On the stability of Wright-convex functions
type: Item
instance of: scholarly article
title: On the stability of Wright-convex functions
zbMATH Open document ID: 1023.39025
author: Jacek Mrowiec
published in: Aequationes Mathematicae
publication date: 2003-07-03T00:00:00Z
review text: Let \(D\) be a convex subset of a linear space. A function \(f:D\to{\mathbb R}\) is called Wright-convex if \[ f\bigl(tx+(1-t)y\bigr)+f\bigl(ty+(1-t)x\bigr)\leq f(x)+f(y) \] for all \(x,y\in D\), \(t\in[0,1]\). The main result of the paper is the following Theorem: Let \(D\subset{\mathbb R}^N\) be a nonempty open and convex set and let \(\delta\geq 0\). Assume that a function \(f:D\to{\mathbb R}\) fulfils the inequality \[ f\bigl(tx+(1-t)y\bigr)+f\bigl(ty+(1-t)x\bigr)\leq f(x)+f(y)+2\delta \] for all \(x,y\in D\), \(t\in[0,1]\). Then there exists a Wright-convex function \(\varphi:D\to{\mathbb R}\) such that \[ \bigl|f(x)-\varphi(x)\bigr|\leq 2C_N\delta \] for all \(x\in D\), where \(C_N\) is a constant depending only on \(N\).
Mathematics Subject Classification ID: 39B82
Mathematics Subject Classification ID: 26A51
Mathematics Subject Classification ID: 39B62
zbMATH DE Number: 1945918
zbMATH Keywords: Wright-convex functions
zbMATH Keywords: approximately convex functions
zbMATH Keywords: approximately Wright-convex functions
zbMATH Keywords: stability
reviewed by: Szymon Wąsowicz
MaRDI profile type: Publication
full work available at URL: s000100300010
OpenAlex ID: W1603411212
DOI: 10.1007/S000100300010
description: scientific article; zbMATH DE number 1945918