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) |
Comments
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].
References
[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. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Jensen_inequality&oldid=16975