Jensen's inequality

Statement

Let $I$ be an interval and let $f \colon I \to \mathbb{R}$ be convex. Let $x_1, \dots, x_n \in I$, and let $w_1, \dots, w_n \ge 0$ with $w_1 + \cdots + w_n = 1$. Then $$f\Big( \sum_{i=1}^{n} w_i x_i \Big) \le \sum_{i=1}^{n} w_i f(x_i).$$

Remarks

The case $n = 2$ is the definition of convexity. Induction on $n$ gives the general case. For a concave function, the inequality is reversed. With $f = \ln$ this gives the AM–GM inequality. Mathlib states it as ConvexOn.map_sum_le.

History

Johan Jensen proved the inequality in 1906.

Explanations

L1 human

The value of a convex function at an average is at most the average of its values. The weights $w_i$ say how much each point counts in the average.