@ARTICLE{À venir, 

TITLE={[FORTHCOMING] D’Ingham à l’inégalité de Nazarov : un survey sur quelques inégalités trigonométriques}, 
  
AUTHOR={Philippe Jaming
, Chadi Saba, }, 
	
JOURNAL={Avancées en Mathématiques Pures et Appliquées}, 
	
VOLUME={}, 

NUMBER={Articles à paraître<br>}, 
	
YEAR={2024}, 

URL={http://openscience.fr/D-Ingham-a-l-inegalite-de-Nazarov-un-survey-sur-quelques-inegalites}, 
	
DOI={À venir}, 
	
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, &#8220;multi-step arithmetic sequences&#8221;). Most proofs are close to existing ones and some open questions are mentionned at the end.}}