Related concept for this equation?

One friend of mine has a cup with a lot of mathematical expressions printed on it. I would like to know more about one of these.

What is the history behind the following inequality, and what is the related concept? Where would this inequality be used, or what natural phenomena does it represent?

$$\sqrt {a\times b}\le\frac {a+b}{2}$$

• Wikipedia calls it "inequality of arithmetic and geometric means", it was known before even Euclid. The tradition ascribes the discovery (along with the harmonic mean) to Pythagoreans. Mar 7, 2017 at 21:56
• That last part needs care when written. Write $\frac{a+b}{2}$ or $(a+b)/2$ so it is clear that it is not $a+(b/2)$. Mar 8, 2017 at 1:06
• @GeraldEdgar This is an answer! Post and I vote you up. On the other hand, thanks for your answer.
– mrc
Mar 9, 2017 at 7:38