Jensen inequality

The Jensen inequality is the following theorem:

If the real function f is convex on the interval I and x_i is a member of I for i = 1, 2, ..., n, and a₁ + a₂ + ... + a_n = 1

SUM(k=1, n)(a_kf(x_k)) ≤ f(SUM(k=1, n)(a_kf(x_k)))

or in words: the arithmetic mean of the function values is less than the function value of the arithmetic mean. If f is concave instead of convex the inequality is reversed.
This theorem is normally proved by induction.

This inequality is very useful since it applies to such a wide range of functions: polynomial, exponential, logarithmic, trigonometric to give some examples. If we eg consider the logarithmic function we get the AM-GM inequality.

More generally (and in probability-speak), for any probability measure, any random variable X taking value in some interval I and any convex function f defined on I,

Ef(X) ≤ f(EX)

The same also holds if we replace the interval I with any convex region in Rⁿ.

See a proof of the Jensen inequality, if you like.