Jensen inequality

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in the simplest discrete form

The inequality


where is a convex function on some set in (see Convex function (of a real variable)), , , , and

Equality holds if and only if or if is linear. Jensen's integral inequality for a convex function is:


where , for and

Equality holds if and only if either on or if is linear on . If is a concave function, the inequality signs in (1) and (2) must be reversed. Inequality (1) was established by O. Hölder, and (2) by J.L. Jensen [2].

With suitable choices of the convex function and the weights or weight function , inequalities (1) and (2) become concrete inequalities, among which one finds the majority of the classical inequalities. For example, if in (1) one sets , , then one obtains an inequality between the weighted arithmetic mean and the geometric mean:


for , inequality (3) takes the form


[1] O. Hölder, "Ueber einen Mittelwertsatz" Göttinger Nachr. (1889) pp. 38–47
[2] J.L. Jensen, "Sur les fonctions convexes et les inégualités entre les valeurs moyennes" Acta Math. , 30 (1906) pp. 175–193
[3] G.H. Hardy, J.E. Littlewood, G. Pólya, "Inequalities" , Cambridge Univ. Press (1934)


Jensen's inequality (2) can be generalized by taking instead a probability measure on a -algebra in a set , a bounded real-valued function in and a convex function on the range of ; then

For another generalization cf. [a2].


[a1] W. Rudin, "Real and complex analysis" , McGraw-Hill (1978) pp. 24
[a2] P.S. Bullen, D.S. Mitrinović, P.M. Vasić, "Means and their inequalities" , Reidel (1988) pp. 27ff
How to Cite This Entry:
Jensen inequality. E.K. Godunova (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098