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.

• Mean
• Logarithmic mean

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