Functional inequalities associated with additive, quadratic and Drygas functional equations

Let G be an abelian group, A a C-& lowast;-algebra and M a pre-Hilbert A-module with an A-valued inner product (sic)., .(sic). We show if a function f : G -> M satisfies the inequality(sic)f(x) + f(y), f(x) + f(y)) <= (sic)f(x + y), f(x + y)(sic), x, y is an element of G, then f is additive. We also prove that for functions f : G -> M, the inequality (sic)2f(x) + 2f(y) - f(x - y), 2f(x) + 2f(y) - f(x - y)) <= (sic)f(x + y), f(x + y)(sic), x, y is an element of G,implies f is quadratic. These results enable us to prove the equivalence of a functional inequality and the Drygas functional equation. In addition, we investigate the stability problem associated with these functional inequalities. Finally, we give some examples of quadratic and Drygas functions.