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.