A universal Cauchy functional equation over the positive reals (Q615946)

The authors study the so-called \textit{universal Cauchy functional equation} \[ af(xy)+bf(x)f(y)+cf(x+y)+d(f(x)+f(y))=0, \] where \(f\) is the unknown function mapping from the positive real numbers to complex numbers, and \(a,b,c,d\) are constants. This equation was studied by Dhombres before, but the domain of \(f\) contains zero. The methods in this paper are also different from those used by Dhombres, and based on solving some difference equations.