On linear combinations of weighted quasi-arithmetic means
Let \(CM(I)\) denote the set of all continuous and strictly monotone real functions on the interval \(I\). A mean \(M\) on \(I\) is called a weighted quasi--arithmetic mean if there exists \(\phi \in CM(I)\) such that \[ M(x,y)=\phi^{-1}(\lambda\phi(x)+(1-\lambda)\phi(y))=:A_{\phi}(x,y;\lambda) \qquad (x,y \in I). \] For \(\phi=Id\), \(A_{\phi}(x,y;\lambda)\) is the weighted arithmetic mean and is denoted by \(A(x,y;\lambda)\). The authors consider the following problem. Let \(0<\lambda<1\), \(\mu \neq 0,1\). Find all \(\phi, \psi \in CM(I)\) such that \[ \mu A_{\phi}(x,y;\lambda)+(1-\mu)A_{\psi}(x,y;\lambda)=A(x,y;\lambda) \qquad (x,y \in I). \] After summarizing the known results about this functional equation, they prove an extension theorem for its solutions. The authors claim that this extension theorem can be useful to determine the general solution of the previous equation.
scientific article; zbMATH DE number 2162497