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On the equality problem of two-variable Bajraktarevi\'c means under first-order differentiability assumptions

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Zsolt Páles, Amr Zakaria

Classical Analysis and ODEs · Submitted 14 Dec 2021

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim 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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arXiv ID2112.07396 (version 1)
Pith IDBRNSNPBS, Pith's permanent record of this paper and its review
CitationAmr Zakaria, Zsolt P\'ales. "On the equality problem of two-variable Bajraktarevi\'c means under first-order differentiability assumptions." arXiv:2112.07396 (2021). https://pith.science/paper/BRNSNPBS
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The equality problem of the two-variable Bajraktarevi\'c means can be expressed as the functional equation $$ \left(\frac{f}{g}\right)^{-1}\bigg(\frac{f(x)+f(y)}{g(x)+g(y)}\bigg) =\left(\frac{h}{k}\right)^{-1}\bigg(\frac{h(x)+h(y)}{k(x)+k(y)}\bigg)\qquad(x,y\in I), $$ where $I$ is a nonempty open real interval, $f,g,h,k:I\to{\mathbb R}$ 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, P\'ales 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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#sentencerolethe authors' own workstates the main conclusionstrength
s0 The equality problem of the two-variable Bajraktarevi\'c means can be expressed as the functional equation $$ \left(\frac{f}{g}\right)^{-1}\bigg(\frac{f(x)+f(y)}{g(x)+g(y)}\bigg) =\left(\frac{h}{k}\right)^{-1}\bigg(\frac{h(x)+h(y)}{k(x)+k(y)}\bigg)\qquad(x,y\in I), $$ where $I$ is a nonempty open real interval, $f,g,h,k:I\to{\mathbb R}$ are continuous functions, $g$, $k$ are positive and $f/g$, $h/k$ are strictly monotone. background 1.00 0.12 0.04 asserted 1.00
s1 This functional equation, for the first time, was solved by Losonczi in 1999 under 6th-order continuous differentiability assumptions. background 1.00 0.02 0.03 asserted 1.00
s2 Additional and new characterizations of this equality problem have been found recently by Losonczi, P\'ales and Zakaria under the same regularity assumptions in 2021. background 1.00 0.04 0.03 asserted 1.00
s3 In this paper it is shown that the same conclusion can be obtained under substantially weaker regularity conditions, namely, assuming only first-order differentiability. contribution 0.70 0.95 0.96 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.99.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

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Original abstract

The equality problem of the two-variable Bajraktarevi\'c means can be expressed as the functional equation $$ \left(\frac{f}{g}\right)^{-1}\bigg(\frac{f(x)+f(y)}{g(x)+g(y)}\bigg) =\left(\frac{h}{k}\right)^{-1}\bigg(\frac{h(x)+h(y)}{k(x)+k(y)}\bigg)\qquad(x,y\in I), $$ where $I$ is a nonempty open real interval, $f,g,h,k:I\to{\mathbb R}$ 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, P\'ales 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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