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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.

##### MSC:
 39B22 Functional equations for real functions 26E60 Means