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:

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

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 $f(x)=x^{p}$.

See also

• Mean