## 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 -
carother@bgnet.bgsu.edu