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:

$\begin{array}{cc}{S}_p(x,y)& =\underset{(\xi ,\eta )\to (x,y)}{\lim }{\left(\frac{{\xi }^p-{\eta }^p}{p(\xi -\eta )}\right)}^{1/(p-1)}\\ & =\{\begin{array}{cc}x& \text{if }x=y\\ {\left(\frac{x^p-y^p}{p(x-y)}\right)}^{1/(p-1)}& \text{else}\end{array}\end{array}$

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 $[x,y]$.

$\mathrm{\exists }\xi \in [x,y]f^{\prime }(\xi )=\frac{f(x)-f(y)}{x-y}$

The Stolarsky mean is obtained by

$\xi ={\left[f^{\prime }\right]}^{-1}\left(\frac{f(x)-f(y)}{x-y}\right)$

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

Special cases

• $\underset{p\to -\mathrm{\infty }}{\lim }{S}_p(x,y)$ is the minimum.
• $S_{-1}(x,y)$ is the geometric mean.
• $\underset{p\to 0}{\lim }{S}_p(x,y)$ 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 $\frac{1}{2}$.
• $\underset{p\to 1}{\lim }{S}_p(x,y)$ 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.
• $\underset{p\to \mathrm{\infty }}{\lim }{S}_p(x,y)$ 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 $f(x)=x^p$.

See also

• Mean