eng Science and Education Publishing Journal of Mathematical Sciences and Applications 2333-8792 2017-06-09 5 1 19 23 10.12691/jmsa-5-1-3 JMSA2017513 article Khalid Boutahir 1 Ali Hafidi hafidiali28@gmail.com 2 Département de Mathématiques & Informatique, Université My Ismail, B. P. 11 201 Zitoune, Meknès, MAROC Faculté des Sciences et Techniques, B.P.509, Boutalamine Errachidia, MAROC We consider on the interval [-1,1] the heat semigroup generated by the Legendre operator acting on the Hilbert space with respect to the uniform measure By means of a simple method involving some semigroup techniques, we describe a large family of optimal integral inequalities with the Poincaré and logarithmic Sobolev inequalities as particular cases. http://pubs.sciepub.com/jmsa/5/1/3/jmsa-5-1-3.pdf heat semigroup legendre operator spectral gap poincaré inequality sobolev inequality logarithmic sobolev inequality φ-entropy inequality