A Class of Littlewood Polynomials that are Not Lα-Flat

Abstract

Abstract We exhibit a class of Littlewood polynomials that are not Lα-flat for any α ≥ 0. Indeed, it is shown that the sequence of Littlewood polynomials is not Lα-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 Lα-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 Lα-flat for any α ≥ 0, and we provide a lemma on the existence of c-flat polynomials.