## On Functions with the Cauchy Difference Bounded by a Functional

### Volume 52 / 2004

#### Abstract

K. Baron and Z. Kominek [2] have studied the functional inequality
$$ f(x+y) - f(x) - f(y) \geq \phi (x,y), \hskip 1em x, y \in X , $$
under the assumptions that $X$ is a real linear space, $\phi $ is homogeneous with respect to the second variable and $f$ satisfies certain regularity conditions. In particular, they have shown that $\phi $ is bilinear and symmetric and $f$ has a representation of the form $f(x) = {1\over 2}\phi (x,x) + L(x)$ for $x \in X$, where $L$ is a linear function.

The purpose of the present paper is to consider this functional inequality under different assumptions upon $X$, $f$ and $\phi $. In particular we will give conditions which force biadditivity and symmetry of $\phi $ and the representation $f(x) = {1\over 2}\phi (x,x) - A(x)$ for $x \in X$, where $A$ is a subadditive function.