A Class of Littlewood Polynomials that are Not L^α-Flat

We exhibit a class of Littlewood polynomials that are not L^α-flat for any α ≥ 0. Indeed, it is shown that the sequence of Littlewood polynomials is not L^α-flat, α ≥ 0, when the frequency of −1 is not in the interval ]14{1 \over 4}, 34{3 \over 4}[ We further obtain a generalization of Jensen-Jensen-Hoholdt’s result by establishing that the sequence of Littlewood polynomials is not L^α-flat for any α> 2 if the frequency of −1 is not 12{1 \over 2}. Finally, we prove that the sequence of palindromic Littlewood polynomials with even degrees are not L^α-flat for any α ≥ 0, and we provide a lemma on the existence of c-flat polynomials.