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Cauchy–Schwarz inequality (Modified Schwarz inequality for 2-positive maps [27]) — For a 2-positive map between C*-algebras, for all , in its domain, () ‖ ‖ (), ‖ ‖ ‖ ‖ ‖ ‖. Another generalization is a refinement obtained by interpolating between both sides of the Cauchy–Schwarz inequality:
As an application of the above estimate, we can obtain the Stieltjes–Vitali theorem, [5] which says that a sequence of holomorphic functions on an open subset that is bounded on each compact subset has a subsequence converging on each compact subset (necessarily to a holomorphic function since the limit satisfies the Cauchy–Riemann equations).
In mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis.It expresses the fact that a holomorphic function defined on a disk is completely determined by its values on the boundary of the disk, and it provides integral formulas for all derivatives of a holomorphic function.
The special case p = q = 2 gives a form of the Cauchy–Schwarz inequality. [1] Hölder's inequality holds even if ‖ fg ‖ 1 is infinite, the right-hand side also being infinite in that case. Conversely, if f is in L p (μ) and g is in L q (μ), then the pointwise product fg is in L 1 (μ).
Many important inequalities can be proved by the rearrangement inequality, such as the arithmetic mean – geometric mean inequality, the Cauchy–Schwarz inequality, and Chebyshev's sum inequality. As a simple example, consider real numbers : By applying with := for all =, …,, it follows that + + + + + + for every permutation of , …,.
The Fisher information matrix plays a role in an inequality like the isoperimetric inequality. [29] Of all probability distributions with a given entropy, the one whose Fisher information matrix has the smallest trace is the Gaussian distribution. This is like how, of all bounded sets with a given volume, the sphere has the smallest surface area.
where the second form can be obtained by expanding the square and using the fact that the integral of a probability density over its domain equals 1. The Hellinger distance H(P, Q) satisfies the property (derivable from the Cauchy–Schwarz inequality) (,)
Cauchy–Schwarz inequality – Mathematical inequality relating inner products and norms Hölder's inequality – Inequality between integrals in Lp spaces Mahler's inequality – inequality relating geometric mean of two finite sequences of positive numbers to the sum of each separate geometric mean Pages displaying wikidata descriptions as a ...