We proved in the last post that a test for the convexity of continuous function is that , for all . I claim that we can use this test to show that is strictly convex. Observe that , for and , by considering the quadratic . Hence,
for all and . We use the strict convexity of to prove a weighted form of the arithmetic mean-geometric mean (AM-GM) inequality.
Proposition 1. If and such that , then
Proof. Fix and such that .
The necessity and sufficiency of the strictness condition is immediate from the strict convexity of .
Suppose and satisfy the same hypotheses as above. Then replacing by , for , we obtain that
Lemma 2. Let be a convex function and (). Then
Proof. Observe that
We use the preceding lemma to prove more inequalities involving arithmetic and geometric means. For and , define
If we apply the above lemma to , then
The monotonicity of the exponential implies the inequalities
Now let . Then
The preceding two inequalities are apparently due to T. Popoviciu and R. Rado, respectively.