A previous page showed from the monotone convergence theorem that in the reals, all positive have an th root, , for each . This page investigates these th roots further by discussing an important and powerful inequality connecting arithmetic and geometric means in an ordered field.
Recall that given numbers their mean is the number . This is often called the arithmetic mean to distinguish it from the geometric mean which is obtained by replacing addition with multiplication (and division by by th roots). I.e. the geometric mean of is . For non-negative numbers , the arithmetic and geometric means are related by the inequality
We will prove this, and the proof will be by induction.
The following lemma will be the base case of the induction.
Given and with , we have: if and only if . In particular
with a similar equality for . So is the maximum possible value for for a given value for . Also, holds if and only if holds, which is if and only if , completing the proof.
We can now prove the famous arithmetic-mean/geometric-mean inequality.
Let and . Then
By induction on , the induction hypothesis being the statement of the theorem. Note that the case is easy and the case is true by the previous lemma.
Assume and assume inductively that
the theorem is true for in place of .
If then each
and it is easy to see that in fact equality holds
in the above
inequality, so we would be done. Otherwise
we may assume (by re-ordering the if necessary) that . So so
. We set
Then if we
as ; also, if we
as ; either way we have
and hence by
the lemma we have
and by the induction hypothesis
giving as required.
We give as a corollary a useful inequality, reminiscent of Bernoulli's inequality which we saw earlier, but the inequality sign goes the other way round and the result is only valid for exponents at most one.
The Exponential Inequality.
Suppose with . Then for all with we have .
With and integers apply the previous result to and . Then