We can use the Jensen inequality for an easy proof of the arithmetic-geometric-harmonic means inequality.

Let $x_1,\ldots,x_n > 0$ ; we shall first prove that $$ \sqrt[n]{x_1\cdot\ldots\cdot x_n} \le \frac{x_1+\ldots+x_n}{n}. $$

Note that $\log$ is a concave function. Applying it to the arithmetic mean of $x_1,\ldots, x_n$ and using Jensen's inequality, we see that

The proof that the geometric mean is at least as large as the harmonic mean is the usual one (see ``proof of arithmetic-geometric-harmonic means inequality'').