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
$$ \def\lo{\longrightarrow} f:G\lo E,\; 2f(x)+2f(y)=f(xy)+f(xy^{-1})\qquad(\forall x,y\in G).\qquad {\rm (JvN)} $$
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Manuscript received: January 18, 2002 and, in final form, August 20, 2002
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Rätz, J. On inequalities associated with the Jordan-von Neumann functional equation . Aequat. Math. 66, 191–200 (2003). https://doi.org/10.1007/s00010-003-2684-8
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DOI: https://doi.org/10.1007/s00010-003-2684-8