A sandwich theorem and Hyers-Ulam stability of affine functions
type: Item
instance of: scholarly article
title: A sandwich theorem and Hyers-Ulam stability of affine functions
zbMATH Open document ID: 0815.39010
DOI: 10.1007/BF01827935
published in: Aequationes Mathematicae
publication date: 1995-06-27T00:00:00Z
full work available at URL: https://eudml.org/doc/182478
review text: 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.
reviewed by: Karol Baron
Mathematics Subject Classification ID: 39B72
Mathematics Subject Classification ID: 26A51
Mathematics Subject Classification ID: 52A35
zbMATH DE Number: 729152
zbMATH Keywords: Hyers-Ulam stability
zbMATH Keywords: system of functional inequalities
zbMATH Keywords: sandwich theorem
zbMATH Keywords: affine function
zbMATH Keywords: convex function
zbMATH Keywords: concave function
author: Kazimierz Nikodem
author: Szymon Wąsowicz
MaRDI profile type: Publication
cites work: Q3196722
cites work: Eine abstrakte Version des Satzes von Hahn-Banach
cites work: Q5338702
OpenAlex ID: W2372728626
description: scientific article; zbMATH DE number 729152