Stolarsky mean

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The Stolarsky mean of two positive real numbers $x, y$ is defined as:

$\begin{matrix} S_p(x,y) &=& \lim_{(\xi,\eta)\to(x,y)} \sqrt[p-1]{\frac{\xi^p-\eta^p}{p\cdot(\xi-\eta)}} \\ &=& \begin{cases} x & \mbox{if }x=y \\ \sqrt[p-1]{\frac{x^p-y^p}{p\cdot(x-y)}} & \mbox{else} \end{cases} \end{matrix}$ .

It is derived from the mean value theorem, which states that the secant of a function $f$ in an interval $[x, y]$ has the same slope like a tangent of $f$ in that interval.

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

The Stolarsky mean is obtained by

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

when chosing $f (x) = x p$ .

1 Special cases
2 Generalizations
3 See also
4 References

[edit] 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 chosing $f (x) = ln x$ .
$S_{\frac{1}{2}}(x,y)$ 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 chosing $f(x) = x\cdot \ln x$ .
$S 2 (x, y)$ is the arithmetic mean.
$S 3 (x, y) = Q M (x, y, G M (x, y))$ is a connection to the quadratic mean and the geometric mean.
$\lim_{p\to\infty} S_p(x,y)$ is the maximum.