Jensen's Inequality is an inequality discovered by Danish mathematician Johan Jensen in 1906.
Let be a convex function of one real variable. Let and let satisfy . Then
If is a concave function, we have:
We only prove the case where is concave. The proof for the other case is similar.
Let . As is concave, its derivative is monotonically decreasing. We consider two cases.
If , then If , then By the fundamental theorem of calculus, we have Evaluating the integrals, each of the last two inequalities implies the same result: so this is true for all . Then we have as desired.
One of the simplest examples of Jensen's inequality is the quadratic mean - arithmetic mean inequality. Taking , which is convex (because and ), and , we obtain
Similarly, arithmetic mean-geometric mean inequality (AM-GM) can be obtained from Jensen's inequality by considering .
In fact, the power mean inequality, a generalization of AM-GM, follows from Jensen's inequality.
Prove AM-GM using Jensen's Inequality
Prove that when
- Prove that for any , we have .
- Show that in any triangle we have
- Let be positive real numbers. Prove that