Existence of a non–quasi-arithmetic continuous symmetric strict solution to the σ-balancing equation

Determine whether there exists a mean M: I^n -> R on a nontrivial interval I that is continuous, symmetric, and strict, and that satisfies the σ-balancing fixed point equation M(M(u_{σ(1)}(x, u)), ..., M(u_{σ(n)}(x, u))) = u for all x in I^n with u = M(x), where u_k(x, t) denotes the vector obtained from x by replacing its k-th coordinate by t, but such that M is not a quasi-arithmetic mean (i.e., M cannot be represented as φ^{-1}((1/n) Σ_{i=1}^n φ(x_i)) for any continuous strictly monotone φ: I -> R).

Background

The paper studies σ-balanced means, defined via a fixed point equation involving coordinate replacements and a permutation σ of indices. Quasi-arithmetic means are shown to satisfy the σ-balancing property and, within the class of generalized quasi-arithmetic means, the main theorem proves that σ-balancedness characterizes the standard quasi-arithmetic subclass.

Beyond generalized quasi-arithmetic means, it remains unsettled whether there exist continuous, symmetric, strict σ-balanced means that are not quasi-arithmetic. The authors explicitly highlight this as an open question, indicating a gap between known sufficient constructions (quasi-arithmetic means) and a potential broader family of σ-balanced means with the same regularity properties.