Jensen, multi-Jensen and polynomial functions on arbitrary abelian groups

Abstract

In contrast to Cauchy’s functional equation, the consideration of Jensen’s equation makes no sense for functions f : G → H if G, H are arbitrary abelian groups. This difficulty may be circumvented by implementing a third variable into the equation in such a way that there is no interference with the desired result that the solutions should be of the form “additive function plus constant”. Proceeding analogously with the multi-Jensen equation we can add certain possibilities to define polynomial functions on abelian groups to possibilities given earlier by Laczkovich.