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

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

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.

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