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Characterization of quasi-arithmetic means without regularity condition


We show that bisymmetry, which is an algebraic property, has a regularity improving feature. More precisely, we prove that every bisymmetric, partially strictly monotonic, reflexive and symmetric function \(F \colon I ^2\to I\) is continuous. As a consequence, we obtain a finer characterization of quasi-arithmetic means than the classical results of Aczél [1], Kolmogoroff [18], Nagumo [20] and de Finetti [11].

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Correspondence to P. Burai.

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The second author was supported by Premium Postdoctoral Fellowship of the Hungarian Academy of Sciences and by the Hungarian NationalFoundation for Scientific Research, Grant No. K124749.

The third author was supported by the Hungarian Academy of Sciences.

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Burai, P., Kiss, G. & Szokol, P. Characterization of quasi-arithmetic means without regularity condition. Acta Math. Hungar. 165, 474–485 (2021).

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