The arithmetic mean of two numbers a and b is (a + b)/2.
The geometric mean of a and b is √(ab).
The harmonic mean of a and b is 2/(1/a + 1/b).
This post will generalize these definitions of means and state a general inequality relating the generalized means.
Let x be a vector of non-negative real numbers, x = (x1, x2, x3…, xn). Define Mr( x ) to be
unless r = 0 or r is negative and one of the xi is zero. If r = 0, define Mr( x ) to be the limit of Mr( x ) as r decreases to 0 . And if r is negative and one of the xi is zero, define Mr( x ) to be zero. The arithmetic, geometric, and harmonic means correspond to M1, M0, and M-1 respectively.
Define M∞( x ) to be the limit of Mr( x ) as r goes to ∞. Similarly, define M-∞( x ) to be the limit of Mr( x ) as r goes to –∞. Then M∞( x ) equals max(x1, x2, x3…, xn) and M-∞( x ) equals min(x1, x2, x3…, xn).
In summary, the minimum, harmonic mean, geometric mean, arithmetic mean and maximum are all special cases of Mr( x ) corresponding to r = –∞, –1, 0, 1, and ∞ respectively. Of course other values of r are possible; these five are just the most familiar. Another common example is the root-mean-square (RMS) corresponding to r = 2.
A famous theorem says that the geometric mean is never greater than the arithmetic mean. This is a very special case of the following theorem.
If r ≤ s then Mr( x ) ≤ Ms( x ).
In fact we can say a little more. If r < s then Mr( x ) < Ms( x ) unless x1 = x2 = x3 = … = xn or s ≤ 0 and one of the xi is zero.
We could generalize the means Mr a bit more by introducing positive weights pi such that p1 + p2 + p3 + … + pn = 1. We could then define Mr( x ) as
with the same fine print as in the previous definition. The earlier definition reduces to this new definition with pi = 1/n. The above statements about the means Mr( x ) continue to hold under this more general definition.
For more on means and inequalities, see Inequalities by Hardy, Littlewood, and Pólya.
Update: Analogous results for means of functions, replacing sums with integrals. Also, physical examples of harmonic mean with springs and resistors.
Related post: Old math books
Of all those the one I find more counter-intuitive is M0.
This is a great post. Keep up with the good work.
These are just generalised p-norms, right? What is the “famous theorem” you refer to? Is it a consequence of Hölder’s inequality?
On the topic of inequalities, if you haven’t got it already I strongly recommend Steele’s The Cauchy-Schwarz Master Class. It’s a wonderfully readable tour through inequalities and their history.
Mark,
These means correspond to p-norms if r ≥ 1, but not for smaller values of r.
The famous theorem I refer to is the geometric mean – arithmetic mean inequality.
I agree about Steele’s book. It’s one of my favorites.
Actually, all p-norms require an absolute value: $latex left(sum_{i=1}^n |x_i|^rright)^{1/r}$
That’s pretty cool about r = 0. I had to plot it to convince myself it was true. The limit comes in from both directions, too. Now I’m trying to prove it for fun.
By “generalised” I meant exactly those cases for when r < 1. The wikipedia article on L_p spaces talks about these generalisations for 0 ≤ r < 1. I hadn’t seen the case of r < 0 before, however.
Another useful generalization is the concept of Chisini mean: Chisini was a less-known Italian mathematician. You can read the idea here:
http://en.wikipedia.org/wiki/Chisini_mean
A really excellent source of these types of inequalities is Chapter 2 of Bela Bollobas’s “Linear Analysis”. It summarizes quite a lot of Hardy, Littlewood and Polya, but with rather more up-to-date notation.