Namespaces
Variants
Actions

Jensen inequality

From Encyclopedia of Mathematics
Jump to: navigation, search
The printable version is no longer supported and may have rendering errors. Please update your browser bookmarks and please use the default browser print function instead.


in the simplest discrete form

The inequality

$$ \tag{1 } f ( \lambda _ {1} x _ {1} + \dots + \lambda _ {n} x _ {n} ) \leq \ \lambda _ {1} f ( x _ {1} ) + \dots + \lambda _ {n} f ( x _ {n} ), $$

where $ f $ is a convex function on some set $ C $ in $ \mathbf R $( see Convex function (of a real variable)), $ x _ {i} \in C $, $ \lambda _ {i} \geq 0 $, $ i = 1 \dots n $, and

$$ \lambda _ {1} + \dots + \lambda _ {n} = 1. $$

Equality holds if and only if $ x _ {1} = \dots = x _ {n} $ or if $ f $ is linear. Jensen's integral inequality for a convex function $ f $ is:

$$ \tag{2 } f \left ( \int\limits _ { D } \lambda ( t) x ( t) dt \right ) \leq \int\limits _ { D } \lambda ( t) f ( x ( t)) dt, $$

where $ x ( D) \subset C $, $ \lambda ( t) \geq 0 $ for $ t \in D $ and

$$ \int\limits _ { D } \lambda ( t) dt = 1. $$

Equality holds if and only if either $ x ( t) = \textrm{ const } $ on $ D $ or if $ f $ is linear on $ x ( D) $. If $ f $ 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 $ f $ and the weights $ \lambda _ {i} $ or weight function $ \lambda $, inequalities (1) and (2) become concrete inequalities, among which one finds the majority of the classical inequalities. For example, if in (1) one sets $ f( x) = - \mathop{\rm ln} x $, $ x > 0 $, then one obtains an inequality between the weighted arithmetic mean and the geometric mean:

$$ \tag{3 } x _ {1} ^ {\lambda _ {1} } \dots x _ {n} ^ {\lambda _ {n} } \leq \ \lambda _ {1} x _ {1} + \dots + \lambda _ {n} x _ {n} ; $$

for $ \lambda _ {1} = \dots = \lambda _ {n} = 1/n $, inequality (3) takes the form

$$ ( x _ {1} \dots x _ {n} ) ^ {1/n} \leq \ \frac{x _ {1} + \dots + x _ {n} }{n} . $$

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 $ \mu $ on a $ \sigma $- algebra $ {\mathcal M} $ in a set $ D \subset \mathbf R $, $ x $ a bounded real-valued function in $ L _ {1} ( \mu ) $ and $ f $ a convex function on the range of $ x $; then

$$ f \left ( \int\limits _ { D } x d \mu \right ) \leq \ \int\limits _ { D } ( f \circ x) d \mu . $$

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
How to Cite This Entry:
Jensen inequality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Jensen_inequality&oldid=47465
This article was adapted from an original article by E.K. Godunova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article