Equality of two variable weighted means: Reduction to differential equations
In this remarkable paper, the author considers the functional equation in a real interval \[ \Phi^{-1} \left( {\Phi (x) F(x) + \Phi (y) F(y)} \over {F(x)+ F(y) } \right) = \Psi^{-1} \left( {\Psi (x) G (x)+ \Psi (y) G(y)} \over {G(x)+G(y) } \right) \tag{*} \] of the equality of two quasiarithmetic means weighted by some weightfuctions \(F\) and \(G\). He shows that if the functions involved are six times differentiable then one can reduce the study of (*) to that of a sixth order differential equation for \( \Phi \circ \Psi^{-1}\), obtaining 32 new families of solutions.
scientific article; zbMATH DE number 1383751