On a functional equation appearing on the margins of a mean invariance problem - Jarczyk, Justyna; Jarczyk, Witold (Annales Mathematicae Silesianae, 2020)
Given a continuous strictly monotonic real-valued function α, defined on an interval I, and a function ω:I→(0,+∞) we denote by Bωα the Bajraktarević mean generated by α and weighted by ω: Bωα(x,y) = α-1(\frac{ω(x)}{ω(x)+ω(y)}α(x) + \frac{ω(y)}{ω(x)+ω(y)}α(y)), x,y∈I. We find a necessary integral formula for all possible three times differentiable solutions (ϕ,ψ) of the functional equation r(x)Bsϕ(x,y) + r(y)Btψ{t}(x,y) = r(x)x + r(y)y, where r, s,t:I→(0,+∞) are three times differentiable functions and the first derivatives of ϕ,ψ and r do not vanish. However, we show that not every pair (ϕ,ψ) given by the found formula actually satisfies the above equation.