Contraharmonic mean quirk

A few weeks ago I wrote about the contraharmonic mean. Given two positive numbers a and b, their contraharmonic mean is

C = \frac{a^2 + b^2}{a + b}

This mean has the unusual property that increasing one of the two inputs could decrease the mean. If you take the partial derivative with respect to a you can see that it is zero when a = (√2 − 1)b. So when x is small relative to b the contraharmonic mean is a decreasing function of a.

The Gini mean with parameters r and s is given by

G(a, b; r, s) = \left(\frac{a^{r+s} + b^{r+s}}{a^s + b^s}\right)^{1/r}

The contraharmonic mean corresponds to r = s = 1 and the Lehmer p mean corresponds to r = 1 and s = p − 1.

Since the contraharmonic mean is a special case of the Lehmer and Gini means, these means are not always increasing functions of their arguments.

Incidentally, the harmonic mean corresponds to the Gini mean r = 1 and s = −1, and the harmonic mean is always in increasing function of its argument.

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One thought on “Contraharmonic mean quirk

  1. If my calculations are right (which they _might_ be) G(a,b;r,s) is an increasing function of its arguments iff s <= 0.

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