Jensen's Inequality

Weighted Jensen’s Inequality

Let be an interval on the reals be Convex on , with and real weights (In many scenarios the weights will be normalised such that ), Then

If is concave then the inequality is flipped. The equality holds if or is affine on the on a domain containing .

Proof

1. W.L.O.G let .
2. Trivially for so and the (in)equality holds.
3. The definition of convexity means . This is the case.
4. Assume that Jensen’s holds for , then for with weights (arbitrary, not necessarily the same as the case) such that . If then all other terms are zero so the inequality trivially holds. If it is equivalent to the induction hypothesis so assume
5. Let
6. So we can rewrite
8. By convexity (same as case),
9. By our inductive hypothesis
10. So we get that
11. The exact same argument follows for concave functions but the inequality of the case flips .

Intuition: The way this proof works is that we view the last point separately and the rest as one convex combination. We then renormalise with to insure the weights sum to (ensuring that the one point a convex combination) so we can apply the two-point Jensen that states that and we’ve just used . And then invoke our induction hypothesis on . ( acts as a shortcut for all input as it forms a new point which lies in the interval as )