Results in Mathematics

, Volume 71, Issue 1–2, pp 397–410 | Cite as

Explicit Solutions of the Invariance Equation for Means

  • Janusz Matkowski
  • Monika Nowicka
  • Alfred WitkowskiEmail author
Open Access


Extending the notion of projective means we first generalize an invariance identity related to the Carlson log given in Kahlig and Matkowski (Math Inequal Appl 18(3):1143–1150, 2015), and then, more generally, given a bivariate symmetric, homogeneous and monotone mean M, we give explicit formula for a rich family of pairs of M-complementary means. We prove that this method cannot be extended for higher dimension. Some examples are given and two open questions are proposed.


Invariant means homogeneous means 

Mathematics Subject Classification



  1. 1.
    Aczél J.: Lectures on Functional Equations and Their Applications. Academic Press, New York (1966)zbMATHGoogle Scholar
  2. 2.
    Gauss, C.F.: Bestimmung der Anziehung eines elliptischen Ringen. Ostwalds Klassiker Exakt. Wiss. Akademische Verlagsgesellschaft, Leipzig (1927)Google Scholar
  3. 3.
    Kahlig P., Matkowski J.: Logarithmic complementary means and an extension of Carlson’s log. Math. Inequal. Appl. 18(3), 1143–1150 (2015)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Matkowski J.: Iterations of mean-type mappings and invariant means. Ann. Math. Silesianae 13, 211–226 (1999)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Matkowski J.: Invariant and complementary means. Aequ. Math. 57, 87–107 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Matkowski, J.: Iterations of the mean-type mappings (ECIT’ 08). In: Sharkovsky, A.N., Susko, I.M. (eds.) Grazer Math. Ber., Bericht Nr. 354, pp. 158–179 (2009)Google Scholar
  7. 7.
    Matkowski J.: Iterations of the mean-type mappings and uniqueness of invariant means. Ann. Univ. Sci. Bp. Sect. Comput. 41, 145–158 (2013)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Sándor J.: On certain subhomogeneous means. Octogon Math. Mag. 8(1), 156–160 (2000)Google Scholar
  9. 9.
    Sándor, J., Toader, Gh.: On some exponential means. In: Seminar on Mathematical Analysis, Preprint vol. 3, pp. 35–40. Babeş-Bolyai University, Cluj (1990)Google Scholar
  10. 10.
    Toader, Gh.: An exponential mean. In: Seminar on Mathematical Analysis, Preprint vol. 5, pp. 51–54. Babeş-Bolyai University, Cluj (1988)Google Scholar
  11. 11.
    Witkowski A.: On Seiffert-like means. J. Math. Inequal. 9(4), 1071–1092 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Witkowski, A.: On two- and four-parameter families. RGMIA 12(1), Article 3 (2009)Google Scholar

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© The Author(s) 2016

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  • Janusz Matkowski
    • 1
  • Monika Nowicka
    • 2
  • Alfred Witkowski
    • 2
    Email author
  1. 1.Faculty of Mathematics, Computer Sciences and EconometricsUniversity of Zielona GoraZielona GóraPoland
  2. 2.Institute of Mathematics and PhysicsUTP University of Science and TechnologyBydgoszczPoland

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