Characterization of quadratic mappings through a functional inequality
type: Item
instance of: scholarly article
title: Characterization of quadratic mappings through a functional inequality
zbMATH Open document ID: 1106.39024
author: Włodzimierz Fechner
published in: Journal of Mathematical Analysis and Applications
publication date: 2006-12-07T00:00:00Z
review text: Motivated by some results of \textit{K. Baron} and \textit{Z. Kominek} [Bull. Pol. Acad. Sci., Math. 51, No. 3, 301--307 (2003; Zbl 1046.39021)] the author studies the functional inequality \[ f(x+y)-\frac{f(2x)+f(2y)}{4} \geq \phi(x,y) \tag{I} \] where \(f:X \to \mathbb R\), \(\phi:X \times X \to \mathbb R\), \(X\) is either an abelian group or a real inner product space. Among others, the following two theorems are proved. Theorem 1. Assume that \((X,+)\) is a uniquely \(2\)-divisible abelian group, and \(f\) and \(\phi\) satisfy (I). If \(f(0)\leq 0\) and \[ \begin{aligned} \phi(x,-y) &\geq -\phi(x,y),\\ \limsup_{k \to +\infty} 4^k \phi(2^{-k}x,2^{-k}x)&< +\infty,\\ \limsup_{k \to +\infty} 4^k \phi(2^{-k}x,2^{-k}y) &\geq \phi(x,y), \end{aligned} \] then \(f(x)=\frac{1}{2}\phi(x,x)\), \(\phi\) is biadditive and symmetric. Theorem 2. Assume that \(X\) is a real inner product space with \(\dim X \geq 3\), and \(f\) and \(\phi\) satisfy (I). If \(f(0)\leq 0\) and \[ \begin{aligned} &\phi(2x,2x) \geq 4\phi(x,x),\\ &\phi(x,-x) \geq -\phi(x,x),\\ &\phi(x,y) \geq 0 \quad \text{if} \quad x \perp y, \end{aligned} \] then there exists a sublinear map \(P:\mathbb R_+ \to \mathbb R\) such that \(f(x)=\frac{1}{2}\phi(x,x)=-P(\| x \|^2)\).
Mathematics Subject Classification ID: 39B62
Mathematics Subject Classification ID: 39B52
zbMATH DE Number: 5078074
zbMATH Keywords: functional inequality
zbMATH Keywords: quadratic and biadditive functionals
zbMATH Keywords: abelian group
zbMATH Keywords: inner product space
reviewed by: Gian Luigi Forti
MaRDI profile type: Publication
full work available at URL: j.jmaa.2005.12.024
OpenAlex ID: W1996314935
cites work: Q4459856
cites work: On Functions with the Cauchy Difference Bounded by a Functional
cites work: On functions with the Cauchy difference bounded by a functional. II
cites work: \(\Phi\)-convex functions defined on metric spaces
DOI: 10.1016/J.JMAA.2005.12.024
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description: scientific article; zbMATH DE number 5078074