Pythagorean means

A geometric construction of the quadratic mean and the Pythagorean means (of two numbers a and b). Harmonic mean denoted by   H, geometric by   G, arithmetic by   A and quadratic mean (also known as root mean square) denoted by   Q.

Comparison of the arithmetic, geometric and harmonic means of a pair of numbers. The vertical dashed lines are asymptotes for the harmonic means.

In mathematics, the three classical Pythagorean means are the arithmetic mean (AM), the geometric mean (GM), and the harmonic mean (HM). These means were studied with proportions by Pythagoreans and later generations of Greek mathematicians because of their importance in geometry and music.

Definition

They are defined by:

${\begin{aligned}\operatorname {AM} \left(x_{1},\;\ldots ,\;x_{n}\right)&={\frac {1}{n}}\left(x_{1}+\;\cdots \;+x_{n}\right)\\[9pt]\operatorname {GM} \left(x_{1},\;\ldots ,\;x_{n}\right)&={\sqrt[{n}]{\left\vert x_{1}\times \,\cdots \,\times x_{n}\right\vert }}\\[9pt]\operatorname {HM} \left(x_{1},\;\ldots ,\;x_{n}\right)&={\frac {n}{\displaystyle {\frac {1}{x_{1}}}+\;\cdots \;+{\frac {1}{x_{n}}}}}\end{aligned}}$

Properties

Each mean, ${\textstyle \operatorname {M} }$, has the following properties:

Value preservation

$\operatorname {M} (x,x,\,\ldots ,\,x)=x$

First order homogeneity

$\operatorname {M} (bx_{1},\,\ldots ,\,bx_{n})=b\operatorname {M} (x_{1},\,\ldots ,\,x_{n})$

Invariance under exchange

$\operatorname {M} (\ldots ,\,x_{i},\,\ldots ,\,x_{j},\,\ldots )=\operatorname {M} (\ldots ,\,x_{j},\,\ldots ,\,x_{i},\,\ldots )$

for any $i$ and $j$.

Averaging

$\min(x_{1},\,\ldots ,\,x_{n})\leq \operatorname {M} (x_{1},\,\ldots ,\,x_{n})\leq \max(x_{1},\,\ldots ,\,x_{n})$

The harmonic and arithmetic means are reciprocal duals of each other for positive arguments:

$\operatorname {HM} \left({\frac {1}{x_{1}}},\,\ldots ,\,{\frac {1}{x_{n}}}\right)={\frac {1}{\operatorname {AM} \left(x_{1},\,\ldots ,\,x_{n}\right)}}$

while the geometric mean is its own reciprocal dual:

$\operatorname {GM} \left({\frac {1}{x_{1}}},\,\ldots ,\,{\frac {1}{x_{n}}}\right)={\frac {1}{\operatorname {GM} \left(x_{1},\,\ldots ,\,x_{n}\right)}}$

Inequalities among means

There is an ordering to these means (if all of the $x_{i}$ are positive)

$\min \leq \operatorname {HM} \leq \operatorname {GM} \leq \operatorname {AM} \leq \max$

with equality holding if and only if the $x_{i}$ are all equal.

This is a generalization of the inequality of arithmetic and geometric means and a special case of an inequality for generalized means. The proof follows from the arithmetic-geometric mean inequality, $\operatorname {AM} \leq \max$, and reciprocal duality ($\min$ and $\max$ are also reciprocal dual to each other).

The study of the Pythagorean means is closely related to the study of majorization and Schur-convex functions. The harmonic and geometric means are concave symmetric functions of their arguments, and hence Schur-concave, while the arithmetic mean is a linear function of its arguments, so both concave and convex.

See also

• Arithmetic-geometric mean
• Average
• Generalized mean

External links

• Cantrell, David W. "Pythagorean Means". MathWorld.
• Nice comparison of Pythagorean means with emphasis on the harmonic mean.