On the equality problem of two-variable Bajraktarević means under first-order differentiability assumptions

2021/12/14 by Zsolt Páles, Páles, Zsolt, Amr Zakaria +1
Mathematics · #39B22 #39B52 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Equations Stability Results #Mathematical Inequalities and Applications

paper · pdf · doi:10.48550/arxiv.2112.07396

openalex publication_date 2021/12/14 · openalex created_date 2022/05/05 · openalex updated_date 2026/07/28

Abstract

The equality problem of the two-variable Bajraktarević means can be expressed as the functional equation ((f)/(g))^-1((f(x)+f(y))/(g(x)+g(y))) =((h)/(k))^-1((h(x)+h(y))/(k(x)+k(y))) (x,y∈ I), where I is a nonempty open real interval, f,g,h,k:I→\mathbb R are continuous functions, g, k are positive and f/g, h/k are strictly monotone. This functional equation, for the first time, was solved by Losonczi in 1999 under 6th-order continuous differentiability assumptions. Additional and new characterizations of this equality problem have been found recently by Losonczi, Páles and Zakaria under the same regularity assumptions in 2021. In this paper it is shown that the same conclusion can be obtained under substantially weaker regularity conditions, namely, assuming only first-order differentiability.

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