The Bohr radius and its modifications for linearly invariant families of analytic functions

Abstract

In 1914, Harald Bohr published an article in which he considered the class B of all analytic functions in the unit disk |z|<1, bounded in absolute value by 1. The article contains proofs that for any function f(z)=∑n=0∞anzn from B, the inequality ∑n=0∞|anzn|≤1 holds in the disk of radius 1/3 centered at the origin, and the value 1/3 is optimal. Also, Bohr himself in this article proved the corresponding result in a circle of radius 1/6 and adds, following the request of Wiener, Wiener's later proof for the disk of radius 1/3. Since then, the constant 1/3 in this problem has been called the Bohr radius. This was followed by a series of papers dealing with analogues of the Bohr radius or its estimates in other classes of functions. In this article, we give estimates for the Bohr radius in some classes of analytic functions in D, associated with linearly invariant families of finite order.