On the Comparison of Cauchy Mean Values

László Losonczi

Suppose that $x_1\le\dots\le x_n$ and $f^{(n-1)}, g^{(n-1)}$ exist, with $g^{(n-1)}\ne 0,$ on $[x_1,x_n]$. Then there is a $t\in [x_1,x_n]$ (moreover $t\in (x_1,x_n)$ if $x_1<x_n$) such that

$$\displaystyle{\frac{[x_1,\dots,x_n]_f}{[x_1,\dots,x_n]_g}}=\displaystyle{\frac{f^{(n-1)}(t)}{g^{(n-1)}(t) }}$$

where $[x_1,\dots,x_n]_f$ denotes the divided difference of $f$ at the points $x_1,\dots,x_n$. This is the Cauchy Mean Value Theorem for divided differences (see e.g. E. Leach and M. Sholander, Multi-variable extended mean values, J. Math. Anal. Appl., 104 1984, 390-407).

If the function $${\frac{f^{(n-1)}}{g^{(n-1)}}}$$ is invertible then

$$t=\( \displaystyle{\frac{f^{(n-1)}}{g^{(n-1)}}}\)^{-1} \( \displaystyle{\frac{[x_1,\dots,x_n]_f}{[x_1,\dots,x_n]_g}}\)$$

is a mean value of $x_1,\dots,x_n$. It is called the Cauchy mean of the numbers $x_1,\dots,x_n$ and will be denoted by $D_{fg}(x_1,\dots,x_n)$ .

Here we completely solve the comparison problem of Cauchy means

$$D_{fg}(x_1,x_2,\dots,x_n)\le D_{FG}(x_1,x_2,\dots,x_n)$$

where $x_1,x_2,\dots,x_n\in I, n\ge 2$ is fixed, in the special cases $g=G$, $f=F$ and $${\frac{f^{(n-1)}}{g^{(n-1)}}}=\displaystyle{\frac{F^{(n-1)}}{G^{(n-1)}}}$$. In the general case we find necessary conditions (which are not sufficient) and also sufficient conditions (which are not necessary).