Math 2.0

Real functions Classical inequalities Theorem

Jensen's inequality

Builds on Convex function

Statement

Let II be an interval and let f ⁣:I→ℝf \colon I \to \mathbb{R} be convex. Let x1,…,xn∈Ix_1, \dots, x_n \in I, and let w1,…,wn≥0w_1, \dots, w_n \ge 0 with w1+⋯+wn=1w_1 + \cdots + w_n = 1. Then f(∑i=1nwixi)≤∑i=1nwif(xi).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=2n = 2 is the definition of convexity. Induction on nn gives the general case. For a concave function, the inequality is reversed. With f=ln⁡f = \ln this gives the AM–GM inequality. Mathlib states it as ConvexOn.map_sum_le.

History

Johan Jensen proved the inequality in 1906.

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Where this leads

17 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Jensen%27s_inequality, Finite form
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