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

We establish that the Rudin-Shapiro polynomials are not $L^{\alpha }$-flat, for any $\alpha \ge 0$. We further prove that the “truncated” Rudin-Shapiro sequence cannot generate a sequence of $L^{\alpha }$-flat polynomials, for any $\alpha \ge 0$. 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 $\alpha \ge 0$.