On Functions with the Cauchy Difference Bounded by a Functional

Włodzimierz Fechner¹

¹ Institute of Mathematics Silesian University Bankowa 14 40-007 Katowice, Poland

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 52 (2004) no. 3, pp. 265-271

Résumé

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.

Détail

DOI : 10.4064/ba52-3-6

Keywords: baron kominek have studied functional inequality geq phi hskip under assumptions real linear space phi homogeneous respect second variable satisfies certain regularity conditions particular have shown phi bilinear symmetric has representation form phi where linear function purpose present paper consider functional inequality under different assumptions phi particular conditions which force biadditivity symmetry phi representation phi where subadditive function