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An inequality, equivalent to the parallelogram equation. (Eine zur Parallelogrammgleichung äquivalente Ungleichung.) (German) Zbl 0992.39026

The following result is proved: Let \(f:G\to E\) satisfy the functional inequality \[ \bigl\|2f(x)+2f(y)- f(xy^{-1})\bigr \|\leq \bigl\|f(xy)\bigr\|, \quad\text{for }x,y\in G, \] where \(G\) is a group and \(E\) is an inner product space. If either \(G\) is a 2-divisible Abelian group or \(G\) is 2-divisible and \(f\) also satisfies \(f(xyz)= f(xzy)\), for \(x,y,z\in G\), then \(f\) satisfies the parallelogram identity or quadratic equation \[ f(xy)+ f(xy^{-1})= 2f(x)+2f(y),\quad \text{for }x,y\in G. \]

MSC:

39B62 Functional inequalities, including subadditivity, convexity, etc.
39B52 Functional equations for functions with more general domains and/or ranges
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