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Such inequalities are of importance in several fields, including communication complexity (e.g., in proofs of the gap Hamming problem [13]) and graph theory. [14] An interesting anti-concentration inequality for weighted sums of independent Rademacher random variables can be obtained using the Paley–Zygmund and the Khintchine inequalities. [15]
Bennett's inequality, an upper bound on the probability that the sum of independent random variables deviates from its expected value by more than any specified amount Bhatia–Davis inequality , an upper bound on the variance of any bounded probability distribution
Inequalities are governed by the following properties. All of these properties also hold if all of the non-strict inequalities (≤ and ≥) are replaced by their corresponding strict inequalities (< and >) and — in the case of applying a function — monotonic functions are limited to strictly monotonic functions.
A great many important inequalities in information theory are actually lower bounds for the Kullback–Leibler divergence.Even the Shannon-type inequalities can be considered part of this category, since the interaction information can be expressed as the Kullback–Leibler divergence of the joint distribution with respect to the product of the marginals, and thus these inequalities can be ...
Carleman's inequality is an inequality in mathematics, named after Torsten Carleman, who proved it in 1923 [1] and used it to prove the Denjoy–Carleman theorem on quasi-analytic classes. [ 2 ] [ 3 ]
Hölder's inequality is used to prove the Minkowski inequality, which is the triangle inequality in the space L p (μ), and also to establish that L q (μ) is the dual space of L p (μ) for p ∈ [1, ∞). Hölder's inequality (in a slightly different form) was first found by Leonard James Rogers .
Compound interest has been described as the "eighth wonder of the world," but many investors don't fully grasp its power. And while it seems ridiculous to think that someone could turn $1 into $1...
When Ω is a ball, the above inequality is called a (p,p)-Poincaré inequality; for more general domains Ω, the above is more familiarly known as a Sobolev inequality. The necessity to subtract the average value can be seen by considering constant functions for which the derivative is zero while, without subtracting the average, we can have ...