On inequalities associated with the Jordan-von Neumann functional equation

Summary.

For a group $(G, \cdot)$ and a real or complex inner product space $(E, \langle\cdot, \cdot\rangle)$ with norm $\def\lr{[\!]} \lr.\lr$ we consider the functional inequality

$$ \def\lr{[\!]} \def\lo{\longrightarrow} f:G\lo E,\;\lr 2f(x) + 2f(y)-f(xy^{-1})\lr\le\lr f(xy)\lr\qquad(\forall x,y\in G)\qquad {\rm (I)} $$

and describe situations in which (I) implies the Jordan-von Neumann parallelogram equation

$$ \def\lo{\longrightarrow} f:G\lo E,\; 2f(x)+2f(y)=f(xy)+f(xy^{-1})\qquad(\forall x,y\in G).\qquad {\rm (JvN)} $$

.