\infty} n^2 \Big(R_n -\frac{1}{3}\Big) =\frac{\pi^2}{3}.NEWLINE\]NEWLINENEWLINENEWLINEFor the entire collection see [Zbl 1318.47003].