Functional equations involving quasi-arithmetic means and their Gauss composition. (English) Zbl 1164.39010

Let \(I\subset\mathbb R\) be a nonvoid open interval, \(M_1,M_2,M_3:I^2\to I\) be weighted quasi-arithmetic means such that \(M_3\) is the Gauss-composition of \(M_1\) and \(M_2\), that is, \(M_3=M_1\otimes M_2\). The author investigates the equivalency of the functional equations \[ f\bigl(M_1(x,y)\bigr)+f\bigl(M_2(x,y)\bigr)=f(x)+f(y) (1) \]
\[ 2f\bigl(M_3(x,y)\bigr)=f(x)+f(y) (2) \] where \(f:I\to\mathbb R\) is an unknown function, and \(x,y\in I\). It can be verified that all solutions of (2) are the solutions of (1). The main result of the paper gives a complete characterization for the means \(M_1,M_2,M_3\) so that arbitrary solutions of (1) also satisfy (2) and reads as follows:
{Theorem}: Let \(M_1,M_2,M_3\) be weighted quasi-arithmetic means on \(I\) such that \(M_3\) is the Gauss-composition of \(M_1\) and \(M_2\). If \((M_1,M_2,M_3)\) is not an exceptional triplet, then the functional equations (1) and (2) are equivalent; if the triplet \((M_1,M_2,M_3)\) is exceptional, there exists a solution of (1) which is not a solution of (2).
Here, a triplet \((M_1,M_2,M_3)\) is said to be exceptional, if the generating function of the means are common, \(M_1\) is weighted by \(p\), \(M_2\) is weighted by \((1-p)\), \(M_3\) is weighted by \(1/2\), where \(p\in]0,1[\) is either transcendental, or algebraic such that \(p/(1-p)\) and \(-p/(1-p)\) are algebraic conjugates.


39B22 Functional equations for real functions
26E60 Means