Functional inequalities associated with additive, quadratic and Drygas functional equations

Abstract

Let $\mathcal{G}$ be an abelian group, $\mathcal{A}$ a $C^*$-algebra and $\mathcal{M}$ a pre-Hilbert $\mathcal{A}$-module with an $\mathcal{A}$-valued inner product $\langle.,.\rangle$. We show if a function $f \colon \mathcal{G}\rightarrow \mathcal{M}$ satisfies the inequality

$$ \langle f(x)+f(y),f(x)+f(y)\rangle \leq \langle f(x+y),f(x+y)\rangle,\quad x,y\in \mathcal{G}, $$

then $f$ is additive. We also prove that for functions $f \colon \mathcal{G}\rightarrow \mathcal{M}$, the inequality

$$\begin{aligned} & \langle 2f(x)+2f(y)-f(x-y),2f(x)+2f(y)-f(x-y)\rangle \\ &\quad\quad\quad\quad \leq \langle f(x+y),f(x+y)\rangle,\quad x,y\in \mathcal{G}, \end{aligned}$$

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.