Functional equations involving means and their Gauss composition

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In this paper the equivalence of the two functional equations\[f(M1(x,y))+f(M2(x,y))=f(x)+f(y)(x,y∈I)f(M_1(x,y))+f(M_2(x,y))=f(x)+f(y) \qquad (x,y\in I)\]and\[2f(M1⊗M2(x,y))=f(x)+f(y)(x,y∈I)2f(M_1\otimes M_2(x,y))=f(x)+f(y) \qquad (x,y\in I)\]is studied, whereM1M_1andM2M_2are two variable strict means on an open real intervalII, andM1⊗M2M_1\otimes M_2denotes their Gauss composition. The equivalence of these equations is shown (without assuming further regularity assumptions on the unknown functionf:I→Rf:I\to \mathbb {R}) for the cases whenM1M_1andM2M_2are the arithmetic and geometric means, respectively, and also in the case whenM1M_1,M2M_2, andM1⊗M2M_1\otimes M_2are quasi-arithmetic means. IfM1M_1andM2M_2are 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.