A note on Bohr's phenomenon for power series

作者：Ali Rosihan M; Barnard Roger W; Solynin Alexander Yu^*

来源：Journal of Mathematical Analysis and Applications, 2017, 449(1): 154-167.

DOI：10.1016/j.jmaa.2016.11.049

• Bohr's radius
• Analytic function

Bohr's phenomenon, first introduced by Harald Bohr in 1914, deals with the largest radius r, 0 < r < 1, such that the inequality Sigma(infinity)(k=0 vertical bar)a(k vertical bar)r(k) <= 1 holds whenever the inequality vertical bar Sigma(infinity)(k=0) a(k)z(k)vertical bar <= 1 holds for all vertical bar z vertical bar < 1. The exact value of this largest radius known as 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 : vertical bar arg w vertical bar < pi alpha/2},1 <= alpha <= 2. In particular, we find Bohr's radius for this class.

• 出版日期2017-5-1