The Jensen inequality
is the following theorem
If the real function f is convex on the interval I and
xi is a member of I for i = 1, 2, ..., n, and a1 + a2 + ... + an = 1
SUM(k=1, n)(akf(xk)) ≤
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.