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# Stolarsky mean

In mathematics, the Stolarsky mean of two positive real numbers x, y is defined as:

\( \begin{align} S_p(x,y) & = \lim_{(\xi,\eta)\to(x,y)} \left({\frac{\xi^p-\eta^p}{p (\xi-\eta)}}\right)^{1/(p-1)} \\[10pt] & = \begin{cases} x & \text{if }x=y \\ \left({\frac{x^p-y^p}{p (x-y)}}\right)^{1/(p-1)} & \text{else} \end{cases} \end{align} \)

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].

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

The Stolarsky mean is obtained by

\( \xi = f'^{-1}\left(\frac{f(x)-f(y)}{x-y}\right) \)

when choosing \( f(x) = x^p. \)

Special cases

\( \lim_{p\to -\infty} S_p(x,y) \) is the minimum.

\( S_{-1}(x,y) \) is the geometric mean.

\( \lim_{p\to 0} S_p(x,y) \) is the logarithmic mean. It can be obtained from the mean value theorem by choosing f(x) = \ln x.

\( S_{\frac{1}{2}}(x,y) \) is the power mean with exponent \frac{1}{2}.

\( \lim_{p\to 1} S_p(x,y) \) 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.

\( \lim_{p\to\infty} S_p(x,y) \) is the maximum.

Generalizations

You can generalize the mean to n + 1 variables by considering the mean value theorem for divided differences for the nth derivative. You obtain

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

See also

mean

References

Stolarsky, Kenneth B.: Generalizations of the logarithmic mean, Mathematics Magazine, volume 48, number 2, March, 1975, pages 87ā92

When k = n or nā1 we saw in the previous section that Vk(Fn) is a principal homogeneous space, and therefore diffeomorphic to the corresponding classical group. These are listed in the table at the right.

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