Stolarsky mean explained

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:

\begin{align} S _{p(x,y) &} = \lim _{(\xi,η)\to(x,y)} \left({ (\xi ^{p-η} ^{p})/(p (\xi -η )) }\right)^ \\[10pt]& = \beginx & \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 [x,y] .\exists \xi\in[x,y] f'(\xi) = (f(x) - f(y))/(x - y)

The Stolarsky mean is obtained by

\xi = \left[f'\right] ^{-1} \left( (f(x) - f(y))/(x - y) \right) when choosing f(x) = x ^{p} .

Special cases

\lim _{p\to} S _{p(x,y)} is the minimum.S _{-1} (x,y) is the geometric mean.\lim _{p\to} S _{p(x,y)} is the logarithmic mean. It can be obtained from the mean value theorem by choosing f(x) = ln x .S _{(1)/(2)} (x,y) is the power mean with exponent (1)/(2) .\lim _{p\to} S _{p(x,y)} is the identric mean. It can be obtained from the mean value theorem by choosing f(x) = x ⋅ 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\toinfty} 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,...,x} _{n)} = {f ^{(n)} }^(n!\cdot f[x_0,\dots,x_n]) for f(x) = x ^{p} .

See also

• Mean

Notes and References

1. 0302.26003 . Stolarsky . Kenneth B. . Generalizations of the logarithmic mean . . 48 . 87–92 . 1975 . 2 . 0025-570X . 2689825 . 10.2307/2689825.