On inequalities associated with the Jordan-von Neumann functional equation. (English) Zbl 1078.39026

Conditions in which the functional inequality \[ f: G \to E,\quad \| 2f(x) + 2f(y) - 2f(xy^{-1})\| \leq \| 2f(xy)\|\;\forall \;x,y \in G \] would imply the Jordan-von Neumann parallelogram equation \[ f: G \to E,\quad 2f(x) + 2f(y) = f(xy) + f(xy^{-1})\;\forall\;x,y \in G \] are established, where \((G,.)\) is a group and \((E,<.,.>)\) is a real or complex inner product space. In addition, the paper further improved the results of A. Gilányi [Aequationes Math. 62, 303–309 (2001; Zbl 0992.39026)] by dropping the 2-divisibility assumption on \(G\) and also replacing the commutativity assumption imposed on \(G\) by \[ f(xyz) = f(xzy)\quad \forall x,y,z\in G. \] Finally, applications of the results obtained are also given.


39B62 Functional inequalities, including subadditivity, convexity, etc.


Zbl 0992.39026
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