Stolarsky mean

In mathematics, the Stolarsky mean is a generalization of the logarithmic mean. It was introduced by Kenneth B. Stolarsky in 1975.{{cite journal | zbl=0302.26003 | last=Stolarsky | first=Kenneth B. | title=Generalizations of the logarithmic mean | journal=Mathematics Magazine | volume=48 | pages=87–92 | year=1975 | issue=2 | issn=0025-570X | jstor=2689825 | doi=10.2307/2689825}}

Definition

For two positive real numbers x, y the Stolarsky Mean is defined as:

: $\begin{align}
S_p(x,y)
& = \lim_{(\xi,\eta)\to(x,y)}
\left({\frac{\xi^p-\eta^p}{p (\xi-\eta)}}\right)^{1/(p-1)} \\[10pt]
& = \begin{cases}
x & \text{if }x=y \\
\left({\frac{x^p-y^p}{p (x-y)}}\right)^{1/(p-1)} & \text{else}
\end{cases}
\end{align}$

Derivation

It is derived from the mean value theorem, which states that a secant line, cutting the graph of a differentiable function $f$ at $( x, f(x) )$ and $( y, f(y) )$ , has the same slope as a line tangent to the graph at some point $\xi$ in the interval $[x,y]$ .

: $\exists \xi\in[x,y]\ f'(\xi) = \frac{f(x)-f(y)}{x-y}$

The Stolarsky mean is obtained by

: $\xi = \left[f'\right]^{-1}\left(\frac{f(x)-f(y)}{x-y}\right)$

when choosing $f(x) = x^p$ .

Special cases

$\lim_{p\to -\infty} S_p(x,y)$ is the minimum.
$S_{-1}(x,y)$ is the geometric mean.
$\lim_{p\to 0} S_p(x,y)$ is the logarithmic mean. It can be obtained from the mean value theorem by choosing $f(x) = \ln x$ .
$S_{\frac{1}{2}}(x,y)$ is the power mean with exponent $\frac{1}{2}$ .
$\lim_{p\to 1} S_p(x,y)$ is the identric mean. It can be obtained from the mean value theorem by choosing $f(x) = x\cdot \ln x$ .
$S_2(x,y)$ is the arithmetic mean.
$S_3(x,y) = QM(x,y,GM(x,y))$ is a connection to the quadratic mean and the geometric mean.
$\lim_{p\to\infty} S_p(x,y)$ is the maximum.

Generalizations

One can generalize the mean to n + 1 variables by considering the mean value theorem for divided differences for the nth derivative.

One obtains

: $S_p(x_0,\dots,x_n) = {f^{(n)}}^{-1}(n!\cdot f[x_0,\dots,x_n])$ for $f(x)=x^p$ .

References

Category:Means