On the comparison of Cauchy mean values

Suppose that and exist, with , on . Then there is (moreover if ) such that where denotes the divided difference of at the points . This is the Cauchy Mean Value Theorem for divided differences (see e.g. [4]). If the function is invertible then is a mean value of . It is called the Cauchy mean of the numbers and will be denoted by ). Here we completely solve the comparison problem of Cauchy means in the special cases and . In the general case we find necessary conditions (which are not sufficient) and also sufficient conditions (which are not necessary).