On functions with the Cauchy difference bounded by a functional

K. Baron and Z. Kominek [2] have studied the functional inequality f(x + y) - f(x) - f(y) is less than or equal [phi](x, y), x, y is an element of X, under the assumptions that X is a real linear space, (phi] is homogeneous with respect to the second variable and f satisfies certain regularity conditions. In particular, they have shown that [phi] is bilinear and symmetric and f has a representation of the form f(x) =1/2[phi](x,x) + L[x) for x is an element of 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 [phi). In particular we will give conditions which force biadditivity and symmetry of (pchi] and the representation f(x) =1/2[phi](x, x) - A(x) for x is an element of X, where A is a subadditive function.