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

,

unless .

*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

Hence,

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

Hence,

The monotonicity of the exponential implies the inequalities

Now let . Then

Hence,

The preceding two inequalities are apparently due to T. Popoviciu and R. Rado, respectively.

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