Stolarsky mean

In mathematics, the Stolarsky mean of two positive real numbers $$x,y is defined as: :$$egin{matrix}S_p(x,y)&=&lim_{(xi,eta) o(x,y)} left({frac{xi^p-eta^p}{p (xi-eta) ight)^{1over p-1}\&=&egin{cases}x & mbox{if }x=y \left({frac{x^p-y^p}{p (x-y) ight)^{1over 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 $$( y, f(y) ), has the same slope as a line tangent to the graph at some point $$xi in the interval $$ [x,y] .:$$exists xiin [x,y] f'(xi) = frac{f(x)-f(y)}{x-y}

The Stolarsky mean is obtained by:$$xi = f'^{-1}left(frac{f(x)-f(y)}{x-y} ight) when choosing $$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 $$n+1 variables by considering the mean value theorem for divided differences for the $$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.

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