Communications of the
Korean Mathematical Society CKMS

In this paper, it is shown that if $f$ satisfies the following functional inequality \begin{equation} \| \sum_{i,j=1}^3 f(x_i, y_j)\| \le \| f(x_1 + x_2 + x_3, y_1 + y_2 + y_3) \| \end{equation} then $f$ is a bi-additive mapping. We moreover prove that if $f$ satisfies the following functional inequality \begin{equation} \|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) \| \end{equation} 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