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

Justyna Jarczyk; Witold Jarczyk

Published: 2020-07-09

Vol. 34 No. 1 (2020)

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

Justyna Jarczyk , Witold Jarczyk

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

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Jarczyk, J., & Jarczyk, W. (2020). On a functional equation appearing on the margins of a mean invariance problem. Annales Mathematicae Silesianae, 34(1), 96–103. Retrieved from https://journals.us.edu.pl/index.php/AMSIL/article/view/13635

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Justyna Jarczyk

Affiliation:

Instytut Matematyki, Uniwersytet Zielonogórski Poland

Witold Jarczyk

wjarczyk@kul.lublin.pl
https://orcid.org/0000-0002-5866-0517

Affiliation:

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

Vol. 34 No. 1 (2020)
Published: 2020-07-20

ISSN: 0860-2107

eISSN: 2391-4238

10.2478/amsil

Publisher

University of Silesia Press

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