On Functions with the Cauchy Difference Bounded by a Functional

Volume 52 / 2004

Włodzimierz Fechner Bulletin Polish Acad. Sci. Math. 52 (2004), 265-271 MSC: Primary 39B62, 39B72. DOI: 10.4064/ba52-3-6


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.


  • Włodzimierz FechnerInstitute of Mathematics
    Silesian University
    Bankowa 14
    40-007 Katowice, Poland

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