We exhibit a class of Littlewood polynomials that are not \( L^\alpha \)-flat for any \( \alpha \geq 0 \). Indeed, it is shown that the sequence of Littlewood polynomials is not \( L^\alpha \)-flat, \( \alpha \geq 0 \), when the frequency of \( -1 \) is not in the interval \( \left[\dfrac{1}{4}, \dfrac{3}{4}\right] \). We further obtain a generalization of Jensen-Jensen-Hoholdt's result by establishing the sequence of Littlewood polynomials is not \( L^\alpha \)-flat for any \( \alpha > 2 \) if the frequency of \( -1 \) is not \( \dfrac{1}{2} \). Finally, we prove that the sequence of palindromic Littlewood polynomials with even degrees are not \( L^\alpha \)-flat for any \( \alpha \geq 0 \), and we provide a lemma on the existence of \( c \)-flat polynomials.