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单词 JensensInequality
释义

Jensen’s inequality


If f is a convex function on the interval [a,b], for each {xk}k=1n[a,b] and each {μk}k=1n with μk0 one has:

f(k=1nμkxkknμk)k=1nμkf(xk)knμk.

A common situation occurs when μ1+μ2++μn=1; in this case, the inequalityMathworldPlanetmath simplifies to:

f(k=1nμkxk)k=1nμkf(xk)

where 0μk1.

If f is a concave function, the inequality is reversed.

Example:
f(x)=x2
is a convex function on [0,10].Then

(0.24+0.53+0.37)20.2(42)+0.5(32)+0.3(72).

A very special case of this inequality is when μk=1n because then

f(1nk=1nxk)1nk=1nf(xk)

that is, the value of the function at the mean of the xk is less or equal than the mean of the values of the function at each xk.

There is another formulation of Jensen’s inequality used in probability:
Let X be some random variableMathworldPlanetmath, and let f(x) be a convex function (defined at least on a segment containing the range of X). Then the expected valueMathworldPlanetmath of f(X) is at least the value of f at the mean of X:

E[f(X)]f(E[X]).

With this approach, the weights of the first form can be seen as probabilities.

TitleJensen’s inequality
Canonical nameJensensInequality
Date of creation2013-03-22 11:46:30
Last modified on2013-03-22 11:46:30
OwnerAndrea Ambrosio (7332)
Last modified byAndrea Ambrosio (7332)
Numerical id13
AuthorAndrea Ambrosio (7332)
Entry typeTheorem
Classificationmsc 81Q30
Classificationmsc 26D15
Classificationmsc 39B62
Classificationmsc 18-00
Related topicConvexFunction
Related topicConcaveFunction
Related topicArithmeticGeometricMeansInequality
Related topicProofOfGeneralMeansInequality
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更新时间:2025/5/4 9:04:17