The paper is devoted to the functional inequality (called by us Hlawka’s functional inequality) $$f(x+y)+f(y+z)+f(x+z)\leq f(x+y+z)+f(x)+f(y)+f(z)$$ for the unknown mapping f defined on an Abelian group, on a linear space or on the real line. The study of the foregoing inequality is motivated by Hlawka’s inequality: $$\|x+y\|+\|y+z\|+\|x+z\|\leq\|x+y+z\|+\|x\|+\|y\|+\|z\|,$$ which in particular holds true for all x, y, z from a real or complex inner product space.

In this paper we prove that if a generalized polynomial function f satisfies the condition f(x) f(y) = 0 for all solutions of the equation x ² + y ² = 1, then f is identically equal to 0.