Editing Quasi-arithmetic mean (section)

== Characterization ==
There are several different sets of properties that characterize the quasi-arithmetic mean (i.e., each function that satisfies these properties is an ''f''-mean for some function ''f'').

* '''Mediality''' is essentially sufficient to characterize quasi-arithmetic means.<ref name=":0">{{Cite book|title=Functional equations in several variables. With applications to mathematics, information theory and to the natural and social sciences. Encyclopedia of Mathematics and its Applications, 31.|author=Aczél, J.|author2=Dhombres, J. G.|publisher=Cambridge Univ. Press|year=1989|location=Cambridge}}</ref>{{Rp|chapter 17}}
* '''Self-distributivity''' is essentially sufficient to characterize quasi-arithmetic means.<ref name=":0" />{{Rp|chapter 17}}
* '''Replacement''': Kolmogorov proved that the five properties of symmetry, fixed-point, monotonicity, continuity, and replacement fully characterize the quasi-arithmetic means.<ref>{{Cite web|url=https://math.stackexchange.com/a/3261514/29780|title=Characterization of the quasi-arithmetic mean|last=Grudkin|first=Anton|date=2019|website=Math stackexchange}}</ref>
* Continuity is superfluous in the characterization of two variables quasi-arithmetic means. See [10] for the details.
* '''Balancing''': An interesting problem is whether this condition (together with symmetry, fixed-point, monotonicity and continuity properties) implies that the mean is quasi-arithmetic. [[Georg Aumann]] showed in the 1930s that the answer is no in general,<ref>{{cite journal|last=Aumann|first=Georg|year=1937|title=Vollkommene Funktionalmittel und gewisse Kegelschnitteigenschaften|journal=[[Journal für die reine und angewandte Mathematik]]|volume=1937|issue=176|pages=49–55|doi=10.1515/crll.1937.176.49|s2cid=115392661}}</ref> but that if one additionally assumes <math>M</math> to be an [[analytic function]] then the answer is positive.<ref>{{cite journal|last=Aumann|first=Georg|year=1934|title=Grundlegung der Theorie der analytischen Analytische Mittelwerte|journal=Sitzungsberichte der Bayerischen Akademie der Wissenschaften|pages=45–81}}</ref>