Some articles on inequality:
... They then show that for t > 0 For k = 2 we obtain Chebyshev's inequality ... and assuming that the kth moment exists, this bound is tighter than Chebyshev's inequality ...
... Selberg derived an inequality for P(x) when a ≤ x ≤ b ... related to the mean (μY) and variance (σY) of Y With this notation Selberg's inequality states that These are known to be the best possible bounds ...
... Then Minkowski's integral inequality is (Stein 1970, §A.1), (Hardy, Littlewood Pólya 1988, Theorem 202) with obvious modifications in the case p = ∞ ... measure on a two-point set S1 = {1,2}, then Minkowski's integral inequality gives the usual Minkowski inequality as a special case for putting ƒi(y) = F(i,y) for i = 1,2, the integral inequality ...
... Cantelli's inequality due to Francesco Paolo Cantelli states that for a real random variable (X) with mean (μ) and variance (σ2) where a ≥ 0 ... This inequality can be used to prove a one tailed variant of Chebyshev's inequality with k > 0 The bound on the one tailed variant is known to be sharp ... There is no need to assume that the variance is finite because this inequality is trivially true if the variance is infinite ...
... In mathematical analysis, the Minkowski inequality establishes that the Lp spaces are normed vector spaces ... Then f + g is in Lp(S), and we have the triangle inequality with equality for 1 < p < ∞ if and only if f and g are positively linearly dependent, i.e ... p = ∞ by the essential supremum The Minkowski inequality is the triangle inequality in Lp(S) ...
Famous quotes related to inequality:
“A man willing to work, and unable to find work, is perhaps the saddest sight that fortune’s inequality exhibits under this sun.”
—Thomas Carlyle (1795–1881)
“Love is a great thing. It is not by chance that in all times and practically among all cultured peoples love in the general sense and the love of a man for his wife are both called love. If love is often cruel or destructive, the reasons lie not in love itself, but in the inequality between people.”
—Anton Pavlovich Chekhov (1860–1904)
“Nature is unfair? So much the better, inequality is the only bearable thing, the monotony of equality can only lead us to boredom.”
—Francis Picabia (1878–1953)
“However energetically society in general may strive to make all the citizens equal and alike, the personal pride of each individual will always make him try to escape from the common level, and he will form some inequality somewhere to his own profit.”
—Alexis de Tocqueville (1805–1859)
“The doctrine of equality!... But there exists no more poisonous poison: for it seems to be preached by justice itself, while it is the end of justice.... ‘Equality for equals, inequality for unequals’Mthat would be the true voice of justice: and, what follows from it, ‘Never make equal what is unequal.’”
—Friedrich Nietzsche (1844–1900)