In this paper we consider the equality problem of generalized Bajraktarević means, i.e., we are going to solve the functional 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 \(f,g:I\rightarrow \mathbb {R}\) are strictly monotone, \(f^{(-1)}\) and \(g^{(-1)}\) denote their generalized left inverses, respectively, and the vector-valued weight functions \(p=(p_1,\dots ,p_n):I\rightarrow \mathbb {R}_{+}^n\) and \(q=(q_1,\dots ,q_n):I\rightarrow \mathbb {R}_{+}^n\) 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 the 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 (*) holds if and only if there exist four constants \(a,b,c,d\in \mathbb {R}\) with \(ad\ne bc\) such that \(\begin{aligned} cf+d>0,\qquad g=\frac{af+b}{cf+d},\qquad \text{ and }\qquad q_\ell =(cf+d)p_\ell \qquad (\ell \in \{1,\dots ,n\}). \end{aligned}\) The main goal of this paper is to establish the same conclusion under first-order differentiability.
