Identric mean

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


\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}

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

References

  1. Lua error in package.lua at line 80: module 'strict' not found.

Weisstein, Eric W., "Identric Mean", MathWorld.