On the equality of generalized Bajraktarevi\'c means under first-order differentiability assumptions

Abstract

In this paper we consider the equality problem of generalized Bajraktarevi\'c 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 $p_1,\dots,p_n$, Gr\"unwald and P\'ales 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.