Definition

For two positive real numbers x and y the Stolarsky Mean is defined as:

: S_p(x,y)

\left { \begin{array}{l l} x, & \text{if }x=y, \ \left({\frac{x^p-y^p}{p (x-y)}}\right)^{1/(p-1)}, & \text{otherwise}. \end{array} \right .

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) = \frac{f(x)-f(y)}{x-y}

The Stolarsky mean is obtained by : \xi = \left[f'\right]^{-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

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

References

References

1. Stolarsky, Kenneth B.. (1975). "Generalizations of the logarithmic mean". [[Mathematics Magazine]].