proof of arithmetic-geometric-harmonic means inequality

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

Let $x_1,\mathrm{\dots },x_n>0$; we shall first prove that

$$\sqrt[n]{x_1\cdot \mathrm{\dots }\cdot x_n}\le \frac{x_1+\mathrm{\dots }+x_n}{n}.$$

Note that $\log$ is a concave function. Applying it to thearithmetic mean of $x_1,\mathrm{\dots },x_n$ and using Jensen’s inequality, we see that

$\log ({\displaystyle \frac{x_1+\mathrm{\dots }+x_n}{n}})$$\ge {\displaystyle \frac{\log (x_1)+\mathrm{\dots }+\log (x_n)}{n}}$

$={\displaystyle \frac{\log (x_1\cdot \mathrm{\dots }\cdot x_n)}{n}}$

$=\log \sqrt[n]{x_1\cdot \mathrm{\dots }\cdot x_n}.$

Since $\log$ is also a monotone function, it follows that the arithmetic mean is at least as large as the geometric mean.

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”).