## Monday Math 70

The harmonic mean of a set of positive real numbers  is given by . (So , and the reciprocal of the harmonic mean is the arithmetic mean of the reciprocals). This mean finds uses in problems involving averages of rates, and also in some electronic circuit problems (resistors or inductors in parallel, capacitors in series).

Now, let us apply the AM-GM inequality to the reciprocal of the harmonic mean:
,
where  is the geometric mean. Thus,
. Therefore, for any set of positive numbers, we see the arithmetic, geometric, and harmonic means obey , with equality if and only if the numbers in the set are all equal.

For an example, one can use this result to prove a classic inequality called Nesbitt’s inequality: for positive real numbers x, y, z
.

Let . Then the AM-HM inequality above tells us that:

so

expanding the product on the left:
.
Q.E.D.