Stolarsky mean

In mathematics, the Stolarsky mean is a generalization of the logarithmic mean. It was introduced by Kenneth B. Stolarsky in 1975.^[1]

Definition

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

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

Derivation

It is derived from the mean value theorem, which states that a secant line, cutting the graph of a differentiable function ${\displaystyle f}$ at ${\displaystyle (x,f(x))}$ and ${\displaystyle (y,f(y))}$, has the same slope as a line tangent to the graph at some point ${\displaystyle \xi }$ in the interval ${\displaystyle [x,y]}$.

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

The Stolarsky mean is obtained by

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

when choosing ${\displaystyle f(x)=x^{p}}$.

Special cases

• ${\displaystyle \lim _{p\to -\infty }S_{p}(x,y)}$ is the minimum.
• ${\displaystyle S_{-1}(x,y)}$ is the geometric mean.
• ${\displaystyle \lim _{p\to 0}S_{p}(x,y)}$ is the logarithmic mean. It can be obtained from the mean value theorem by choosing ${\displaystyle f(x)=\ln x}$.
• ${\displaystyle S_{\frac {1}{2}}(x,y)}$ is the power mean with exponent ${\displaystyle {\frac {1}{2}}}$.
• ${\displaystyle \lim _{p\to 1}S_{p}(x,y)}$ is the identric mean. It can be obtained from the mean value theorem by choosing ${\displaystyle f(x)=x\cdot \ln x}$.
• ${\displaystyle S_{2}(x,y)}$ is the arithmetic mean.
• ${\displaystyle S_{3}(x,y)=QM(x,y,GM(x,y))}$ is a connection to the quadratic mean and the geometric mean.
• ${\displaystyle \lim _{p\to \infty }S_{p}(x,y)}$ 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

${\displaystyle S_{p}(x_{0},\dots ,x_{n})={f^{(n)}}^{-1}(n!\cdot f[x_{0},\dots ,x_{n}])}$ for ${\displaystyle f(x)=x^{p}}$.

See also

• Mean

References