On functional inequalities associated with Drygas functional equation

Abstract

In the paper, the equivalence of the functional inequality $$\Vert 2f(x)+f(y)+f(-y)-f(x-y)\Vert \le \Vert f(x+y)\Vert (x,y\in G)$$ and the Drygas functional equation $$f(x+y)+f(x-y)=2f(x)+f(y)+f(-y)(x,y\in G)$$ is proved for functions $f:G\to E$ where $(G,+)$ is an abelian group, $(E,<\cdot ,\cdot >)$ is an inner product space, and the norm is derived from the inner product in the usual way.

5 pages