# On a functional equation appearing on the margins of a mean invariance problem

### Abstract

Given a continuous strictly monotonic real-valued function α, defined on an interval I, and a function ω:I→(0,+∞) we denote by B_{ω}^{α} the Bajraktarević mean generated by α and weighted by ω:

B_{ω}^{α}(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)B_{s}^{ϕ}(x,y) + r(y)B_{t}^{ψ}{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.

### Keywords

functional equation; mean; invariance; Bajraktarević mean

### References

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*Annales Mathematicae Silesianae*,

*34*(1), 96-103. Retrieved from https://www.journals.us.edu.pl/index.php/AMSIL/article/view/13635

Instytut Matematyki, Uniwersytet Zielonogórski Poland

Instytut Matematyki, Informatyki i Architektury Krajobrazu, Katolicki Uniwersytet Lubelski Jana Pawła II Poland

https://orcid.org/0000-0002-5866-0517

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