Improved version of Bohr's inequality
type: Item
instance of: scholarly article
title: Improved version of Bohr's inequality
zbMATH Open document ID: 1432.30002
author: Saminathan Ponnusamy
author: I. R. Kayumov
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2018-03-12T00:00:00Z
review text: Let \(f\) be a bounded analytic function on the unit disk \(\mathbb{D}=\{z\in\mathbb{C}: |z|<1\}\) with Taylor expansion \(\displaystyle \sum_{k=0}^{\infty} a_kz^k\). The classical Bohr inequality states that: \[M_f(r)=\displaystyle\sum_{n=0}^{\infty}|a_n|r^n\leq||f||_{\infty}, \ \text{for} \ 0\leq r\leq\frac{1}{3}\] and the constant \(\frac{1}{3}\) is sharp. In this interesting article, the authors give several different improved versions of the aforementioned inequality. Let us state one of them: Let \(f\) be a bounded analytic function on the unit disk \(\mathbb{D}\) with \(||f||_{\infty}\leq 1\). Then \[|a_0|+\sum_{k=1}^{\infty}\biggl( |a_k|+\frac{1}{2}|a_k|^2\biggl)r^k\leq 1, \ \text{for} \ 0\leq r\leq\frac{1}{3}\] and the numbers \(\frac{1}{3}\) and \(\frac{1}{2}\) cannot be improved.
reviewed by: V. Vlachou
Mathematics Subject Classification ID: 30A10
Mathematics Subject Classification ID: 30H05
Mathematics Subject Classification ID: 30B10
zbMATH DE Number: 6849184
zbMATH Keywords: Bohr's inequality
zbMATH Keywords: bounded holomorphic functions
zbMATH Keywords: Taylor series
MaRDI profile type: Publication
full work available at URL: j.crma.2018.01.010
OpenAlex ID: W2791426992
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DOI: 10.1016/J.CRMA.2018.01.010
description: scientific article; zbMATH DE number 6849184