| Περιγραφή: |
Let \(X\) be a vector space and \(Y\) be a normed space. A function \(f: X\to Y\) is called quasi-Jensen function if there exists an \(x_0\in X\) and \(\varepsilon> 0\) such that the inequality \[ \Biggl|f \Biggl({x+ y\over 2}\Biggr)- {f(x)+ f(y)\over 2}\Biggr|\leq \varepsilon\min\Biggl\{\Biggl|f \Biggl({x+ y\over 2}\Biggr)- f(x_0)\Biggr|, \Biggl|{f(x)+ f(y)\over 2}- f(x_0)\Biggr|\Biggr\} \] for \(x, y\in X\) is satisfied. However, functions satisfying the inequality \[ |f(x+ y)- f(x)- f(y)|\leq \varepsilon\min \{|f(x+ y)|, |f(x)+ f(y)|\}, \] for \(x, y\in X\) are called quasi-additive. The main result of the paper is to show a strong relationship between quasi-Jensen and quasi-additive functions. |