The Bohr radius of the \(n\)-dimensional polydisk is equivalent to \(\sqrt{(\log n) / n}\)
type: Item
instance of: scholarly article
title: The Bohr radius of the \(n\)-dimensional polydisk is equivalent to \(\sqrt{(\log n) / n}\)
zbMATH Open document ID: 1331.46037
author: J. B. Seoane-Sepúlveda
author: Frédéric Bayart
author: Daniel Pellegrino
published in: Advances in Mathematics
publication date: 2014-08-26T00:00:00Z
full work available at URL: https://arxiv.org/abs/1310.2834
Mathematics Subject Classification ID: 46G25
zbMATH DE Number: 6334316
zbMATH Keywords: Blei's inequality
zbMATH Keywords: Bohnenblust-Hille inequality
zbMATH Keywords: Bohr radius
zbMATH Keywords: interpolation
MaRDI profile type: Publication
OpenAlex ID: W2964256713
arXiv ID: 1310.2834
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DOI: 10.1016/J.AIM.2014.07.029
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review text: This paper provides at least four striking results: a surprising and far reaching improvement of an inequality due to Ron Blei, the best known constants of the multilinear Bohnenblust--Hille inequality, the subexponentiality of the complex polynomial Bohnenblust-Hille inequality, and the final solution to the famous Bohr radius problem.NEWLINENEWLINEFor two positive integers \(n,m\), let NEWLINE\[NEWLINE \mathcal{M}(m,n) =\big\{\mathbf{i}=(i_{1},\dots ,i_{m});\;i_{1},\dots ,i_{m}\in \{1,\dots ,n\}\big\}NEWLINE\]NEWLINE NEWLINE\[NEWLINE \mathcal{J}(m,n) =\big\{\mathbf{i}\in \mathcal{M}(m,n);\;i_{1}\leq i_{2}\leq \dots \leq i_{m}\big\}. NEWLINE\]NEWLINE For \(1\leq k\leq m\), let \(\mathcal{P}_{k}(m)\) denote the set of subsets of \( \{1,\dots ,m\}\) with cardinality \(k\). For \(S=\{s_{1},\dots ,s_{k}\}\) in \( \mathcal{P}_{k}(m)\), let \(\hat{S}\) denote its complement in \(\{1,\dots ,m\}\), and \(\mathbf{i}_{S}\) denotes \((i_{s_{1}},\dots ,i_{s_{k}})\in \mathcal{M} (k,n)\).NEWLINENEWLINEThe first main result of the paper is a beautiful and surprising generalization of Blei's inequality, which is of independent interest. It reads as follows:NEWLINENEWLINE{Theorem 1.} Let \(m,n\geq 1\) and \(1\leq k\leq m\). Then, for all families \((a_{ \mathbf{i}})_{\mathbf{i}\in \mathcal{M}(m,n)}\) of complex numbers, NEWLINE\[NEWLINE \left( \sum_{\mathbf{i}\in \mathcal{M}(m,n)}|a_{\mathbf{i}}|^{\frac{2m}{m+1} }\right) ^{\frac{m+1}{2m}}\leq \prod_{S\in \mathcal{P}_{k}(m)}\left( \sum_{ \mathbf{i}_{S}}\left( \sum_{\mathbf{i}_{\hat{S}}}|a_{\mathbf{i}}|^{2}\right) ^{\frac{1}{2}\times \frac{2k}{k+1}}\right) ^{\frac{k+1}{2k}\times \frac{1}{ \binom{m}{k}}}. NEWLINE\]NEWLINE The proof uses a very interesting interpolative approach that relies on the mixed Hölder inequality for mixed sums.NEWLINENEWLINERecall that the multilinear Bohnenblust-Hille inequality, proved in [\textit{H. F. Bohnenblust} and \textit{E. Hille}, Annals of Math. (2) 32, 600--622 (1931; JFM 57.0266.05)], asserts that…
description: scientific article; zbMATH DE number 6334316