Invariance of the Bajraktarević means with respect to the Beckenbach-Gini means (Q2843875)

Let \(I \subset \mathbb R\) be a real interval and \(f, g\!: I\rightarrow \mathbb R\) be continuous functions such that \(g(x) \not = 0\) on \(I\) and \(f/g\) is one-to-one. The functions \(B^{[f,g]}\!:I^2\rightarrow I,\, B^{[g]}\!:I^2\rightarrow I\), defined by NEWLINE\[NEWLINE B^{[f,g]}(x,y)=\left (\frac {f}{g}\right)^{-1}\left (\frac {f(x)+f(y)}{g(x)+g(y)}\right), \qquad B^{[g]}(x,y)=\frac {xg(x)+yg(y)}{g(x)+g(y)}, NEWLINE\]NEWLINE are called Bajraktarević mean of generators \(f\) and \(g\) and Bechenbach-Gini mean of generator \(g\), respectively.NEWLINENEWLINEIn the paper, the invariance equation is considered, namely the author solves the problem NEWLINE\[NEWLINE B^{[f,g]}\circ \left (B^{[f]},B^{[g]}\right)=B^{[f,g]}, NEWLINE\]NEWLINE assuming that unknown functions \(f,g\) are three times differentiable and \(f/g\) is one-to-one.