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:


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


where 0μk1.

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

is a convex function on [0,10]. Then


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


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:


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

Title Jensen’s inequality
Canonical name JensensInequality
Date of creation 2013-03-22 11:46:30
Last modified on 2013-03-22 11:46:30
Owner Andrea Ambrosio (7332)
Last modified by Andrea Ambrosio (7332)
Numerical id 13
Author Andrea Ambrosio (7332)
Entry type Theorem
Classification msc 81Q30
Classification msc 26D15
Classification msc 39B62
Classification msc 18-00
Related topic ConvexFunction
Related topic ConcaveFunction
Related topic ArithmeticGeometricMeansInequality
Related topic ProofOfGeneralMeansInequality