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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f8db0526800a446260c59fd7036b318d0e5db4f)
Derivation
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
.
Special cases
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.
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
for
.
See also
References
- ^ Stolarsky, Kenneth B. (1975). "Generalizations of the logarithmic mean". Mathematics Magazine. 48 (2): 87–92. doi:10.2307/2689825. ISSN 0025-570X. JSTOR 2689825. Zbl 0302.26003.