# Jensen inequality

in the simplest discrete form

The inequality

 (1)

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:

 (2)

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:

 (3)

for , inequality (3) takes the form

#### References

 [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)