The harmonic mean of a set of positive numbers is equal to the inverse of the average of the inverses of the numbers. The geometric mean of a set of positive numbers is equal to the nth root of the product of the numbers, where n is the size of the set. The arithmetic mean is equal to the average of the numbers.

It is known that the arithmetic mean is always greater than or equal to the geometric mean, given a set of positive numbers. But where does the harmonic mean fit in with these two other means? Is it greater, lesser, or inbetween? Prove it. Note that in this problem, a "set" allows repeated numbers.

The formation of the geometric mean commutes with inverting the numbers, that is, the geometric mean of the inverses of the numbers is the inverse of the geometric mean of the original numbers.

From what is said in the problem, the arithmetic mean is >= the geometric of any set of numbers.  Now replace the set with the set of the numbers' inverses.  Then the arithmetic mean of the inverse numbers is also >= the geometric mean of the inverse numbers.  So the inverse of the arithmetic mean of the inverses is <= the inverse of the geometric mean of the inverses.  The former is the harmonic mean of the original set of numbers, the latter just the geometric mean of the original numbers.  So the harmonic mean is <= the geometric mean.

Posted by vswitchs on 2006-10-25 13:47:46