On the invariance of the arithmetic mean with respect to generalized Bajraktarević means

Abstract

The purpose of this paper is to investigate the following invariance equation involving two 2-variable generalized Bajraktarević means, i.e., we aim to solve the functional equationwhere I is a nonempty open real interval and f, g : I → R are continuous, strictly monotone and p 1 , p 2 , q 1 , q 2 : I → R + are unknown functions. The main result of the paper shows that, assuming four times continuous differentiability of f , g, twice continuous differentiability of p 1 and p 2 and assuming that p 1 differs from p 2 on a dense subset of I, a necessary and sufficient condition for the equality above is that the unknown functions are of the formwhere u, v, w, z : I → R are arbitrary solutions of the second-order linear differential equation F ′′ = γF (γ ∈ R is arbitrarily fixed) such that v > 0 and z > 0 holds on I and {u, v} and {w, z} are linearly independent.

Means in money exchange operations

“…The harmonic mean is applied, for instance, in kinematic where, in some special situations, it provides the correct averages of speed of a point, where it is also employed to determine the density of a homogeneous mass formed with components of given masses and densities, and it is also used in electricity and finance. Taking into account that the invariance of the geometric mean with respect to the arithmetic-harmonic mean-type mapping and its (unique) finite dimensional extension [2] (see also [3][4][5][6][7][8][9][10][11][12]), it can be expected that the considered simple model related to the currency market can be also used in technical sciences.…”