Cauchy--Schwarz-type inequalities for additive functions

The main goal of this paper is to show that if a real valued function defined on a groupoid satisfies a certain Levi–Civita-type functional equation, then it also fulfills a Cauchy–Schwarz-type functional inequality. In particular, if the groupoid is the multiplicative structure of commutative ring, then we can establish the existence of nontrivial additive functions satisfying inequalities connected to the multiplicative structure.