Identric mean

The identric mean of two positive real numbers x, y is defined as:

$\begin{array}{rl}I(x,y)& =\frac{1}{e}\cdot \underset{(\xi ,\eta )\to (x,y)}{\lim }\sqrt[\xi -\eta ]{\frac{{\xi }^{\xi }}{{\eta }^{\eta }}}\\ & =\underset{(\xi ,\eta )\to (x,y)}{\lim }\exp (\frac{\xi \cdot \ln \xi -\eta \cdot \ln \eta }{\xi -\eta }-1)\\ & =\{\begin{array}{ll}x& \text{if }x=y\\ \frac{1}{e}\sqrt[x-y]{\frac{x^x}{y^y}}& \text{else}\end{array}\end{array}$

It can be derived from the mean value theorem by considering the secant of the graph of the function $x\mapsto x\cdot \ln x$. It can be generalized to more variables according by the mean value theorem for divided differences. The identric mean is a special case of the Stolarsky mean.

See also

• Mean
• Logarithmic mean

References

• Weisstein, Eric W.. "Identric Mean". http://mathworld.wolfram.com/IdentricMean.html.