Means (Averages)

Everyone's familiar with the average or arithmetic mean of two (positive) numbers and :

but there are many other useful "averages" we might consider. For example,

is called the harmonic mean; it's related to the arithmetic mean by

In other words, the harmonic mean is the reciprocal of the average of the reciprocals! While there are many other "averages" we might consider, there is only one more that we will define here, called the geometric mean:

Now each of these expressions is an "average" in the sense that each lies between and . But we can say a bit more: It's not hard to check that the various means we've defined satisfy

for all positive numbers and .

At least one of these inequalities is both well-known and useful. It's called the arithmetic-geometric mean inequality:

We can give a quick proof by means of a simple observation: Since and are positive, we can write each as a square. That is, we can find positive numbers and satisfying

Substituting these expressions into the arithmetic-geometric mean inequality yields

and this inequality if easy to verify. Indeed, moving all the terms to the right-hand side, this inequality is equivalent to

which is (nearly) obvious, since

Finally, the "harmonic-geometric mean inequality" follows from the arithmetic-geometric mean inequality by taking reciprocals:

Neal Carothers -