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Identric mean

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

I ( x , y ) = 1 e ⋅ lim ( ξ , η ) → ( x , y ) ξ ξ η η ξ − η = lim ( ξ , η ) → ( x , y ) exp ⁡ ( ξ ⋅ ln ⁡ ξ − η ⋅ ln ⁡ η ξ − η − 1 ) = { x if  x = y 1 e x x y y x − y else {\displaystyle {\begin{aligned}I(x,y)&={\frac {1}{e}}\cdot \lim _{(\xi ,\eta )\to (x,y)}{\sqrt{\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&{\text{if }}x=y\\{\frac {1}{e}}{\sqrt{\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 ↦ x ⋅ ln ⁡ x {\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.

See also

References

  1. RICHARDS, KENDALL C; HILARI C. TIEDEMAN (2006). "A NOTE ON WEIGHTED IDENTRIC AND LOGARITHMIC MEANS" (PDF). Journal of Inequalities in Pure and Applied Mathematics. 7 (5). Archived (PDF) from the original on 21 September 2013. Retrieved 20 September 2013.
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