Jensen's inequality (nonfiction): Difference between revisions
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In [[Mathematics (nonfiction)|mathematics]], '''Jensen's inequality''', named after the Danish mathematician [[Johan Jensen (nonfiction)|Johan Jensen]], relates the value of a convex function of an integral to the integral of the convex function. It was proved by Jensen in 1906. Given its generality, the inequality appears in many forms depending on the context | In [[Mathematics (nonfiction)|mathematics]], '''Jensen's inequality''', named after the Danish mathematician [[Johan Jensen (nonfiction)|Johan Jensen]], relates the value of a convex function of an integral to the integral of the convex function. It was proved by Jensen in 1906. Given its generality, the inequality appears in many forms depending on the context. | ||
In its simplest form the inequality states that the convex transformation of a mean is less than or equal to the mean applied after convex transformation; it is a simple corollary that the opposite is true of concave transformations. | |||
Jensen's inequality generalizes the statement that the secant line of a convex function lies above the graph of the function, which is Jensen's inequality for two points. | Jensen's inequality generalizes the statement that the secant line of a convex function lies above the graph of the function, which is Jensen's inequality for two points. | ||
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TO_DO: math | TO_DO: math | ||
== In the News == | == In the News == | ||
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* [[Gnomon algorithm]] | * [[Gnomon algorithm]] | ||
* [[Gnomon Chronicles]] | * [[Gnomon Chronicles]] | ||
* [[Mathematics]] | * [[Mathematics]] | ||
== Nonfiction cross-reference == | == Nonfiction cross-reference == | ||
* [[ | * [[A proof without words of Jensen's inequality (nonfiction)]] | ||
* [[Inequality (nonfiction)]] | |||
* [[Jensen's inequality (nonfiction)|Jensen's inequality]] - in [[Complex analysis (nonfiction)|complex analysis]], Jensen's formula, introduced by Johan Jensen (1899), relates the average magnitude of an analytic function on a circle with the number of its zeros inside the circle. It forms an important statement in the study of entire functions. | |||
* [[Karamata's inequality for a more general inequality (nonfiction)]] | |||
* [[Law of averages (nonfiction)]] | |||
* [[Mathematics (nonfiction)]] | * [[Mathematics (nonfiction)]] | ||
* [[Popoviciu's inequality (nonfiction)]] | |||
* [[Rao–Blackwell theorem (nonfiction)]] - in [[Statistics (nonfiction)|statistics]], a result which characterizes the transformation of an arbitrarily crude estimator into an estimator that is optimal by the mean-squared-error criterion or any of a variety of similar criteria. | |||
== External links == | == External links == | ||
Latest revision as of 06:47, 9 May 2020
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. It was proved by Jensen in 1906. Given its generality, the inequality appears in many forms depending on the context.
In its simplest form the inequality states that the convex transformation of a mean is less than or equal to the mean applied after convex transformation; it is a simple corollary that the opposite is true of concave transformations.
Jensen's inequality generalizes the statement that the secant line of a convex function lies above the graph of the function, which is Jensen's inequality for two points.
TO_DO: math.
Statements
The classical form of Jensen's inequality involves several numbers and weights. The inequality can be stated quite generally using either the language of measure theory or (equivalently) probability. In the probabilistic setting, the inequality can be further generalized to its full strength.
TO_DO: math.
Proofs
Jensen's inequality can be proved in several ways, and three different proofs corresponding to the different statements above will be offered. Before embarking on these mathematical derivations, however, it is worth analyzing an intuitive graphical argument based on the probabilistic case where X is a real number (see figure) ...
TO_DO: math
In the News
Fiction cross-reference
Nonfiction cross-reference
- A proof without words of Jensen's inequality (nonfiction)
- Inequality (nonfiction)
- Jensen's inequality - in complex analysis, Jensen's formula, introduced by Johan Jensen (1899), relates the average magnitude of an analytic function on a circle with the number of its zeros inside the circle. It forms an important statement in the study of entire functions.
- Karamata's inequality for a more general inequality (nonfiction)
- Law of averages (nonfiction)
- Mathematics (nonfiction)
- Popoviciu's inequality (nonfiction)
- Rao–Blackwell theorem (nonfiction) - in statistics, a result which characterizes the transformation of an arbitrarily crude estimator into an estimator that is optimal by the mean-squared-error criterion or any of a variety of similar criteria.
External links
- Jensen's inequality @ Wikipedia