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.