László Losonczi

Abstract

Suppose that x1≤⋯≤xn and f(n−1),g(n−1) exist, with g(n−1)≠0, on [x1,xn]. Then there is t∈[x1,xn] (moreover t∈[x1,xn] if x1<xn) such that [x1,⋯,xn]f[x1,⋯,xn]g=f(n−1)(t)g(n−1)(t) where [x1,⋯,xn]f denotes the divided difference of f at the points x1,⋯,xn. This is the Cauchy Mean Value Theorem for divided differences (see e.g. [4]).

If the function f(n−1)/g(n−1) is invertible then t=(f(n−1)g(n−1))−1([x1,…,xn]f[x1,…,xn]g) is a mean value of x1,⋯,xn. It is called the Cauchy mean of the numbersx1,⋯,xn and will be denoted by Dfg(x1,…,xn)).

Here we completely solve the comparison problem of Cauchy means Dfg(x1,x2,…,xn)≤DFG(x1,x2,…,xn)(x1,x2,…,xn∈I,n≥2 is fixed) in the special cases g=G,f=F and f(n−1)/g(n−1)=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).