# Gamma factor between means

Consider the mean between two quantities:

$c+v\;&space;\mathrm{and}\;&space;c-v.$

The arithmetic mean is

$\frac{c+v+c-v}{2}=c.$

The harmonic mean is

$\left&space;(\frac{(c+v)^{-1}+(c-v)^{-1}}{2}&space;\right&space;)^{-1}=\frac{c^2-v^2}{c}=\frac{c}{\gamma&space;^{2}}.$

where

$\gamma&space;=&space;\left&space;(&space;1-\frac{v^2}{c^2}\right&space;)^{-1/2}$

which equals the gamma factor of the Lorentz transformation.

The geometric mean is

$((c+v)(c-v))^{1/2}=(c^{2}-v^{2})^{1/2}=\frac{c}{\gamma&space;}$

Then the factor γ2 transforms a harmonic mean into an arithmetic mean:

$\gamma^2&space;\left&space;(&space;\frac{c^2-v^2}{c}&space;\right&space;)=c$

The inverse γ factor transforms an arithmetic mean into a geometric mean:

$\lambda^2&space;c&space;=&space;\left&space;(&space;\frac{c^2-v^2}{c}&space;\right&space;)$

so that

$\lambda&space;=&space;\left&space;(&space;1-\frac{v^2}{c^2}\right&space;)^{1/2}=\gamma^{-1}$

The inverse γ2 factor transforms an arithmetic mean into a harmonic mean.

Or consider the mean between these two:

$\frac{1}{c+v}\;&space;\mathrm{and}\;&space;\frac{1}{c-v}.$

The arithmetic mean is

$\frac{1}{2}\left&space;(\frac{1}{c+v}+\frac{1}{c-v}&space;\right&space;)=&space;\frac{c}{c^2-v^2}=\frac{\gamma&space;^{2}}{c}.$

The harmonic mean is

$\left&space;(\frac{1}{2}\left&space;(\left&space;(\frac{1}{c+v}&space;\right&space;)^{-1}+\left&space;(\frac{1}{c-v}&space;\right&space;)^{-1}&space;\right&space;)&space;\right&space;)^{-1}=\frac{1}{c}.$

The geometric mean is

$\left&space;(\left&space;(\frac{1}{c+v}&space;\right&space;)\left&space;(\frac{1}{c-v}&space;\right&space;)&space;\right&space;)^{1/2}=\frac{\gamma&space;}{c}.$

The factor γ transforms a geometric mean into an arithmetic mean. The factor γ2 transforms a harmonic mean into an arithmetic mean.

Or consider the mean between these two with β = v/c:

$\frac{1}{1+\beta}\;&space;\mathrm{and}\;&space;\frac{1}{1-\beta}.$

The arithmetic mean is

$\frac{1}{2}\left&space;(\frac{1}{1+\beta}+\frac{1}{1-\beta}&space;\right&space;)=&space;\frac{1}{1-\beta^2}=\gamma&space;^{2}.$

The harmonic mean is

$\left&space;(\frac{1}{2}\left&space;(\left&space;(\frac{1}{1+\beta}&space;\right&space;)^{-1}+\left&space;(\frac{1}{1-\beta}&space;\right&space;)^{-1}&space;\right&space;)&space;\right&space;)^{-1}=1.$

The geometric mean is

$\left&space;(\left&space;(\frac{1}{1+\beta}&space;\right&space;)\left&space;(\frac{1}{1-\beta}&space;\right&space;)&space;\right&space;)^{1/2}=\gamma.$

The gamma factor relates the arithmetic mean and the harmonic mean, and the geometric mean combines them. An arithmetic mean and harmonic mean can be replaced by a geometric mean, in a sense.

Revised 1/2024.