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Scales of quasi-arithmetic means determined by invariance property
It is well known that if denotes a set of power means then
the mapping is
both 1-1 and onto for any non-constant sequence of
positive numbers. Shortly: the family of power means is a scale.
If is an interval and is a
continuous, strictly monotone function then
is a natural generalization of power means, so called quasi-arithmetic mean
generated by .
A famous folk theorem says that the only homogeneous, quasi-a\-rith\-me\-tic
means are power means. We prove that, upon replacing the homogeneity
requirement by an invariant-type axiom, one gets a family of quasi-arithmetic
means building up a scale, too.Comment: 11 page