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On inequalities associated with the Jordan-von Neumann functional equation

Rätz, Jürg

In: Aequationes mathematicae, 2003, vol. 66, no. 1-2, p. 191-200

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    Title
    • On inequalities associated with the Jordan-von Neumann functional equation
    Author
    • Rätz, Jürg. Mathematisches Institut, Universität Bern, Sidlerstrasse 5, 3012, Bern, Switzerland
    Document Type
    • Postprint
    Language
    • English
    Published in
    • Aequationes mathematicae, 2003, vol. 66, no. 1-2, p. 191-200. Springer International Publishing; www.birkhauser.ch
    Other Version
    OAI-PMH Identifier
    • oai:doc.rero.ch:316380
    Summary
    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 $$ \def\lo{\longrightarrow} f:G\lo E,\; 2f(x)+2f(y)=f(xy)+f(xy^{-1})\qquad(\forall x,y\in G).\qquad {\rm (JvN)} $$