ON AN ALTERNATIVE FUNCTIONAL EQUATION RELATED TO THE JENSEN’S FUNCTIONAL EQUATION

Abstract

Given an integer λ = 1, we study the alternative Jensen’s functional equation f(xy−1)−2f(x)+f(xy) = 0 or f(xy−1)−2f(x)+λf(xy)=0 , where f is a mapping from a group (G,·) to a uniquely divisible abelian group (H,+). We prove that for λ = −3, the above functional equation is equivalent to the classical Jensen’s functional equation. Furthermore, if G is a 2-divisible group, then we can strengthen the results by the showing that the equivalence is valid for all integers λ = 1.