en>Isaac.rs200: /* External links */

2014-01-18T20:59:59Z

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{{Refimprove|date=January 2010}}
The '''identric mean''' of two positive [[real number]]s ''x'', ''y'' is defined as:<ref name=RICHARDS2006>{{cite journal|last=RICHARDS|first=KENDALL C|coauthors=HILARI C. TIEDEMAN|title=A NOTE ON WEIGHTED IDENTRIC AND LOGARITHMIC MEANS|journal=Journal of Inequalities in Pure and Applied Mathematics|year=2006|volume=7|issue=5|url=http://www.kurims.kyoto-u.ac.jp/EMIS/journals/JIPAM/images/202_06_JIPAM/202_06_www.pdf|accessdate=20 September 2013}}</ref>
:<math>
\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 line|secant]] of the graph of the function <math>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==
* [[Mean]]
* [[Logarithmic mean]]

==References==
{{reflist}}
{{MathWorld|title=Identric Mean|urlname=IdentricMean}}

{{DEFAULTSORT:Identric Mean}}
[[Category:Means]]

Andreas Wirth - Revision history

en>Isaac.rs200: /* External links */