On two-variable means with variable weights (Q1888659)

The two-variable equality problem of two weighted quasi-arithmetic means was solved by \textit{L. Losonczi} [Aequationes Math. 58, 223--241 (1999; Zbl 0939.39015)] under the supposition of six times differentiability of the functions involved. The authors first prove a regularity theorem saying that every solution of a Pexider type functional equation with two unknown functions -- one strictly monotonic and continuous, the other positive (only) -- is infinitely many times differentiable. Therefore, Losonczi's result can be applied, yielding six families of solutions of the problem in the just described class of functions.