Aequationes mathematicae

, Volume 92, Issue 5, pp 801–872 | Cite as

Invariance of means

  • Justyna JarczykEmail author
  • Witold Jarczyk
Open Access


We give a survey of results dealing with the problem of invariance of means which, for means of two variables, is expressed by the equality \(K\circ \left( M,N\right) =K\). At the very beginning the Gauss composition of means and its strict connection with the invariance problem is presented. Most of the reported research was done during the last two decades, when means theory became one of the most engaging and influential topics of the theory of functional equations. The main attention has been focused on quasi-arithmetic and weighted quasi-arithmetic means, also on some of their surroundings. Among other means of great importance Bajraktarević means and Cauchy means are discussed.


Mean Invariance Weighted quasi-arithmetic mean Cauchy mean Lagrangian mean Bajraktarević mean Gauss composition Convergence of successive iterates 

Mathematics Subject Classification

Primary: 26E60 Secondary: 39B22 



The authors are especially indebted to Krzysztof Ciepliński for his strong encouragement to write this survey.


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Authors and Affiliations

  1. 1.Faculty of Mathematics, Computer Science and EconometricsUniversity of Zielona GóraZielona GóraPoland

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