On an equation involving weighted quasi-arithmetic means (Q626013)

The main theorem of this paper gives a full solution of the Matkowski-Sutô type functional equation \[ \kappa x+(1-\kappa)y=\lambda \varphi^{-1}(\mu\varphi(x)+(1-\mu)\varphi(y)) +(1-\lambda)\psi^{-1}(\nu\psi(x)+(1-\nu)\psi(y)). \] The unknown functions \(\varphi\) and \(\psi\) are assumed to be continuous and strictly monotone on an interval (these are natural regularity conditions), \(\kappa,\lambda\) are real numbers different from zero and one, \(\mu,\nu\in]0,1[\) are weights.