Stolarsky mean

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

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

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 $ξ$ 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 $f (x) = x p$ .

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 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.
$S 3 (x, y) = Q M (x, y, G M (x, y))$ is a connection to the quadratic mean and the geometric mean.
$\lim_{p\to\infty} S_p(x,y)$ is the maximum.