Remarks on the Bohr phenomenon
type: Item
instance of: scholarly article
title: Remarks on the Bohr phenomenon
zbMATH Open document ID: 1067.30094
DOI: 10.1007/BF03321051
author: Catherine Bénéteau
author: Anders Dahlner
author: Dmitry Khavinson
published in: Computational Methods and Function Theory
publication date: 2004-09-28T00:00:00Z
review text: In 1914 Harald Bohr published the following surprising result: If \(f(z)=\sum a_n z^n\) is an analytic functions on the unit disc such that \(| f(z)| \leq 1\) for each \(z\) in the disc, then \(\sum | a_n z^n | \leq 1\) when \(| z| \leq 1/3\), and moreover the radius \(1/3\) is best possible. The authors note that no such Bohr phenomenon occurs if the space \(H^{\infty}\) is replaced by a Hardy space \(H^q, 0 < q < \infty\). In fact, there is no \(0<r<1\) such that \(\sum | a_n| r^n \leq 1\) for each \(f(z)=\sum a_n z^n\) in the closed unit ball of \(H^q\). \textit{P. B. Djakov} and \textit{M. S. Ramanujan}, [J.\ Anal.\ 8, 65--77 (2000; Zbl 0969.30001)], studied the growth of the series \(\sum | a_n| ^p r^n\) for bounded analytic functions on the disc. In Section 2 of the present article, the authors study the growth of \(\sum | a_n| ^p r^n\) as \(r\) tends to \(1\) for analytic functions in the unit ball of \(H^s, s>0\), particularly when \(0<p<2\). They observe that the case \(p \geq 2\) is open except for those trivial cases covered by the Hausdorff-Young theorem. In Section 3 they analyze the Bohr phenomenon for general Banach spaces \(X\) of analytic functions such that the polynomials are dense in \(X\), the point evaluation at each point of the disc is continuous and every non-constant function \(f\) in the closed unit ball of \(X\) satisfies that \(| f(0)| < 1\). A nice characterization is obtained and several examples are presented. Finally, the results are extended to several variables and, as a consequence, general Schwarz-Pick type estimates for bounded analytic functions are presented. Recently several authors have studied Bohr's theorem in higher dimension: given a finite dimensional Banach space \(X=(C^k,| | .| | )\), find the largest radius \(K(B_X)\), called the Bohr radius of the unit ball of \(X\), such that if \(| \sum a_{\alpha} z^{\alpha} | \leq 1\) for \(| | z| | <1\), then \(\sum | a_{\alpha} z^{\alpha}| \leq 1\) when \(| | z| | <K(B_X) \). A…
reviewed by: José Bonet
Mathematics Subject Classification ID: 30H05
Mathematics Subject Classification ID: 32A05
Mathematics Subject Classification ID: 32A35
Mathematics Subject Classification ID: 46E15
zbMATH DE Number: 2104110
zbMATH Keywords: Bohr's theorem
zbMATH Keywords: Bohr radius
zbMATH Keywords: Hardy spaces
zbMATH Keywords: Taylor series
zbMATH Keywords: Schwarz-Pick estimates
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2104110