in the simplest discrete form
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 .
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
|||O. Hölder, "Ueber einen Mittelwertsatz" Göttinger Nachr. (1889) pp. 38–47|
|||J.L. Jensen, "Sur les fonctions convexes et les inégualités entre les valeurs moyennes" Acta Math. , 30 (1906) pp. 175–193|
|||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|
Jensen inequality. E.K. Godunova (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Jensen_inequality&oldid=16975