K. Baron and Z. Kominek  have studied the functional inequality
f(x+y) - f(x) - f(y) ≥ ϕ (x,y), x, y ∈ X,
under the assumptions that X is a real linear space, ϕ is homogeneous with respect to the second variable and f satisfies certain regularity conditions. In particular, they have shown that ϕ is bilinear and symmetric and f has a representation of the form f(x) = ½ ϕ(x,x) + L(x) for x ∈ X, where L is a linear function.
The purpose of the present paper is to consider this functional inequality under different assumptions upon X, f and ϕ. In particular we will give conditions which force biadditivity and symmetry of ϕ and the representation f(x) = ½ ϕ(x,x) - A(x) for x ∈ X, where A is a subadditive function.