New Inequalities for the Coefficients of Unimodular Bounded Functions

The classical inequality of Bohr asserts that if a power series converges in the unit disk and its sum has modulus less than or equal to 1, then the sum of absolute values of its terms is less than or equal to 1 for the subdisk | z| < 1 / 3 and 1/3 is the best possible constant. Recently, there has been a number of investigations on this topic. In this article, we present related inequalities using ∑n=0∞|an|2r2n that generalize for example the well known inequality ∑n=0∞|an|2r2n≤1.