A note on the Bohr's phenomenon for power series

Rosihan M. Ali, Roger Barnard, Alexander Solynin

Research output: Contribution to journalArticlepeer-review

Abstract

Bohr's phenomenon, first introduced by Harald Bohr in 1914, deals with the largest radius $r$, $0<r<1$, such that the inequality $\sum_{k=0}^\infty |a_k|r^k\le 1$ holds whenever the inequality $|\sum_{k=0}^\infty a_kz^k|\le 1$ holds for all $|z|<1$. The exact value of this largest radius known as the \emph{Bohr's radius}, which is $r_b=1/3$, 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 $W_\alpha =\{w:\,|\arg w|<\pi\alpha/2\}$, $1\le \alpha \le 2$. In particular, we find the Bohr's radius for this class.
Original languageEnglish
Pages (from-to)154-167
JournalJournal of Mathematical Analysis and Applications
StatePublished - May 1 2017

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