On a class of means of several variables (Q2745651)

Given an open interval \(I\), a mean (of \(n\geq 2\) variables) is any function \(M: I^n\to I\) which is continuous, symmetric and intermediary.NEWLINENEWLINENEWLINETo each strictly monotonic continuous \(\varphi: I\to \mathbb{R}\) one can attach a new mean of \(n\) variables, NEWLINE\[NEWLINEM^*_\varphi(x_1,\dots, x_n)= \varphi^{-1}\Biggl({\varphi(x_1)+\cdots+ \varphi(x_n)- \varphi(M(x_1,\dots, x_n))\over n-1}\Biggr),NEWLINE\]NEWLINE called the conjugate mean. The main result of the paper asserts that \(M^*_\varphi\leq M^*_\psi\) if and only if \(\varepsilon_\psi \psi\circ\varphi^{-1}\) is convex, where \(\varepsilon_\psi= 1\) if \(\psi\) is increasing and \(\varepsilon_\psi= -1\) if \(\psi\) is decreasing. This extends a previous result of \textit{Z. Daróczy} [Publ. Math. 55, No. 1-2, 177-197 (1999; Zbl 0932.39019)]. The paper also contains a complete description of all conjugate means which are homogeneous or quasi-arithmetic.