@ARTICLE{TBA, 

TITLE={[FORTHCOMING] From Ingham to Nazarov’s inequality: a survey on some trigonometric inequalities}, 
  
AUTHOR={Philippe Jaming
, Chadi Saba, }, 
	
JOURNAL={Advances in Pure and Applied Mathematics}, 
	
VOLUME={}, 

NUMBER={Forthcoming papers<br>}, 
	
YEAR={2024}, 

URL={https://www.openscience.fr/From-Ingham-to-Nazarov-s-inequality-a-survey-on-some-trigonometric-inequalities}, 
	
DOI={TBA}, 
	
ISSN={1869-6090}, 
	
ABSTRACT={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 McGehee, 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”). Most proofs are close to existing ones and some open questions are mentionned at the end.}}