Identric mean

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

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

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