A note on Bohr's phenomenon for power series
type: Item
instance of: scholarly article
title: A note on Bohr's phenomenon for power series
zbMATH Open document ID: 1360.30002
author: Roger W. Barnard
author: A. Yu. Solynin
author: Rosihan M. Ali
published in: Journal of Mathematical Analysis and Applications
publication date: 2017-01-17T00:00:00Z
review text: Let \(f(z)=\sum_{n=1}^{\infty}a_n z^n\) be a power series with radius of convergence 1 and \(\mathbb{D}=\{z\in\mathbb{C}: |z|<1\}\). For \(z\in\mathbb{D}\), the authors use the notation: \[ M_f(z)=\sum_{n=1}^{\infty}|a_n| |z|^n \;\text{ and } A_f(z)=\sum_{n=1}^{\infty}(-1)^n|a_n| |z|^n. \] Let \(\mathcal{S}=\{f\in H(\mathbb{D}): f(\mathbb{D})\subset \overline{\mathbb{D}}\}\). It is known [\textit{H. Bohr}, Proc. Lond. Math. Soc. (2) 13, 1--5 (1914; JFM 44.0289.01)] that \[ \sup\big\{r>0: M_f(r)\leq 1 \;\forall f\in\mathcal{S} \big\}=\frac{1}{3}. \] Using the above fact and basic properties of power series, the authors prove that \[ \sup\big\{r>0: M_f(r)\leq 1 \;\forall f\in\mathcal{S}, \;f \text{ even}\big\}=\frac{1}{\sqrt{3}} \] \[ \sup\big\{r>0: |A_f(r)|\leq 1 \;\forall f\in\mathcal{S}\big\}=\frac{1}{\sqrt{3}}, \] and they give a lower and an upper bound for \[ \sup\big\{r>0: M_f(r)\leq 1 \;\forall f\in\mathcal{S}, \;f \text{ odd}\big\}. \] In the second part of the article, they prove that, for \(\alpha\in[1,2]\), \[ \sup\big\{r>0: M_f(z)\leq \mathrm{dist}(f(0), \partial W_{\alpha}), \;\forall f\in \mathcal{S_{\alpha}}\big\}=\frac{2^{\frac{1}{\alpha}}-1}{2^{\frac{1}{\alpha}}+1}, \] where \(W_{\alpha}=\{w\in\mathbb{C}: |\mathrm{arg }w|<\frac{\pi \alpha}{2}\}\) and \(\mathcal{S_{\alpha}}\) contains all analytic functions \(f\) that map \(\mathbb{D}\) into \(W_{\alpha}\).
reviewed by: V. Vlachou
Mathematics Subject Classification ID: 30B10
zbMATH DE Number: 6675583
zbMATH Keywords: Bohr radius
zbMATH Keywords: power series
zbMATH Keywords: conformal mappings
MaRDI profile type: Publication
full work available at URL: j.jmaa.2016.11.049
OpenAlex ID: W2550065328
cites work: Bohr's phenomenon in subordination and bounded harmonic classes
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cites work: Bohr radius for subordinating families of analytic functions and bounded harmonic mappings
cites work: Bohr's phenomenon for analytic functions and the hyperbolic metric
cites work: Q5673143
cites work: Generalization of results about the Bohr radius for power series
cites work: Brannan’s conjecture and trigonometric sums
cites work: Remarks on the Bohr phenomenon
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cites work: Q3845489
cites work: Q4055949
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cites work: ON BOHR'S INEQUALITY
cites work: ON BRANNAN'S COEFFICIENT CONJECTURE AND APPLICATIONS
DOI: 10.1016/J.JMAA.2016.11.049
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description: scientific article; zbMATH DE number 6675583