From Ingham to Nazarov’s inequality: a survey on some trigonometric inequalities

The aim of this paper is to give an overview of some inequalities about L p -norms (p = 1 or p = 2) of harmonic (periodic) and non-harmonic trigonometric polynomials.Among the material covered, we mention Ingham's Inequality about L 2 norms of non-harmonic trigonometric polynmials, the proof of the Littlewood conjecture by Mc Gehee, Pigno and Smith on the lower bound of the L 1 norm of harmonic trigonometric polynomials as well as its counterpart in the non-harmonic case due to Nazarov.For the later one, we give a quantitative estimate that completes our recent result with an estimate of L 1 -norms over small intervals.We also give some stronger lower bounds when the frequencies satisfy some more restrictive conditions (lacunary Fourier series, "multi-step arithmetic sequences").