Equality of Cauchy mean values (Q2707231)

If \(I\) is a real interval, \(f\) and \(g\) are real valued \(n-1\) times differentiable functions on \(I\) such that \(g^{(n-1)}\) is non-vanishing, and \(x_1,x_2,\dots ,x_n\in I\), then, according to a theorem by \textit{E. Leach} and \textit{M. Sholander} [J. Math. Anal. Appl. 104, 390-407 (1984; Zbl 0558.26014)], there exists a \(t\) between \(\min(x_1,\dots ,x_n)\) and \(\max(x_1,\dots ,x_n)\) such that NEWLINE\[NEWLINE\frac{[x_1,\dots ,x_n]_f}{[x_1,\dots ,x_n]_g}=\frac{f^{(n-1)}(t)}{g^{(n-1)}(t)},NEWLINE\]NEWLINE where \([x_1,\dots ,x_n]_f\) denotes the divided difference of \(f\) at the points \(x_1,\dots ,x_n\). If the function \(\frac{f^{(n-1)}}{g^{(n-1)}}\) is invertible, this unique \(t\) is denoted by \(D_{f,g}(x_1,x_2,\dots ,x_n)\). Supposing that \(n\geq 3\) is fixed and the real valued functions \(f,g,F,G\) are \(n+2\) times continuously differentiable on \(I\) such that \(g^{(n-1)}\), \(G^{(n-1)}\), \((\frac{f^{(n-1)}}{g^{(n-1)}})'\) and \((\frac{F^{(n-1)}}{G^{(n-1)}})'\) are non-vanishing, the functional equation NEWLINE\[NEWLINED_{f,g}(x_1,x_2,\dots ,x_n)=D_{F,G}(x_1,x_2,\dots ,x_n)\;\;(x_1,x_2,\dots ,x_n\in I)NEWLINE\]NEWLINE holds if, and only if, the linear hull of the pair \(\{f^{(n-1)},g^{(n-1)}\}\) coincides with that of \(\{F^{(n-1)}\), \(G^{(n-1)}\}\).