The Matkowski-Sutô problem for weighted quasi-arithmetic means
Let \(I\subset\mathbb{R}\) be a non-void open interval and let \(\mathcal{CM}(I)\) denote the class of all continuous and strictly monotone real-valued functions defined on the interval \(I\). A function \(M:I\times I \to I\) is called a weighted quasi-arithmetic mean on \(I\) if there exist a number \(0<\lambda<1\) and a function \(\varphi\in \mathcal{CM}(I)\) such that \[ M(x,y)=\varphi^{-1}\left(\lambda \varphi (x)+(1-\lambda)\varphi (y)\right)=:A_{\varphi}(x,y;\lambda) \] for all \(x,y\in I.\) If \(\varphi(x)=x, \quad x\in I\) then \(A_{\varphi}(x,y;\lambda)\) is denoted simply by \(A(x,y;\lambda).\) The authors solve the invariance equation \[ A(A_{\varphi}(x,y;\lambda),A_{\psi}(x,y;\lambda);\lambda) =A(x,y;\lambda) \qquad (x,y \in I) \] where \(0<\lambda<1\) is fixed and \(\varphi, \psi :I \to \mathbb{R}\) are continuously differentiable on \(I\) with \(\varphi'(x) \psi'(x)\neq 0\) if \(x\in I.\) The case \(\lambda =\frac{1}{2}\), \(\varphi, \psi \in \mathcal{CM}(I)\) is the original Matkowski-Sutô problem that was solved also by the authors in an earlier paper [in: Functional equations -- results and advances, Adv. Math. Dordr. 3, 189-200 (2002; Zbl 1012.39019)].
scientific article; zbMATH DE number 2098988