Invariant Means, Complementary Averages of Means, and a Characterization of the Beta-Type Means

Janusz Matkowski and Paweł Pasteczka
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Janusz Matkowski: Institute of Mathematics, University of Zielona Góra, Szafrana 4a, PL-65-516 Zielona Góra, Poland
Paweł Pasteczka: Institute of Mathematics, Pedagogical University of Krakow, Podchorążych 2, PL-30-084 Kraków, Poland

Mathematics, 2020, vol. 8, issue 10, 1-9

Abstract: We prove that whenever the selfmapping ( M 1 , … , M p ) : I p ? I p , ( p ? N and M i -s are p -variable means on the interval I ) is invariant with respect to some continuous and strictly monotone mean K : I p ? I then for every nonempty subset S ? { 1 , … , p } there exists a uniquely determined mean K S : I p ? I such that the mean-type mapping ( N 1 , … , N p ) : I p ? I p is K -invariant, where N i : = K S for i ? S and N i : = M i otherwise. Moreover min ( M i : i ? S ) ? K S ? max ( M i : i ? S ) . Later we use this result to: (1) construct a broad family of K -invariant mean-type mappings, (2) solve functional equations of invariant-type, and (3) characterize Beta-type means.

Keywords: invariant means; complementary averages of means; characterizations; beta-type means (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
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