Jensen inequality

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