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Asymmetric covariance estimates of Brascamp–Lieb type and related inequalities for log-concave measures

Eric A. Carlen, Dario Cordero-Erausquin, Elliott H. Lieb (2013)

Annales de l'I.H.P. Probabilités et statistiques

An inequality of Brascamp and Lieb provides a bound on the covariance of two functions with respect to log-concave measures. The bound estimates the covariance by the product of the L 2 norms of the gradients of the functions, where the magnitude of the gradient is computed using an inner product given by the inverse Hessian matrix of the potential of the log-concave measure. Menz and Otto [Uniform logarithmic Sobolev inequalities for conservative spin systems with super-quadratic single-site potential....

Best Constant in the Weighted Hardy Inequality: The Spatial and Spherical Version

Samko, Stefan (2005)

Fractional Calculus and Applied Analysis

Mathematics Subject Classification: 26D10.The sharp constant is obtained for the Hardy-Stein-Weiss inequality for fractional Riesz potential operator in the space L^p(R^n, ρ) with the power weight ρ = |x|^β. As a corollary, the sharp constant is found for a similar weighted inequality for fractional powers of the Beltrami-Laplace operator on the unit sphere.

Bounds for Convex Functions of Čebyšev Functional Via Sonin's Identity with Applications

Silvestru Sever Dragomir (2014)

Communications in Mathematics

Some new bounds for the Čebyšev functional in terms of the Lebesgue norms f - 1 b - a ∫ a b f ( t ) d t [ a , b ] , p and the Δ -seminorms ∥ f ∥ p Δ : = ∫ a b ∫ a b | f ( t ) - f ( s ) | p d t d s 1 p are established. Applications for mid-point and trapezoid inequalities are provided as well.

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