Jensen inequality (idea) by krimson

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.

proof of the Jensen inequality	AM-GM inequality	Why equal distribution of income is good	Convex
physical proof that the arithmetic mean is at least as large as the harmonic mean	Cauchy-Schwarz inequality	Fatou's lemma	factor theorem
Proof of the Cauchy-Schwarz inequality	proof of the AM-GM inequality	Hölder's inequality	noding limericks
Friedrich Schiller	Inequality	Concave	quadratic formula
NURBS	Jensen	convex combination	Trigonometric
Regression	Polynomial	Logarithmic	exponential