Stolarsky mean

In mathematics, the Stolarsky mean is a generalization of the logarithmic mean. It was introduced by Kenneth B. Stolarsky in 1975.^[1]

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 = f'^{-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

^ Stolarsky, Kenneth B. (1975). "Generalizations of the logarithmic mean". Mathematics Magazine. 48: 87–92. doi:10.2307/2689825. ISSN 0025-570X. JSTOR 2689825. Zbl 0302.26003.

[1] Stolarsky, Kenneth B. (1975). "Generalizations of the logarithmic mean". Mathematics Magazine. 48: 87–92. doi:10.2307/2689825. ISSN 0025-570X. JSTOR 2689825. Zbl 0302.26003.

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