Generalized weighted quasi-arithmetic means and the Kolmogorov-Nagumo theorem (Q2866745)

If in the definition of the weighted quasi-arithmetic mean \(\mathfrak A_n(\underline a, \underline w) = f^{-1}\Bigl(\sum_{i=1}^nw_i f(a_i)\Bigr),\) [\(f\) is continuous, strictly monotonic and \(\sum_{i=1}^nw_i=1\), \(n\geq2\),], we write instead \(\mathfrak A^{[f_1, \dots,f_n]}(\underline a, \underline w) = f^{-1}\Bigl(\sum_{i=1}^n f_i(a_i)\Bigr)\) where \(f=\sum_{i=1}^nf_i\), and all the functions \(f_i\) are continuous and strictly monotonic in the same sense we obtain the generalised mean discussed in this paper. Many basic properties are obtained: \(m\)-associativity for all \(1\leq m\leq n\), symmetry if and only if \(\mathfrak A^{[f_1, \ldots,f_n]}\) is the equal weighted quasi-arithmetic mean, and a comparability condition is given. These means satisfy all the condition of the Kolomogorov-Nagumo theorem except for symmetry and it is conjectured that they are the only such means.