Stolarsky mean

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In mathematics, the Stolarsky mean of two positive real numbers x, y is defined as:

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

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

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

$S_{p}(x_{0},\dots ,x_{n})={f^{{(n)}}}^{{-1}}(n!\cdot f[x_{0},\dots ,x_{n}])$ for $f(x)=x^{p}$ .

References

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

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Stolarsky mean

Special cases

Generalizations

See also

References