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 a_{1} + a_{2} + ... + a_{n} = 1

SUM(k=1, n)(a_{k}f(x_{k})) ≤
f(SUM(k=1, n)(a_{k}f(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.