On stability for the Jensen equation on intervals
The paper discusses the stability properties of the Jensen equation \[ f\left (\frac{x+y}{2}\right)=\frac{f(x)+f(y)}{2}, \qquad x,y\in D \tag{1} \] for real valued functions. These functions are assumed to be defined on the domain \(D=I\cap (G+\gamma)\), where \(G\) is a \(2\)-divisible subgroup of \(\mathbb R\), \(I\subset\mathbb R\) is an interval and \(\gamma\in I\). A function \(f\) is called a Jensen function if \(f\) satisfies (1) and it is called an \(\varepsilon\) -Jensen function (\(\varepsilon\geq 0\)) if \[ \left |\frac{f(x)+f(y)}{2}-f\left (\frac{x+y}{2}\right)\right |\leq\varepsilon ,\qquad x,y,\in D . \] As the main stability result it is proved that if \(f:D\to\mathbb R\) is an \(\varepsilon\)-Jensen function, then there exists a Jensen function \(g:D\to\mathbb R\) such that \[ |f(x)-g(x)|\leq\varepsilon\qquad\text{ for every }x\in D . \] This result yields the best constant (\(K=1\)) in the problem of the stability of (1) in the Hyers-Ulam sense for real valued functions defined on an arbitrary real interval.
scientific article; zbMATH DE number 1556604