Stolarsky mean

Last updated August 14, 2023

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{aligned}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{aligned}}

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}

.

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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.

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[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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