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Any quasi-arithmetic mean of two variables has the balancing property .
The quasi-arithmetic mean is invariant with respect to offsets and scaling of :
Moreover, any of those properties is essentially sufficient to characterize quasi-arithmetic means; see Aczél-Dhombres, Chapter 17.
Indeed, the only homogeneous quasi-arithmetic means are the power means and the geometric mean; see Hardy-Littlewood-Pólya, page 68.