On means that are both quasi-arithmetic and conjugate arithmetic (Q5948304)

The authors determine all means of two variables that are simultaneously of the form \[ \psi^{-1}\biggl({\psi(x)+\psi(y)\over 2}\biggr)\quad\text{and}\quad \varphi^{-1}(\varphi(x)+\varphi(y)-\varphi\Bigl({x+y\over 2}\Bigl)). \] The functions \(\psi\) and \(\varphi\) are strictly monotonic, continuous, defined on an open real interval, and one of them is continuously differentiable. The result obtained improves that of \textit{J. Matkowski} see e.g. [Aequationes Math. 57, No, 1, 87-107 (1999; Zbl 0930.26014)].