A Family of Integral Inequalities on the Interval \([-1,1]\)

  • Ali Hafidi
  • Moulay Rchid Sidi Ammi
  • Praveen Agarwal
Part of the Trends in Mathematics book series (TM)


We study the heat semigroup \((P^{n}_{t})_{t\ge 0}=\{\exp (tL_{n})\}_{t\ge 0}\) generated by the Gegenbauer operator \(L_{n}:=(1-x^{2})\frac{d^{2}}{dx^{2}}-nx\frac{d}{dx}\), on the interval \([-1,1]\) equipped with the probability measure    \(\mu _{n}(dx):=c_{n}(1-x^{2})^{\frac{n}{2}-1}\), where \(c_{n}\) the normalization constant and n is a strictly positive real number. By means of a simple method involving essentially a commutation property between the semigroup and derivation, we describe a large family of optimal integral inequalities with logarithmic Sobolev and Poincaré inequalities as particular cases.


Heat semigroup Gegenbauer operator Spectral gap Poincaré’s inequality Sobolev’s inequality Logarithmic Sobolev inequality \(\varphi \)-entropy inequality 

Mathematics Subject Classification 2010

39B62 39B72 44A15 46E35 60J25 


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Copyright information

© Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  • Ali Hafidi
    • 1
  • Moulay Rchid Sidi Ammi
    • 1
  • Praveen Agarwal
    • 2
    • 3
  1. 1.Department of Mathematics, AMNEA Group, Faculty of Sciences and TechniquesMoulay Ismail UniversityErrachidiaMorocco
  2. 2.Department of MathematicsAnand International College of EngineeringJaipurIndia
  3. 3.International Centre for Basic and Applied SciencesJaipurIndia

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