Solution of a problem of Z. Daróczy, J. Jarczyk and W. Jarczyk and generalization of a theorem of R. Ger and T. Kochane (Q1669652)

The following is a problem of \textit{Z. Daróczy} et al., [Aequationes Math. 90, No. 1, 211--233 (2016; Zbl 1351.26050)]: let \(I\subseteq \mathbb R\) be an interval and let \(M:I\mapsto I\) be a mean: is it true that if \(M\) is not quasi-arithmetic then every solution \(f:I\mapsto \mathbb R\) of \(f\bigl(M(x,y)\bigr)=\bigl(f(x) + f)(y) \bigr)\big/2\) is constant? This was answered in the negative as was also the case if \(M\) assumed to be strict. That left open the question of the \(M\) being in addition continuous. This is answered here by the following theorem, generalizing one of \textit{R. Ger} and \textit{T. Kochanek} [Colloq. Math. 115, No. 1, 87--99 (2009; Zbl 1168.39006)]: let \(I ,J\subseteq\mathbb R, M:I^2\mapsto I\) be a strict mean continuous with respect to each variable, \(K:J^2\mapsto I\) a quasi-arithmetic mean. If \(f\bigl(M(x,y)\bigr)=K\bigl(f(x),f(y)\bigr)\) has a non-constant solution \(f:I\mapsto J\) then \(M\) is a quasi-arithmetic mean.