A note on Bohr's phenomenon for power series

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=0∞akzk|≤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.