A sandwich theorem and Hyers-Ulam stability of affine functions
Let \(f\) and \(g\) be real functions defined on a real interval \(I\). It is proved that the following three conditions are equivalent: (i) there exists an affine function \(h : I \to \mathbb{R}\) such that \(f \leq h \leq g\); (ii) there exists a convex function \(h_ 1 : I \to \mathbb{R}\) and a concave function \(h_ 2 : I \to \mathbb{R}\) such that \(f \leq h_ 1 \leq g\) and \(f \leq h_ 2 \leq g\); (iii) \(f(\lambda x + (1 - \lambda) y) \leq \lambda g(x) + (1 - \lambda) g(y)\) and \(g(\lambda x + (1 - \lambda)y) \geq \lambda f(x) + (1 - \lambda) f(y)\) for all \(x,y \in I\) and \({\lambda \in [0,1]}\). As a simple corollary the authors obtain from this a Hyers-Ulam type stability for affine functions.
scientific article; zbMATH DE number 729152