Dependence among cauchy-type functional equations

Abstract

In 1988, Jean Dhombres investigated various kinds of independence among the four forms of the classical Cauchy functional equation as well as solved completely a functional equation, called the universal Cauchy functional equation, which contains all the four forms of the Cauchy functional equation. He took a ring which is divisible by 2 and possesses a unit as the domain of solution functions and a skew-field as their range. In 1999 and 2005, Konrad J. Heuvers and Palanippan Kannappan showed that the three functional equations: f(x+y) - f(x) - f(y) = f(1/x + 1/y), f(x+y) - f(xy)= f(1/x + 1/y) and f(xy) = f(x) + f(y) with f:\R^+ -> R, are equivalent in the sense that a solution of one equation is also a solution of another. The work of Dhombres does not include the case of logarithmic function because his domain of solution functions contains 0, the condition which plays a vital role in his work. Our first objective is to complement the work of Dhombres by re-investigating all his results for solution functions whose domain is the set of positive real numbers and whose range is the complex numbers. Our second objective is to solve an extension of one of the functional equations considered by Heuvers.