Identric mean explained

The identric mean of two positive real numbers x, y is defined as:^[1]

\begin{align} I(x,y) &=	1
	e

⋅ \lim_{(\xi,η)\to(x,y)}\sqrt[\xi-η]{

	\xi^\xi
	η^η

}\\[8pt]&=\lim_\exp\left(\frac-1\right)\\[8pt]&=\beginx & \textx=y \\[8pt]\frac \sqrt[x-y] & \text\end\end

It can be derived from the mean value theorem by considering the secant of the graph of the function

x\mapstox ⋅ lnx

. 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.

Notes and References

RICHARDS. KENDALL C. HILARI C. TIEDEMAN. A NOTE ON WEIGHTED IDENTRIC AND LOGARITHMIC MEANS. Journal of Inequalities in Pure and Applied Mathematics. 2006. 7. 5. 20 September 2013. 21 September 2013. https://web.archive.org/web/20130921055354/http://www.kurims.kyoto-u.ac.jp/EMIS/journals/JIPAM/images/202_06_JIPAM/202_06_www.pdf. live.