Let be an interval, and let be twice-differentiable. For a parameter , consider the function

We want to figure the values of for which is multiplicatively convex (i.e. , for all . I leave it as an exercise to the reader to verify that is multiplicatively convex if and only if is convex on . Define by

Since is twice-differentiable, convexity of is equivalent to , which implies that is multiplicatively convex if and only if

Equivalently, is convex if and only if

Define . Since for the choice , we have that on and therefore is concave. We obtain the inequalities

which implies that

and

which implies that

In particular, if restricted to the interval , where , then .

We can use the preceding estimate of the AM-GM inequality to obtain an inequality for nonnegative random variables. This inequality sheds light on the probabilistic connections of the AM-GM inequality. Let and be as above. If is the discrete random variable with distribution for , then and

So we can restate the above inequality in the language of probability theory as

Naturally, we ask if the inequality holds for not necessarily discrete random variables taking values in . The answer is yes, as we now show.

**Proposition. **Let be a probability space, and let be a random variable taking values in , where . Then

*Proof. *Consider the simple function , where

for all . If we can show that and , then the desired inequality follows from our work above and a limiting argument. But a.s. and by continuity, a.s. Also, and . So the desired convergence results follow from the dominated convergence theorem.

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