Jordan-von Neumann type functional inequalities

Abstract

It is shown that $f: \mathbb R \rightarrow \mathbb R$ satisfies the following functional inequalities \begin{eqnarray} |f(x)+f(y)| & \le & | f(x+y)| , \\ |f(x)+f(y)| & \le & |2f(\frac{x+y}{2})| , \\ |f(x)+f(y)-2f(\frac{x-y}{2})| & \le & |2f(\frac{x+y}{2})| , \end{eqnarray} 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.