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:

[math]\displaystyle{ \begin{align} 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{align} }[/math]

Derivation

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

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

The Stolarsky mean is obtained by

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

when choosing [math]\displaystyle{ f(x) = x^p }[/math].

Special cases

• [math]\displaystyle{ \lim_{p\to -\infty} S_p(x,y) }[/math] is the minimum.
• [math]\displaystyle{ S_{-1}(x,y) }[/math] is the geometric mean.
• [math]\displaystyle{ \lim_{p\to 0} S_p(x,y) }[/math] is the logarithmic mean. It can be obtained from the mean value theorem by choosing [math]\displaystyle{ f(x) = \ln x }[/math].
• [math]\displaystyle{ S_{\frac{1}{2}}(x,y) }[/math] is the power mean with exponent [math]\displaystyle{ \frac{1}{2} }[/math].
• [math]\displaystyle{ \lim_{p\to 1} S_p(x,y) }[/math] is the identric mean. It can be obtained from the mean value theorem by choosing [math]\displaystyle{ f(x) = x\cdot \ln x }[/math].
• [math]\displaystyle{ S_2(x,y) }[/math] is the arithmetic mean.
• [math]\displaystyle{ S_3(x,y) = QM(x,y,GM(x,y)) }[/math] is a connection to the quadratic mean and the geometric mean.
• [math]\displaystyle{ \lim_{p\to\infty} S_p(x,y) }[/math] 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

[math]\displaystyle{ S_p(x_0,\dots,x_n) = {f^{(n)}}^{-1}(n!\cdot f[x_0,\dots,x_n]) }[/math] for [math]\displaystyle{ f(x)=x^p }[/math].

See also

• Mean