Some inequalities connected with a quadratic functional equation

Let (X, +) be an Abelian group. One can show that a mapping f: X R satisfying the inequality f(x + y) + f(x-y)≤2f(x)+2f(y) (1) for all x, y ∈ X also satisfies the inequalities f(2x + y)≤4f(:c) + f (y) + f (x + y) - f (x - y) and f(2x+y) + f(2x -y)≤ 8f(x) + 2f(y) for all x, y ∈ X. A question of finding sufficient conditions under which the inequalities (1), (2) and (3) are equivalent will be considered. In this note, some properties of the solution of (1) will be proved. We also consider another definition of a subquadratic function given in [1].