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

TL;DR

This paper investigates when two generalized Bajraktarević means are equal, solving a functional equation with minimal differentiability assumptions, and extends previous results to broader conditions.

Contribution

The authors prove that the equality of generalized Bajraktarević means holds under only first-order differentiability, improving upon earlier higher-order regularity requirements.

Findings

The equality condition involves specific linear fractional transformations of the functions.

The result generalizes previous work by reducing differentiability assumptions.

Explicit formulas relate the functions f, g, p, and q under the equality condition.

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.