Functional equations involving means and their Gauss composition

Abstract

In this paper the equivalence of the two functional equations \[ f ( M 1 ( x , y ) ) + f ( M 2 ( x , y ) ) = f ( x ) + f ( y ) ( x , y ∈ I ) f(M_1(x,y))+f(M_2(x,y))=f(x)+f(y) \qquad (x,y\in I) \] and \[ 2 f ( M 1 ⊗ M 2 ( x , y ) ) = f ( x ) + f ( y ) ( x , y ∈ I ) 2f(M_1\otimes M_2(x,y))=f(x)+f(y) \qquad (x,y\in I) \] is studied, where M 1 M_1 and M 2 M_2 are two variable strict means on an open real interval I I , and M 1 ⊗ M 2 M_1\otimes M_2 denotes their Gauss composition. The equivalence of these equations is shown (without assuming further regularity assumptions on the unknown function f : I → R f:I\to \mathbb {R} ) for the cases when M 1 M_1 and M 2 M_2 are the arithmetic and geometric means, respectively, and also in the case when M 1 M_1 , M 2 M_2 , and M 1 ⊗ M 2 M_1\otimes M_2 are quasi-arithmetic means. If M 1 M_1 and M 2 M_2 are weighted arithmetic means, then, depending on the algebraic character of the weight, the above equations can be equivalent and also non-equivalent to each other.