Contraharmonic mean

In mathematics, a contraharmonic mean is a function complementary to the harmonic mean. The contraharmonic mean is a special case of the Lehmer mean, L₂.

Definition

The contraharmonic mean of a set of positive numbers is defined as the arithmetic mean of the squares of the numbers divided by the arithmetic mean of the numbers:

$C(x_1, x_2, \dots , x_n) ={ \left({ x_1^2+x_2^2+\cdots+x_n^2 \over n}\right) \over \left({x_1+x_2+\cdots+x_n \over n}\right)},$

or, more simply,

$C(x_1, x_2, \dots , x_n) ={ { x_1^2+x_2^2+\cdots+x_n^2} \over {x_1+x_2+\cdots+x_n }}.$

Properties

It is easy to show that this satisfies the characteristic properties of a mean:

• $C(x_1, x_2, \dots , x_n) \in [ \min(x_1, x_2, \dots , x_n) , \max(x_1, x_2, \dots , x_n) ]$
• $C(t\cdot x_1, t\cdot x_2, \dots , t\cdot x_n) = t\cdot C(x_1, x_2, \dots , x_n)\text{ for }t > 0$

The first property implies that for all k > 0,

C(k, k, ..., k) = k (fixed point property).

For two variables, a and b, taken as

$0 < a \le b ,$

it is easier to see why this mean is complementary to the harmonic mean. Then the contraharmonic mean C(a, b) is also that mean that is as high above the arithmetic mean as the arithmetic mean is above the harmonic mean:

C(a, b) − A(a, b) = A(a, b) − H(a, b)

or

C(a, b) = 2A(a, b) − H(a, b).

Explication

From the formulas for the arithmetic mean and harmonic mean of two variables we have :

$A(a,b) = {{a+b} \over 2}\text{ and}$

$H(a,b) = {1 \over {{1 \over 2}\cdot {({1 \over a} + {1 \over b})}}} = {{2ab} \over {a+b}}$

$C(a,b) = 2\cdot A(a,b) - H(a,b) = a+b- {{2ab} \over {a+b}}$

$C(a,b) = { {(a+b)^2 - 2ab } \over {a+b}} = {{a^2 + b^2} \over {a+b}}$

Notice that for two variables the average of the harmonic and contraharmonic means is exactly equal to the arithmetic mean:

A(H(a, b), C(a, b) ) = A(a, b)

As a gets closer to 0 then H(a, b) also gets closer to 0. The harmonic mean is very sensitive to low values. On the other hand, the contraharmonic mean is sensitive to larger values, so as a approaches 0 then C(a, b) approaches b (so their average remains A(a, b) ).

The contraharmonic mean is higher in value than the average and also higher than the root mean square :

min(a,b) < H(a,b) < G(a,b) < L(a,b) < A(a,b) < R(a,b) < C(a,b) < max(a,b) for 0 < a < b

where H is the harmonic mean, G is geometric mean, L is the logarithmic mean, A is the arithmetic mean, R is the root mean square and C is the contraharmonic mean. If a could be equal to b then the above < signs can be replaced by $\le$. When a = b the above chain of comparisons collapses to the same value, a.

There are two other notable relationships between 2-variable means. First, the geometric mean of the arithmetic and harmonic means is equal to the geometric mean of the two values :

$G( A(a,b), H(a,b) )=G\left({{a+b}\over 2}, {{2ab}\over {a+b}}\right) = \sqrt {{{a+b}\over 2}\cdot {{2ab}\over {a+b}}} = \sqrt{ab} = G(a,b)$

The second relationship is that the geometric mean of the arithmetic and contraharmonic means is the root mean square:

$G( A(a,b), C(a,b) )=G\left({{a+b}\over 2}, {{a^2+b^2}\over {a+b}}\right)$

$= \sqrt{{{a+b}\over 2}\cdot {{a^2+b^2}\over {a+b}}} = \sqrt{{{a^2+b^2}\over 2}} = R(a,b)$

The contraharmonic mean of two variables can be constructed geometrically using a trapezoid (see [1] ).

Additional constructions

You can also construct it using a circle similar to the way the Pythagorean means of two variables are constructed.
Basically, the Contraharmonic mean is the rest of the diameter on which the Harmonic mean lies. Their sum is always the length of the diameter (a+b) of the circle that shows the Pythagorean means (see Pythagorean means on MathWorld at [2] ). Note that the Contraharmonic mean is also sometimes called the Antiharmonic mean. The article at [3] explains how that came about (based on the view of the means as satisfying special proportions, the contraharmonic proportion being the opposite of the harmonic one).

References

• Essay #3 - Some "mean" Trapezoids, by Shannon Umberger: [4]

• Construction of the Contraharmonic Mean in a Trapezoid: [5]

• Means in the Trapezoid: [6]

• Means of Complex Numbers: [7]

• Proofs without Words / Exercises in Visual Thinking, by Roger B. Nelsen, page 56, ISBN 0-88385-700-6

• Pythagorean Means: [8] (extend the segment that represents the Harmonic mean through the circle's center to the other side, creating a diameter. The length of the diameter segment after the Harmonic segment is the Contraharmonic mean.)

• Contraharmonic Proportion: [9]

• Pahikkala, Jussi (2010), On contraharmonic mean and Pythagorean triples, Elemente der Mathematik 65 (2): 62–67.