# 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)

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