The Rudin-Shapiro polynomials and The Fekete polynomials are not $L^\alpha$-flat

Abstract

We establish that the Rudin-Shapiro polynomials are not $L^\alpha$-flat, for any $\alpha \geq 0$. We further prove that the "truncated" Rudin-Shapiro sequence cannot generate a sequence of $L^\alpha$-flat polynomials, for any $\alpha \geq 0$. In the appendix, we present a simple proof of the fact that the Fekete polynomials and the modified or shifted Fekete polynomials are not $L^\alpha$-flat, for any $\alpha \geq 0$.