P-means and the Solution of a Functional Equation Involving Cauchy Differences

Solutions to the functional equation are sought for the admissible pairs constituted by a strictly monotonic function f and a strictly increasing in both variables mean . A related class of means, P-means, is introduced, studied and then employed in solving (1) under additional hypotheses on . For instance, Ger has proved that the unique P-mean which is also quasiarithmetic is the geometric mean . An elementary proof to this result is given in this paper. Moreover, as a consequence of a fundamental result on the uniqueness of representation of P-means it is proved that the geometric mean G is the unique homogeneous P-mean.