The famous Arithmetic-Geometric Mean (AGM) has found many uses in analysis (e.g., elliptic integrals) and number theory ( e.g.,quadratically convergent algorithms for computing the digits of PI), but there does not seem to have been much attention devoted to higher-order versions of iterated means of more than two arguments.

### Introduction : The Eleven Means of Ancient Greece

The ancient Greeks defined a list of eleven distinct “means”, including most of the well-known means that we still use today. However, oddly enough, they never explicitly defined what we call the root-mean-square (although it can be constructed by composition of some of their means). In Pythagoras‘s time there were just three means, which we call the arithmetic, the geometric, and the harmonic (originally known as the “subcontrary mean”). Later, three more “means” were added, possibly by Eudoxus. These six are described in the article Iterated Means. The last four means were added by two later Pythagoreans, Myonides and Euphranor. We actually have a listing of “The Ten Means” from two different ancient authors ( Nicomachus and Pappus), but the lists are not quite identical. They each give one “mean” that the other left out, so taking the two lists together, we have eleven distinct means in all.

Consider three quantities a,b,c such that a > b > c, where we wish to make b the “mean” of a and c. Notice that we can form three positive differences with these quantities: (a-b), (b-c), and (a-c). The Greeks worked on the idea of equating a ratio of two of these differences to a ratio of two of the original quantities (not necessarily distinct). For example, if we set the ratio (a-b)/(b-c) equal to the ratio a/b, the result is b^2 = ac, which represents the geometric mean.

Of all the possible ways of doing this, several of them are automatically ruled out by the assumed inequalities on a,b,c. The ones that are not (necessarily) ruled out are the ten (actually eleven) means as summarized below:

Some of these are obviously not very intuitively useful definitions of “means”. For example, using the 11th mean we would have m11(5,4) = 1. This mean presumably was included because it doesn’t necessarily violate the assumed inequalities, e.g., m11(5,1) = 4, but it seems only marginally acceptable. On the other hand, it’s interesting to note that m5(2,1) equals f, the golden proportion (1.618…). We might also note that the 2nd mean of the 3rd and 4th means on this list is equivalent to what we call the root-mean-square.

Anyway, it’s not too surprising that only the arithmetic, geometric, and harmonic have survived in common usage. This process of broad abstract definition followed by pragmatic selection reminds me of how Western music originally had seven distinctly identified “modes” ( Ionian, Dorian, Phrygian, Lydian, Mixolydian, Aeolian, Locrian ), and then over time we discarded all but two of them (the Ionian and Aeolian), which we call the “major” and natural “minor” scales. This is doubly fitting, considering that the original concept of numerical “means” among the Pythagoreans and others was closely involved with the study of musical tones and scales.

### The Super-Symmetric Mean

One obvious extension is to consider the

iterated Arithmetic-Geometric-Harmonic mean of three numbers

a[0], b[0], c[0], defined by the recurrence

a[n+1] = A(a[n],b[n],c[n])

b[n+1] = G(a[n],b[n],c[n])

c[n+1] = H(a[n],b[n],c[n])

Similarly, making use of all ten of the “means” defined by the ancient

Greeks, we can have iterations of up to 10 elements. However, this

seems somewhat arbitrary, and the restriction to just a small number

of means is rather artificial.

Another obvious approach would be to use the “Holder means“, which

are essentially just isomorphic to the simple arithmetic mean carried

out in a transformed domain. In other words, for any invertible

transformation function f we can define the mean of the values x1,

x2,..,xn by the formula

f(x_mean) = ( f(x1) + f(x2) + … + f(xn) ) / n

which is just the arithmetic average in the f domain. The Holder

mean M_k() is defined by the above formula with f(x)=x^k, from

which it follows that M_1() is the Arithmetic mean, M_-1() is

the Harmonic mean, M_2() is the root-sum-square, and the limit of

M_k() as k goes to infinity is the Geometric mean. Of course, we

can also get the Geometric mean by simply defining f(x)=log(x).

However, these means are all just (in a sense) weighted version

of a single prototype, and don’t seem to capture the essence of

the desired generalization of the AGM.

By the way, it’s interesting to consider the class of means of

complex numbers given by the above equation with f() taken from

the class of the most general holomorphic functions of the complex

plane, namely, the linear fractional transformations

f(x) = (ax + b)/(cx + d)

This has connection with modular functions, and is very interesting

in its own right, but it still doesn’t seem to capture the desired

generalization.

I think the optimum approach to generalizing the AGM would be to

devise an nth-order mean by observing how our two basic means arise

from two of the elementary symmetric functions of n objects. We could

devise “means” based on all n of the symmetric functions. Recall that

the first elementary symmetric function of x1,x2,..,xn is just the sum

S_1 = x1 + x2 + … + xn

and the “mean” based on this is the single number that, if we set all

the x’s to that value, would give the same S_1. In other words, it

is (S_1)/n. This gives the usual arithmetic mean. Similarly, the

nth elementary symmetric function is just the product of the n values,

i.e.,

S_n = (x1)(x2)…(xn)

and again the mean based on this function is the single value that,

if we set all the x’s to that value, would give the same S_n. In

other words, it is the nth root of S_n.

Now it’s clear how we can proceed to the other symmetric functions.

For example, S_2 is the sum of all products of two of our x values.

To illustrate, with n=4 we would have

S_2 = (x1)(x2) + (x1)(x3) + (x1)(x4) + (x2)(x3) + (x2)(x4) + (x3)(x4)

Obviously the “mean” based on this function would be the square root

of (1/6)th of S_2. In general, we can define the kth “symmetric mean”

of n objects as follows

M_k = ( S_k / C (n,k) ) ^ (1/k)

where C(n,k) is the binomial coefficient (n choose k). M_1 is just

the arithmetic mean and M_n is the geometric mean.

Now, our generalization of the AGM to n elements is the “super-symmetric

mean” of the n values x1[0], x2[0], …, xn[0], defined as the convergent

value given by the braid of iterations

xj[k+1] = M_j(x1[k],x2[k]…,xn[k]) j=1,2,..,n

Of course our original values of xj[0] are the roots of the polynomial

f(x) = x^n – (S_1) x^(n-1) + (S_2) x^(n-2) – … +- (S_n)

and the next set of values, xj[1], are the roots of another polynomial

whose coefficients are the symmetric functions used on the next

iteration, and so on. Thus we have a sequence of polynomials

converging on one of the form F(x) = (x – r)^n, where r is the

super-symmetric mean of the original x’s. This implies that we can

define the polynomials Mj(x) = (x – M_j(x1,x2,..,xn))^n where the

jth such polynomial matches the (n-j)th derivative of f(x) at x=0.

### Conclusion and Generalization

It's also interesting to see how we can construct various
other means from the elementary symmetric means. For example,
the harmonic mean of n values is given by n.M_n/(M_{n-1}).
More generally, we can define the family of means of n numbers
as follows
R_{k,j} = (((M_k)/C(n,k)) / (((M_j)/C(n,j))) ) ^ (1/(k-j)), n>=k>j
In these terms the ordinary arithmetic mean is R_{1,0},
the geometric mean is R_{n,0}, and the harmonic mean is R_{n,n-1}.