Equality of two variable Cauchy mean values (Q1396527)

Given two differentiable real functions \(f,g:I\to \mathbb{R}\) such that \(g'\neq 0\) and \(f'/g'\) is invertible on \(I\), the two variable Cauchy mean \(D_{f,g}\) of two elements \(x\neq y\) of the interval \(I\) is defined as \[ D_{f,g}(x,y):=\left({f'\over g'}\right)^{-1} \left({f(x)-f(y)\over g(x)-g(y)}\right). \] The main result of this paper completely solves the \textit{equality problem} of Cauchy means, i.e., the functional equation \[ D_{f_1,g_1}(x,y)=D_{f_2,g_2}(x,y) \qquad (x,y\in I) \] for seven times continuously differentiable unknown functions \(f_1,g_1:I\to\mathbb{R}\) and \(f_2,g_2:I\to \mathbb{R}\). There are 33 families of solutions. One family, called principal solution, allows two arbitrary functions and four arbitrary constants, while the other 32 families involve only one arbitrary function and several arbitrary constants.