Functional equations involving means

Abstract

In this paper, the functional equation $$ f(px + (1 - p)y) + f((1 - p)x + py) = f(x) + f(y), (x,y \in I) $$ is considered, where 0 < p < 1 is a fixed parameter and f: I → R is an unknown function. The equivalence of this and Jensen’s functional equation is completely characterized in terms of the algebraic properties of the parameter p. As an application, solutions of certain functional equations involving four weighted arithmetic means are also determined.