Unboundedness of the Coefficients of Higher Powers of a Unimodular Power Series

Abstract

Let $R(z)={\sum }_{n=0}^{\infty }{r}_n{z}^n$ be a power series with $|r_n|=1$ for every $n\ge 0$. We show that for each integer $m\ge 2$, the coefficient sequence of $R(z{)}^m$ is unbounded. The proof combines Parseval's identity with Jensen's inequality. As a consequence, Conjecture~3.9 of Gawron, Miska, and Ulas \cite{gmu} is confirmed.