A note on Bohr's phenomenon for power series

Rosihan M. Ali, Roger W. Barnard, Alexander Yu Solynin

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Abstract

Bohr's phenomenon, first introduced by Harald Bohr in 1914, deals with the largest radius r, 0<r<1, such that the inequality ∑k=0|ak|rk≤1 holds whenever the inequality |∑k=0akzk|≤1 holds for all |z|<1. The exact value of this largest radius known as Bohr's radius, which is rb=1/3, was discovered long ago. In this paper, we first discuss Bohr's phenomenon for the classes of even and odd analytic functions and for alternating series. Then we discuss Bohr's phenomenon for the class of analytic functions from the unit disk into the wedge domain Wα={w:|arg⁡w|<πα/2}, 1≤α≤2. In particular, we find Bohr's radius for this class.

Original languageEnglish
Pages (from-to)154-167
Number of pages14
JournalJournal of Mathematical Analysis and Applications
Volume449
Issue number1
DOIs
StatePublished - May 1 2017

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Keywords

  • Analytic function
  • Bohr's radius

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