From Wikipedia, the free encyclopedia
In
mathematics, the Stolarsky mean is a generalization of the
logarithmic mean. It was introduced by
Kenneth B. Stolarsky in 1975.
[1]
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f8db0526800a446260c59fd7036b318d0e5db4f)
It is derived from the
mean value theorem, which states that a
secant line, cutting the graph of a
differentiable function
at
and
, has the same
slope as a line
tangent to the graph at some point
in the
interval
.
![{\displaystyle \exists \xi \in [x,y]\ f'(\xi )={\frac {f(x)-f(y)}{x-y}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de508a525bca5e9fdd9e4f2c66a3b3e7d28b72d0)
The Stolarsky mean is obtained by
![{\displaystyle \xi =\left[f'\right]^{-1}\left({\frac {f(x)-f(y)}{x-y}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fa492264dcee6e015611b1969e018608807c26f5)
when choosing
.
is the
minimum.
is the
geometric mean.
is the
logarithmic mean. It can be obtained from the mean value theorem by choosing
.
is the
power mean with exponent
.
is the
identric mean. It can be obtained from the mean value theorem by choosing
.
is the
arithmetic mean.
is a connection to the
quadratic mean and the
geometric mean.
is the
maximum.
One can generalize the mean to n + 1 variables by considering the
mean value theorem for divided differences for the nth
derivative.
One obtains
for
.