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.