P-means and the solution of a functional equation involving Cauchy differences

Abstract

Solutions to the functional equation f(x+y)−f(x)−f(y)=2f(Φ(x,y)),x,y>0, are sought for the admissible pairs (f,Φ)(f,Φ) constituted by a strictly monotonic function f and a strictly increasing in both variables mean ΦΦ . A related class of means, P-means, is introduced, studied and then employed in solving (1) under additional hypotheses on ΦΦ . For instance, Ger has proved that the unique P-mean which is also quasiarithmetic is the geometric mean G(x,y)=xy−−√G(x,y)=xy . 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.