Identric mean

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

{\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 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. ↑ RICHARDS, KENDALL C; HILARI C. TIEDEMAN (2006). "A NOTE ON WEIGHTED IDENTRIC AND LOGARITHMIC MEANS". Journal of Inequalities in Pure and Applied Mathematics 7 (5). Retrieved 20 September 2013. 

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

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