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:
The proof of Jensen's inequality is very simple. Since the graph of every convex function lies above its tangent line at every point, we can compare the function with the linear function , whose graph is tangent to the graph of at the point . Then the left hand side of the inequality is the same for and , while the right hand side is smaller for . But the equality case holds for all linear functions! (check it yourself)
One of the simplest examples of Jensen's inequality is the quadratic mean - arithmetic mean inequality. Take (verify that and ) and . You'll get . Similarly, arithmetic mean-geometric mean inequality can be obtained from Jensen's inequality by considering .
Prove AM-GM using Jensen's Inequality
- Prove that for any , we have .
- Show that in any triangle we have
- Let be positive real numbers. Prove that