Means and inequalities

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

M_r(x) = \left( \frac{1}{n} \sum_{i = 1}^n x_i^r \right)^{1/r}

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

M_r(x) = \left( \sum_{i = 1}^n p_i x_i^r \right)^{1/r}

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

8 thoughts on “Means and inequalities

  1. Of all those the one I find more counter-intuitive is M0.
    This is a great post. Keep up with the good work.

  2. 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.

  3. 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.

  4. 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.

  5. 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.

  6. 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.

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