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For example, the set of real numbers consisting of 0, 1, and all numbers in between is an interval, denoted [0, 1] and called the unit interval; the set of all positive real numbers is an interval, denoted (0, ∞); the set of all real numbers is an interval, denoted (−∞, ∞); and any single real number a is an interval, denoted [a, a].
The main objective of interval arithmetic is to provide a simple way of calculating upper and lower bounds of a function's range in one or more variables. These endpoints are not necessarily the true supremum or infimum of a range since the precise calculation of those values can be difficult or impossible; the bounds only need to contain the function's range as a subset.
If α is the zero function and u is non-negative, then Grönwall's inequality implies that u is the zero function. The integrability of u with respect to μ is essential for the result. For a counterexample, let μ denote Lebesgue measure on the unit interval [0, 1], define u(0) = 0 and u(t) = 1/t for t ∈ (0, 1], and let α be the zero function.
unstrict inequality signs (less-than or equals to sign and greater-than or equals to sign) 1670 (with the horizontal bar over the inequality sign, rather than below it) John Wallis: 1734 (with double horizontal bar below the inequality sign) Pierre Bouguer
This identity is used in a simple proof of Markov's inequality. In many cases, such as order theory , the inverse of the indicator function may be defined. This is commonly called the generalized Möbius function , as a generalization of the inverse of the indicator function in elementary number theory , the Möbius function .
The extremal equality is one of the ways for proving the triangle inequality ‖ f 1 + f 2 ‖ p ≤ ‖ f 1 ‖ p + ‖ f 2 ‖ p for all f 1 and f 2 in L p (μ), see Minkowski inequality. Hölder's inequality implies that every f ∈ L p (μ) defines a bounded (or continuous) linear functional κ f on L q (μ) by the formula
Jensen's inequality generalizes the statement that a secant line of a convex function lies above its graph. Visualizing convexity and Jensen's inequality. In mathematics, Jensen's inequality, named after the Danish mathematician Johan Jensen, relates the value of a convex function of an integral to the integral of the convex function.
Muirhead's inequality states that [a] ≤ [b] for all x such that x i > 0 for every i ∈ { 1, ..., n} if and only if there is some doubly stochastic matrix P for which a = Pb. Furthermore, in that case we have [ a ] = [ b ] if and only if a = b or all x i are equal.