Stolarsky mean

In mathematics, the Stolarsky mean is a generalization of the logarithmic mean. It was introduced by Kenneth B. Stolarsky in 1975.

Definition
For two positive real numbers x, y the Stolarsky Mean is defined as:



\begin{align} S_p(x,y) & = \lim_{(\xi,\eta)\to(x,y)} \left({\frac{\xi^p-\eta^p}{p (\xi-\eta)}}\right)^{1/(p-1)} \\[10pt] & = \begin{cases} x & \text{if }x=y \\ \left({\frac{x^p-y^p}{p (x-y)}}\right)^{1/(p-1)} & \text{else} \end{cases} \end{align} $$

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

The Stolarsky mean is obtained by
 * $$ \xi = \left[f'\right]^{-1}\left(\frac{f(x)-f(y)}{x-y}\right) $$

when choosing $$f(x) = x^p$$.

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.
 * $$S_3(x,y) = QM(x,y,GM(x,y))$$ is a connection to the quadratic mean and the geometric mean.
 * $$\lim_{p\to\infty} S_p(x,y)$$ is the maximum.

Generalizations
One can generalize the mean to n + 1 variables by considering the mean value theorem for divided differences for the nth derivative. One obtains
 * $$S_p(x_0,\dots,x_n) = {f^{(n)}}^{-1}(n!\cdot f[x_0,\dots,x_n])$$ for $$f(x)=x^p$$.