Drygas functional inequality on restricted domains

Abstract

Let f be a function from an Abelian normed group $\mathcal{G}$ to an inner product space $\mathcal{E}$ such that f(0) = 0. It is proved that if f satisfies the functional inequality

$$\|2f(x)+f(y)+f(-y)-f(x-y)\| \leqslant \|f(x+y)\|,\quad \|x\|+\|y\| \geqslant d$$

for some $d > 0$, then f is a solution of the Drygas functional equation. The equivalence of the functional inequality $\|f(x)+f(y)\|\leqslant\|f(x+y)\|$ and additive functional equation $f(x+y)=f(x)+f(y)$ has also been investigated under some restricted conditions.