aequationes mathematicae

, Volume 45, Issue 2–3, pp 300–321 | Cite as

On the mutual noncompatibility of homogeneous analytic non-power means

  • Marcin E. Kuczma
Research Papers

Summary

Homogeneous symmetric meansµ and\(\tilde \mu \), defined on ℝ + n and ℝ + n+1 , respectively, are calledcompatible if the value of\(\tilde \mu \) remains unchanged upon replacing n of its arguments by theirµ-mean. Power means (of a common exponent) are a model example, which turns out to be unique, given analyticity of at least one of the two means considered. This is proved by fixing all but one argument in bothμ and\(\tilde \mu \), which leads to a functional equation with two unknown functions, involving their mutual superpositions. The equation is solved in the class of analytic functions by comparing the power series coefficients.

AMS (1991) subject classification

39B30 39B40 

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References

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    Aczél, J.,Lectures on functional equations and their applications. Academic Press, New York and London, 1966.Google Scholar
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    Bullen, P. S., Mitrinović, D. S. andVasić, P. M.,Means and their inequalities. Reidel, Dordrecht, 1988.Google Scholar
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    Kolmogorov, A. N.,Sur la notion de moyenne. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., (6)12 (1930), 388–391.Google Scholar
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    Mitrinović, D. S.,Analytic inequalities. Springer, Berlin—Heidelberg—New York, 1970.Google Scholar

Copyright information

© Birkhäuser Verlag 1993

Authors and Affiliations

  • Marcin E. Kuczma
    • 1
  1. 1.Institute of MathematicsUniversity of WarsawWarszawaPoland

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