On the balancing property of Matkowski means (Q2227521)

Let \(I\subset\mathbb{R}\) be a nonempty open interval and let \(M: I\times I\to\mathbb{R}\) be a mean on \(I\) (i.e., \(\min(x,y)\leq M(x,y)\leq\max(x,y)\) for all \(x,y\in I\)). We say that \(M\) is \textit{balanced} iff: \[ M(M(x,M(x,y)),M(M(x,y),y))=M(x,y),\qquad x,y\in I. \] The author characterizes all balanced means in a class \(\mathcal{M}(I)\), containing the class of Matkowski means \(\mathcal{M}_{f,g}\) (i.e., means of the form \(M(x,y)=(f+g)^{-1}(f(x)+g(y))\) where \(f,g: I\to\mathbb{R}\) are continuous and strictly monotone mappings). In particular, it is proved that a Matkowski mean \(\mathcal{M}_{f,g}\) is balanced if and only if it is a quasi-arithmetic mean generated by the function \(f+g\), i.e., \[ \mathcal{M}_{f,g}(x,y)=(f+g)^{-1}\left(\frac{(f+g)(x)+(f+g)(y)}{2}\right),\qquad x,y\in I. \] As opposed to earlier investigations on the subject (going back to \textit{G. Aumann}'s results [Math. Ann. 111, 713--730 (1935; Zbl 0012.25205); J. Reine Angew. Math. 176, 49--55 (1936; Zbl 0015.06202)]) no differentiability of \(f,g\) is assumed.