Alienation of additive and logarithmic equations

Abstract. Let \((R,+, \cdot)\) stand for an Archimedean totally ordered unitary ring and let \((H, +)\) be an Abelian group. Denote by \(C\) the cone of all strictly positive elements in \(R\). We study the solutions \(f,g: C \longrightarrow H\) of a Pexider type functional equation $$ {\rm (E)} \hskip 20mm f(x+y) +g(xy) = f(x) + f(y) + g(x) + g(y) \hskip 100mm $$ resulting from summing up the additive and logarithmic equations side by side. We show that modulo an additive constant equation (E) forces \(f\) and \(g\) to split back to the system of two Cauchy equations $$ \left \{\begin{array}{ll} f(x+y) = f(x) + f(y)\\ g(xy) = g(x) + g(y) \,\, \end{array}\right. $$ for every \(x,y \in C\ \ \) (alienation phenomenon).