Identric mean

From HandWiki

The identric mean of two positive real numbers xy is defined as:[1]

[math]\displaystyle{ \begin{align} I(x,y) &= \frac{1}{e}\cdot \lim_{(\xi,\eta)\to(x,y)} \sqrt[\xi-\eta]{\frac{\xi^\xi}{\eta^\eta}} \\[8pt] &= \lim_{(\xi,\eta)\to(x,y)} \exp\left(\frac{\xi\cdot\ln\xi-\eta\cdot\ln\eta}{\xi-\eta}-1\right) \\[8pt] &= \begin{cases} x & \text{if }x=y \\[8pt] \frac{1}{e} \sqrt[x-y]{\frac{x^x}{y^y}} & \text{else} \end{cases} \end{align} }[/math]

It can be derived from the mean value theorem by considering the secant of the graph of the function [math]\displaystyle{ x \mapsto x\cdot \ln x }[/math]. 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

References

  1. RICHARDS, KENDALL C; HILARI C. TIEDEMAN (2006). "A NOTE ON WEIGHTED IDENTRIC AND LOGARITHMIC MEANS". Journal of Inequalities in Pure and Applied Mathematics 7 (5). http://www.kurims.kyoto-u.ac.jp/EMIS/journals/JIPAM/images/202_06_JIPAM/202_06_www.pdf. Retrieved 20 September 2013. 

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