Stolarsky mean

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

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 $[x,y]$.:$exists\; xiin\; [x,y]\; f\text{'}(xi)\; =\; frac\{f(x)-f(y)\}\{x-y\}$

The Stolarsky mean is obtained by:$xi\; =\; f\text{'}^\{-1\}left(frac\{f(x)-f(y)\}\{x-y\}\; ight)$when choosing $f(x)\; =\; x^p$.

Special cases

*$lim\_\{p\; o\; -infty\}\; S\_p(x,y)$ is the minimum.
*$S\_\{-1\}(x,y)$ is the geometric mean.
*$lim\_\{p\; o\; 0\}\; S\_p(x,y)$ 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 $frac\{1\}\{2\}$.
*$lim\_\{p\; o\; 1\}\; S\_p(x,y)$ is the identric mean. It can be obtained from the mean value theorem by choosing $f(x)\; =\; xcdot\; ln\; x$.
*$S\_2(x,y)$ 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.
*$lim\_\{p\; oinfty\}\; S\_p(x,y)$ is the maximum.

Generalizations

You can generalize the mean to $n+1$ variables by considering the mean value theorem for divided differences for the $n$th derivative.You obtain:$S\_p(x\_0,dots,x\_n)\; =\; \{f^\{(n)^\{-1\}(n!cdot\; f\; [x\_0,dots,x\_n]\; )$ for $f(x)=x^p$.

See also

*mean

References

* Stolarsky, Kenneth B.: " [http://links.jstor.org/sici?sici=0025-570X%28197503%2948%3A2%3C87%3AGOTLM%3E2.0.CO%3B2-6 Generalizations of the logarithmic mean] ", Mathematics Magazine, Vol. 48, No. 2, Mar., 1975, pp 87-92