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 S_p(x,y) & = \lim_ \left(\right)^ \\$0pt& = \begin
x & \textx=y \\
\left(\right)^ & \text
\end
\end

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 ,y/math>.
: \exists \xi\in ,y f'(\xi) = \frac

The Stolarsky mean is obtained by
:$\xi = f'^\left(\frac\right)$
when choosing $f(x) = x^p$.

Special cases

*$\lim_ S_p(x,y)$ is the minimum.
*$S_(x,y)$ is the geometric mean.
*$\lim_ S_p(x,y)$ is the logarithmic mean. It can be obtained from the mean value theorem by choosing $f(x) = \ln x$.
*$S_(x,y)$ is the power mean with exponent $\frac$.
*$\lim_ S_p(x,y)$ 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.
*$\lim_ S_p(x,y)$ is the maximum.

Generalizations

One can generalize the mean to ''n'' + 1 variables by considering the mean value theorem for divided differences for the ''n''th derivative.
One obtains
:S_p(x_0,\dots,x_n) = ^(n!\cdot f _0,\dots,x_n for $f(x)=x^p$.

See also

*Mean

References

{{reflist

Means