• 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.

• Journal of the Chungcheong Mathematical Society
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• v.20 no.3
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• pp.269-277
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• 2007

It is shown that $f:\mathbb{R}{\rightarrow}\mathbb{R}$ satisfies the following functional inequalities (0.1) ${\mid}f(x)+f(y){\mid}{\leq}{\mid}f(x+y){\mid}$, (0.2) ${\mid}f(x)+f(y){\mid}{\leq}{\mid}2f(\frac{x+y}{2}){\mid}$, (0.3) ${\mid}f(x)+f(y)-2f(\frac{x-y}{2}){\mid}{\leq}{\mid}2f(\frac{x+y}{2}){\mid}$, respectively, then the function $f:\mathbb{R}{\rightarrow}\mathbb{R}$ satisfies the Cauchy functional equation, the Jensen functional equation and the Jensen quadratic functional equation, respectively.

• Honam Mathematical Journal
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• v.31 no.2
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• pp.219-231
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• 2009

For a mapping satisfying the inequality ${\parallel}{\lambda}f(x)+2{\lambda}f(y)+2f({\lambda}z){\parallel}{\leq}{\parallel}2f({\lambda}({\frac{2}{x}}+y+z)){\parallel}+{\phi}(x,y,z)$, we will study the stability problem of this mapping.

• Bulletin of the Korean Mathematical Society
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• v.44 no.4
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• pp.851-860
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• 2007

In this paper, we prove that a function satisfying the following inequality $${\parallel}f(x)+2f(y)+2f(z){\parallel}{\leq}{\parallel}2f(\frac{x}{2}+y+z){\parallel}+{\epsilon}({\parallel}x{\parallel}^r{\cdot}{\parallel}y{\parallel}^r{\cdot}{\parallel}z{\parallel}^r)$$ for all x, y, z ${\in}$ X and for $\epsilon{\geq}0$, is Cauchy additive. Moreover, we will investigate for the stability in Banach spaces.