• Communications of the Korean Mathematical Society
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• v.23 no.3
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• pp.371-376
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• 2008

In this paper, it is shown that if f satisfies the following functional inequality (0.1) $${\parallel}\sum\limits_{i,j=1}^3\;f{(xi,yj)}{\parallel}{\leq}{\parallel}f(x_1+x_2+x_3,\;y_1+y_2+y_3){\parallel}$$ then f is a bi-additive mapping. We moreover prove that if f satisfies the following functional inequality (0.2) $${\parallel}2\sum\limits_{j=1}^3\;f{(x_j,\;z)}+2\sum\limits_{j=1}^3\;f{(x_j,\;w)-f(\sum\limits_{j=1}^3\;xj,\;z-w)}{\parallel}{\leq}f(\sum\limits_{j=1}^3\;xj,\;z+w){\parallel}$$ then f is an additive-quadratic mapping.