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 Kcirc left( M,Nright) =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.
Highlights
The idea of a mean is as old in human cognition as that one of a number
Well known already in the antiquity, are the arithmetic mean A : Rp → R: A (x1, . . . , xp) xp, the geometric mean G : (0, +∞)p → (0, +∞): G (x1, . . . , xp) = √p x1 . . . xp, the harmonic mean H : (0, +∞)p → (0, +∞): p
G (A (x, y), H (x, y)) = G(x, y). This is the celebrated equality expressing the invariance of the geometric mean with respect to the pair (A, H) of the arithmetic and harmonic means. This is a good starting point to study the main problem of the paper which can be formulated as follows: Given an interval I and a positive integer p we are interested in means K : Ip → I and M1, . . . , Mp : Ip → I satisfying the invariance equation
Summary
The idea of a mean is as old in human cognition as that one of a number. Given quantities x1, . . . , xp one can intuitively look for a mean of them as any number M (x1, . . . , xp) lying somewhere between the extreme values of x1, . . . , xp: min {x1, . . . , xp} ≤ M (x1, . . . , xp) ≤ max {x1, . . . , xp}. This is the celebrated equality expressing the invariance of the geometric mean with respect to the pair (A, H) of the arithmetic and harmonic means This is a good starting point to study the main problem of the paper which can be formulated as follows: Given an interval I and a positive integer p we are interested in means K : Ip → I and M1, . Most often the invariance problem is studied in classes of means described with the aid of function generators and some parameters. This causes that the invariance Eq (1.3) takes different forms and becomes a functional equation in several variables, with a number of unknown functions (the generators of the means) and parameters to be determined. The selection we made reflects our personal preference only and no doubt this survey does not pretend to be comprehensive in any way
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