Problems and results on generalized
quasi-arithmetic means

Abstract. Let \(M:I^2\to I\) be a non-symmetric generalized quasi-arithmetic mean. We investigate the function \(N:I^2\to \mathbb{R}\) defined by $$ N(x,y):=\alpha x+\beta y+\gamma M(x,y) \qquad (x,y\in I), $$ where \(\alpha\beta\gamma\neq 0\), \(\alpha\neq\beta\).
We answer the following question: Under what conditions is the function \(N\) symmetric?