aequationes mathematicae

, Volume 32, Issue 1, pp 171–194 | Cite as

On the characterization of quasiarithmetic means with weight function

  • Zsolt Páles
Research Papers

Abstract

In the present note we completely solve the characterization problem of quasiarithmetic means with weight function, that is, functions of the form
$$M(x_1 ,...,x_n ) = f^{ - 1} \left( {\sum\limits_{i = 1}^n {p(x_i )} f(x_i )/\sum\limits_{i = 1}^n {p(x_i )} } \right)$$
(f is a strictly monotonic continuous real function andp is a positive valued real function.)

The result obtained gives a partial answer to a 22-year-old problem of Aczél [1] and generalizes the characterization theorem of quasiarithmetic means which is due to Kolmogorov [8], Nagumo [9] and de Finetti [7].

AMS (1980) subject classification

Primary 39C05 Secondary 26B99 26D99 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Aczél, J.,Problem 7. Arch. Math. (Basel)16 (1964), 348.Google Scholar
  2. [2]
    Aczél, J. andDaróczy, Z., Über verallgemeinerte quasilineare Mittelwerte, die mit Gewichtsfunktionen gebildet sind. Publ. Math. Debrecen10 (1963), 171–190.MathSciNetGoogle Scholar
  3. [3]
    Aczél, J. andDaróczy, Z.,On measures of information and their characterizations. Academic Press, New York, 1975.Google Scholar
  4. [4]
    Bajraktarević, M., Sur une èquation fonctionelle aux valeurs moyennes. Glas. Mat. Ser. III13 (1958), 243–248.Google Scholar
  5. [5]
    Daróczy, Z., Über eine Klasse von Mittelwerten. Publ. Math. Debrecen19 (1972), 211–217.MathSciNetGoogle Scholar
  6. [6]
    Daróczy, Z. andPáles, Zs.,Multiplicative mean values and entropies. InFunctions, Series and Operators, Colloq. Math. Soc. János Bolyai35 (1980), 343–359.Google Scholar
  7. [7]
    de Finetti, B., Sul concetto di meddia. Giornale dell' Ist. Ital. d. Attarii2 (1931), 369–396.MATHGoogle Scholar
  8. [8]
    Kolmogorov, A. N.,Sur la notion de la moyenne. Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur. Sez (6)12 (1930), 388–391.Google Scholar
  9. [9]
    Nagumo, M.,Über eine Klasse der Mittelwerte. Japan J. Math. 7 (1930), 235–253.Google Scholar
  10. [10]
    Páles, Zs.,Characterization of quasideviation means. Acta Math. Hungar.40 (1982), 243–260.MATHGoogle Scholar

Copyright information

© Birkhäuser Verlag 1987

Authors and Affiliations

  • Zsolt Páles
    • 1
  1. 1.Department of MathematicsKossuth Lajos University, Pf. 12DebrecenHungary

Personalised recommendations