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

Abstract

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