Midpoint convex functions majorized by midpoint concave functions
type: Item
instance of: scholarly article
title: Midpoint convex functions majorized by midpoint concave functions
zbMATH Open document ID: 0612.26007
DOI: 10.1007/BF02311298
published in: Aequationes Mathematicae
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/137175
review text: The main result is: if \(f(x)\leq g(x),\) \(f[(x+y)/2]\leq [f(x)+f(y)]/2,\) \(g[(x+y)/2]\geq [g(x)+g(y)]/2\) for all x,y on an open convex subset D of \({\mathbb{R}}^ n\), then there exist functions a,h,k additive, convex or concave, respectively, such that \(f(x)=a(x)+h(x),g(x)=a(x)+k(x)\) on D. If \(D={\mathbb{R}}^ n\) then h and k are constant. - There is also a somewhat similar statement on set-valued midpoint-convex functions (without majorant condition), and generalizations to arbitrary vector spaces are mentioned.
Mathematics Subject Classification ID: 26A51
Mathematics Subject Classification ID: 39B99
Mathematics Subject Classification ID: 26B25
Mathematics Subject Classification ID: 28B20
zbMATH DE Number: 3989677
zbMATH Keywords: midpoint concave function
zbMATH Keywords: Cauchy functional equation
zbMATH Keywords: majorization
zbMATH Keywords: theorem of Bernstein and Doetsch
zbMATH Keywords: set-valued midpoint-convex functions
author: Kazimierz Nikodem
MaRDI profile type: Publication
cites work: Q5514010
cites work: Q4164349
OpenAlex ID: W4234767451
description: scientific article; zbMATH DE number 3989677