Identric mean

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

$\begin{matrix} I(x,y) &=& \frac{1}{e}\cdot \lim_{(\xi,\eta)\to(x,y)} \sqrt[\xi-\eta]{\frac{\xi^\xi}{\eta^\eta}} \\ &=& \lim_{(\xi,\eta)\to(x,y)} \exp\left(\frac{\xi\cdot\ln\xi-\eta\cdot\ln\eta}{\xi-\eta}-1\right) \\ &=& \begin{cases} x & \mbox{if }x=y \\ \frac{1}{e} \sqrt[x-y]{\frac{x^x}{y^y}} & \mbox{else} \end{cases} \end{matrix}$ .

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.