On the equality of generalized Bajraktarević means under first-order differentiability assumptions
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2024
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| author | Páles, Zsolt Zakaria, Amr |
| author_facet | Páles, Zsolt Zakaria, Amr |
| contents | In this paper we consider the equality problem of generalized Bajraktarević means, i.e., we are going to solve the functional equation \begin{equation}\label{E0}\tag{*}
f^{(-1)}\bigg(\frac{p_1(x_1)f(x_1)+\dots+p_n(x_n)f(x_n)}{p_1(x_1)+\dots+p_n(x_n)}\bigg)=g^{(-1)}\bigg(\frac{q_1(x_1)g(x_1)+\dots+q_n(x_n)g(x_n)}{q_1(x_1)+\dots+q_n(x_n)}\bigg), \end{equation} which holds for all $x=(x_1,\dots,x_n)\in I^n$, where $n\geq 2$, $I$ is a nonempty open real interval, the unknown functions $f,g:I\to\mathbb{R}$ are strictly monotone, $f^{(-1)}$ and $g^{(-1)}$ denote their generalized left inverses, respectively, and the vector-valued weight functions $p=(p_1,\dots,p_n):I\to\mathbb{R}_{+}^n$ and $q=(q_1,\dots,q_n):I\to\mathbb{R}_{+}^n$ are also unknown. This equality problem in the symmetric two-variable case (i.e., when $n=2$ and $p_1=p_2$, $q_1=q_2$) was solved under sixth-order regularity assumptions by Losonczi in 1999. The authors of this paper improved this result in 2023 by reaching the same conclusion assuming only first-order differentiability. In the nonsymmetric case, assuming third-order differentiability of $f$, $g$ and the first-order differentiability of at least three of the functions $p_1,\dots,p_n$, Grünwald and Páles proved that \eq{0} holds if and only if there exist four constants $a,b,c,d\in\mathbb{R}$ with $ad\neq bc$ such that $$
cf+d>0,\qquad
g=\frac{af+b}{cf+d},\qquad\mbox{and}\qquad q_\ell=(cf+d)p_\ell\qquad (\ell\in\{1,\dots,n\}). $$ The main goal of this paper is to establish the same conclusion under first-order differentiability. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_16074 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the equality of generalized Bajraktarević means under first-order differentiability assumptions Páles, Zsolt Zakaria, Amr Classical Analysis and ODEs 39B30, 39B40, 26E60 In this paper we consider the equality problem of generalized Bajraktarević means, i.e., we are going to solve the functional equation \begin{equation}\label{E0}\tag{*} f^{(-1)}\bigg(\frac{p_1(x_1)f(x_1)+\dots+p_n(x_n)f(x_n)}{p_1(x_1)+\dots+p_n(x_n)}\bigg)=g^{(-1)}\bigg(\frac{q_1(x_1)g(x_1)+\dots+q_n(x_n)g(x_n)}{q_1(x_1)+\dots+q_n(x_n)}\bigg), \end{equation} which holds for all $x=(x_1,\dots,x_n)\in I^n$, where $n\geq 2$, $I$ is a nonempty open real interval, the unknown functions $f,g:I\to\mathbb{R}$ are strictly monotone, $f^{(-1)}$ and $g^{(-1)}$ denote their generalized left inverses, respectively, and the vector-valued weight functions $p=(p_1,\dots,p_n):I\to\mathbb{R}_{+}^n$ and $q=(q_1,\dots,q_n):I\to\mathbb{R}_{+}^n$ are also unknown. This equality problem in the symmetric two-variable case (i.e., when $n=2$ and $p_1=p_2$, $q_1=q_2$) was solved under sixth-order regularity assumptions by Losonczi in 1999. The authors of this paper improved this result in 2023 by reaching the same conclusion assuming only first-order differentiability. In the nonsymmetric case, assuming third-order differentiability of $f$, $g$ and the first-order differentiability of at least three of the functions $p_1,\dots,p_n$, Grünwald and Páles proved that \eq{0} holds if and only if there exist four constants $a,b,c,d\in\mathbb{R}$ with $ad\neq bc$ such that $$ cf+d>0,\qquad g=\frac{af+b}{cf+d},\qquad\mbox{and}\qquad q_\ell=(cf+d)p_\ell\qquad (\ell\in\{1,\dots,n\}). $$ The main goal of this paper is to establish the same conclusion under first-order differentiability. |
| title | On the equality of generalized Bajraktarević means under first-order differentiability assumptions |
| topic | Classical Analysis and ODEs 39B30, 39B40, 26E60 |
| url | https://arxiv.org/abs/2410.16074 |