Invariance of means

Abstract

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 • (M, N) = 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.

Note on generalized Archimedes–Borchardt algorithm

“…The description of the generalized Gauss algorithm and its limit behaviour, presented in Theorem M, originates in the research of Matkowski in his paper [21] (see also [19,20] and, for some further details, [15,Section 2]). In some earlier publications Matkowski assumed that at most one of the means N 1 , .…”

Set-Valued Means

“…is the quasi-arithmetic mean generated by f (see e.g. [8] and the references therein). In a similar way we can define its set-valued counterpart putting, for a given set-valued map F :…”

“…There are books and numerous papers devoted to it (see e.g. [1,3,4,[6][7][8][9][10]12] and the references given there).…”

Gaussian iterative algorithm and integrated automorphism equation for random means

“…Origins of the phenomenon, known today as the Gaussian algorithm, can be found in the Lagrangean treatise [15] written at the end of 18th century and then in Gaussian work [10] published a hundred years later (see also [11], where the results of [10] were reminded by Geppert in 1927). For some other references the reader is referred to the survey [13].…”