JORDAN-VON NEUMANN TYPE FUNCTIONAL INEQUALITIES

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.