Matkowski-Sutô type equation on symmetrized weighted quasi-arithmetic means
Given \(\varphi \in \mathcal{CM}(I)\) (\(\mathcal{CM}(I)\) is the class of continuous and strictly monotone real valued functions defined on the open interval \(I\)), \(A_{\varphi}(x,y;\alpha)\) denotes the weighted quasi-arithmetic mean generated by \(\varphi\) with weight \(\alpha\), \(0<\alpha<1\), i.e., \[ A_{\varphi}(x,y;\alpha)=\varphi^{-1}(\alpha \varphi(x)+(1-\alpha) \varphi(y)), \quad x, y \in I. \] Defined \[ A^*_{\varphi}(x,y;\alpha):=\frac{A_{\varphi}(x,y;\alpha)+A_{\varphi}(x,y;(1-\alpha))}{2}, \] the paper aims to find all pairs of functions \(\varphi, \psi \in \mathcal{CM}(I)\) such that \[ A^*_{\varphi}(x,y;\alpha)+A^*_{\psi}(x,y;\alpha)=x+y, \quad x, y \in I. \tag{E} \] Note that if \(\alpha=1/2\) we have the original Matkowski-Sutô problem. The main result is stated in the following Theorem: Let \((\varphi,\psi) \in \mathcal{CM}(I)^2\) be a solution of equation (E) and assume that there exists a nonempty subinterval \(J\) of \(I\) such that both functions are two times continuously differentiable on \(J\). If at least one of them is six times continuously differentiable on \(J\), then there exists a real number \(p\) such that \[ \varphi(x)=a_1\chi_p(x)+b_1, \quad \psi(x)=a_2\chi_{-p}(x)+b_2, \] for some \(a_1, a_2 \neq 0\) and \(b_1, b_2 \in \mathbb R\), where \[ \chi_r(x)=\begin{cases} e^{rx}, &\text{if }r\neq 0,\\ x, &\text{if }r=0.\end{cases} \]
scientific article; zbMATH DE number 6135254
scientific article