Functional inequalities associated with additive, quadratic and Drygas functional equations.

Let G be an abelian group, A a C ∗ -algebra and M a pre-Hilbert A -module with an A -valued inner product ⟨. ,. ⟩ . We show if a function f : G → M satisfies the inequality ⟨ f (x) + f (y) , f (x) + f (y) ⟩ ≤ ⟨ f (x + y) , f (x + y) ⟩ , x , y ∈ G , then f is additive. We also prove that for functions f : G → M , the inequality ⟨ 2 f (x) + 2 f (y) - f (x - y) , 2 f (x) + 2 f (y) - f (x - y) ⟩ ≤ ⟨ f (x + y) , f (x + y) ⟩ , x , y ∈ 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. [ABSTRACT FROM AUTHOR]