Equality of two variable Cauchy mean values

Summary.

Let I be an interval, $f,g : I \to \mathbb{R}$ be differentiable functions on I. Then, by the Cauchy mean value theorem for every $x,y \in I, x \ne y$ there exists a t between x and y such that $f'(t)[g(x) - g(y)] = g'(t)[f(x) - f(y)]$. If $g' \ne 0$ and $\dfrac{f'}{g'}$ is invertible on I, then t is unique and¶¶ $t = \left(f' \over g'\right)^{-1} \left(f(x) - f(y) \over g(x) - g(y)\right).$¶¶ Continuous extension gives t = x if y = x. The number t is called the Cauchy mean value of x,y and is denoted by $D_{f,g}(x,y).$¶ We solve the equality problem for (two variable) Cauchy means, i.e. we solve the functional equation¶¶ $D_{f_1,g_1}(x,y) = D_{f_2,g_2}(x,y)\qquad(x,y \in I)$¶¶ under the basic assumption that the functions $f_1, g_1, f_2, g_2$ are seven times continuously differentiable. First we reduce the number of unknown functions to 3 by a suitable transformation. Then, using the second, fourth and sixth partial derivatives (the first, third and fifth partial derivatives do not give independent equations) of this transformed equation at equal values of the variables we obtain a system of sixth order ordinary differential equations.¶ We show that the solutions of our system can be obtained from the solutions of a Riccati equation¶¶ $4\bar{h}' - 2\bar{h}^2 = c(G')^{4/3},$¶¶ where c is an arbitrary constant and G is the solution of the nonlinear equation¶¶ $9\frac{G^{1V}}{G'} - 45\frac{G'''}{G'}\,\frac{G''}{G'} + 40\left(\frac{G''}{G'}\right)^3 = 0.$¶¶ There are 33 families of solutions. The principal solution is¶¶ $\begin{array}{l} f'_{2}(x) = \alpha f'_{1}(x) + \beta g'_{1}(x) \\ g'_{2}(x) = \gamma f'_{1}(x) + \delta g'_{1}(x),\end{array}$¶¶ where $\alpha, \beta, \gamma, \delta$ are arbitrary constants with $\alpha\delta - \beta\gamma \neq 0.$ It contains two “arbitrary” functions $f'_1$ and $g'_1$ (and corresponds to the general solution of the same equality problem for $n \geq 3$ variables, see Losonczi [17]).¶ The other 32 families arise only for two variables and they contain only one “arbitrary” function and several arbitrary constants.