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In the context of metric measure spaces, the definition of a Poincaré inequality is slightly different.One definition is: a metric measure space supports a (q,p)-Poincare inequality for some , < if there are constants C and λ ≥ 1 so that for each ball B in the space, ‖ ‖ () ‖ ‖ ().
Two-dimensional linear inequalities are expressions in two variables of the form: + < +, where the inequalities may either be strict or not. The solution set of such an inequality can be graphically represented by a half-plane (all the points on one "side" of a fixed line) in the Euclidean plane. [2]
Hardy's inequality is an inequality in mathematics, named after G. H. Hardy.. Its discrete version states that if ,,, … is a sequence of non-negative real numbers, then for every real number p > 1 one has
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.
In mathematics, the Newton inequalities are named after Isaac Newton. Suppose a 1, a 2, ..., a n are non-negative real numbers and let denote the kth elementary symmetric polynomial in a 1, a 2, ..., a n. Then the elementary symmetric means, given by = (),
Solution set (portrayed as feasible region) for a sample list of inequations. Similar to equation solving, inequation solving means finding what values (numbers, functions, sets, etc.) fulfill a condition stated in the form of an inequation or a conjunction of several inequations.
Grunsky's inequalities; Hanner's inequalities; Hardy's inequality; Hardy–Littlewood inequality; Hardy–Littlewood–Sobolev inequality; Harnack's inequality; Hausdorff–Young inequality; Hermite–Hadamard inequality; Hilbert's inequality; Hölder's inequality; Jackson's inequality; Jensen's inequality; Khabibullin's conjecture on integral ...
A function : is said to be operator convex if for all and all , with eigenvalues in , and < <, the following holds (+ ()) + (). Note that the operator + has eigenvalues in , since and have eigenvalues in .