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 On functional inequalities associated with Jordan--von Neumann type functional equations Commun. Korean Math. Soc. 2008 Vol. 23, No. 3, 371-376 Printed September 1, 2008 Jong Su An Pusan National University Abstract : In this paper, it is shown that if $f$ satisfies the following functional inequality $$\| \sum_{i,j=1}^3 f(x_i, y_j)\| \le \| f(x_1 + x_2 + x_3, y_1 + y_2 + y_3) \|$$ then $f$ is a bi-additive mapping. We moreover prove that if $f$ satisfies the following functional inequality $$\|2 \sum_{j=1}^3 f(x_j, z)+ 2\sum_{j=1}^3 f(x_j, w) - f(\sum_{j=1}^3 x_j, z-w) \| \le \| f(\sum_{j=1}^3 x_j, z+w) \|$$ then $f$ is an additive-quadratic mapping. Keywords : Jordan--von Neumann type bi-additive functional equation, Jordan--von Neumann type additive-quadratic functional equation, Hyers--Ulam--Rassias stability, functional inequality MSC numbers : Primary 39B62, 39B82, 46B03 Downloads: Full-text PDF

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