On the equality problem of generalized Bajraktarević means

Abstract

The purpose of this paper is to investigate the equality problem of generalized Bajraktarević means, i.e., 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 $n\ge 2$, $I$ 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 $p=(p_1,\dots ,p_n):I\to {\mathbb{R}}_{+}^n$ and $q=(q_1,\dots ,q_n):I\to {\mathbb{R}}_{+}^n$ are also unknown functions. This equality problem in the symmetric two-variable (i.e., when $n=2$) case was already investigated and solved under sixth-order regularity assumptions by Losonczi in 1999. In the nonsymmetric two-variable case, assuming three times differentiability of $f$, $g$ and the existence of $i\in \{1,2\}$ such that either $p_i$ is twice continuously differentiable and $p_{3-i}$ is continuous on $I$, or $p_i$ is twice differentiable and $p_{3-i}$ is once differentiable on $I$, we prove that \eqref{E0} holds if and only if there exist four constants $a,b,c,d\in \mathbb{R}$ with $ad\ne bc$ such that \begin{equation*} 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\}). \end{equation*} In the case $n\ge 3$, we obtain the same conclusion with weaker regularity assumptions. Namely, we suppose that $f$ and $g$ are three times differentiable, $p$ is continuous and there exist $i,j,k\in \{1,\dots ,n\}$ with $i\ne j\ne k\ne i$ such that $p_i,p_j,p_k$ are differentiable.