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}](../../../math/f/c/a/fcaa5380a0d1d700b749e1842d8a986b.png)
.
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}](../../../math/1/1/e/11eecaea7f491127953437aceb1a32e5.png)
The Stolarsky mean is obtained by
when chosing f(x) = xp.
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[edit] Special cases
is the minimum.- S − 1(x,y) is the geometric mean.
is the logarithmic mean. It can be obtained from the mean value theorem by chosing f(x) = lnx.
is the power mean with exponent 
.
is the identric mean. It can be obtained from the mean value theorem by chosing 
.- S2(x,y) is the arithmetic mean.
- S3(x,y) = QM(x,y,GM(x,y)) is a connection to the quadratic mean and the geometric mean.
is the maximum.
[edit] 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])](../../../math/f/3/5/f35082a941e311f1a3d3e79866878e7b.png)
for f(x) = xp.
[edit] See also
[edit] References
- Stolarsky, Kenneth B.: Generalizations of the logarithmic mean, Mathematics Magazine, Vol. 48, No. 2, Mar., 1975, pp 87-92
