On conjugate means of \(n\) variables

Abstract. Let \(I\subset {\Bbb R}\) be a nonvoid open interval and let \(n\geq 3\) be a fixed natural number. The question is which conjugate means of \(n\) variables generated by the \(n\) variable arithmetic mean are weighted quasi-arithmetic means of \(n\) variables at the same time? This question is a functional equation problem. We characterize the real valued continuous and strictly monotone functions \(\varphi,\, \psi\) defined on \(I\) and the parameters \(p_1,\ldots,p_n\), \(q_1,\ldots,q_n\) for which the equation $$ \varphi^{-1}\left(\sum_{i=1}^np_i\varphi(x_i)+\left(1-\sum_{i=1}^np_i\right)\varphi\left({{x_1+\ldots+x_n}\over{n}}\right)\right)= $$ $$ =\psi^{-1}\left(\sum_{i=1}^nq_i\psi(x_i)\right) $$ holds for all \(x_1,\ldots,x_n\in I\), where $$ q_i>0\quad (i=1,\ldots,n),\quad \sum_{i=1}^n q_i=1, $$ $$ p_j>0\quad {\rm and }\quad \sum_{i=1}^np_i-p_j\leq1\ (j=1,\ldots,n). $$