Equality characterization for continuous-generator quasi-arithmetic means

Characterize the strictly increasing continuous generators f,g defined on an appropriate domain C in a finite-dimensional Euclidean space for which the corresponding weighted quasi-arithmetic means with fixed weights and a fixed number of variables are equal for every input tuple.

Background

The paper introduces vector-valued weighted quasi-arithmetic means and formulates their equality problem for a fixed number of variables and fixed weights. It notes that the complete equality characterization remains unresolved even in the real-valued setting for non-continuous generators, while the classical continuous scalar case is known.

The stated problem asks specifically for a characterization under the additional assumption that the generators are continuous, and leaves the domain C to be chosen appropriately.

References

Is it possible to solve the equality problem of QAMs assuming that the generators are {\em continuous}? That is, how can we characterize the strictly increasing generators $ f, g : C R $ for which eq-EqualityProblem holds on an appropriate domain $ C \subseteq R $?

— Generalized inverses of strictly monotone transformations  (2608.19997 - Tóth, 20 Aug 2026) in Section 5, second Open Problem

What structural assumptions on the ambient space, the domain, and the generators, in addition to affine equivalence to strictly increasing mappings, yield an analogue of the affine/exponential classification? In general ambient spaces, identifying suitable analogues of the exponential pairs $(\chi_p,\chi_{-p})$ in eq:chi is itself part of the problem.

eq:chi:

χp(z)≔{exp⁡(pz)p≠0zp=0.\chi_p (z) \coloneqq \begin{cases} \exp(pz) & p \ne 0\\ z & p = 0\end{cases}.

— The two-dimensional Matkowski--Sutô equation with holomorphic and strictly increasing generators  (2609.11102 - Okamura, 10 Sep 2026) in Question 1, “Additional assumptions for classification,” Section 6.2 (Open problems)

How does the classification depend on the geometry of the domain? In particular, how does it change when the generators are defined on a proper convex subset of the ambient space?

— The two-dimensional Matkowski--Sutô equation with holomorphic and strictly increasing generators  (2609.11102 - Okamura, 10 Sep 2026) in Question 2, “Dependence on the domain,” Section 6.2 (Open problems)