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

Author

Listed:
  • 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)

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.

Suggested Citation

  • Janusz Matkowski & Paweł Pasteczka, 2020. "Invariant Means, Complementary Averages of Means, and a Characterization of the Beta-Type Means," Mathematics, MDPI, vol. 8(10), pages 1-9, October.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:10:p:1753-:d:426506
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