On a functional inequality connected with quadratic functionals
type: Item
instance of: scholarly article
title: On a functional inequality connected with quadratic functionals
zbMATH Open document ID: 1118.39011
author: Włodzimierz Fechner
published in: Journal of Mathematical Analysis and Applications
publication date: 2007-05-23T00:00:00Z
review text: The author finds functions \(f: X \to \mathbb R\), \(\phi: X \times X \to \mathbb R\) fulfilling the conditions \[ \begin{alignedat}{2} \frac{f(2x + 2y)}{4} - f(x) - f(y) &\geq \phi(x, y),&\quad &x,y \in X,\\ \phi(x, -y) &\geq -\phi(x,y),&\quad &x,y \in X,\\ \limsup_{n \to \infty} \frac{1}{4^{n}} \phi(2^{n}x, 2^{n}x) &< + \infty,&\quad &x \in X,\\ \liminf_{n \to \infty} \frac{1}{4^{n}} \phi(2^{n}x, 2^{n}y) &\geq \phi(x,y),&\quad &x,y \in X, \end{alignedat} \] where \((X,+)\) is an abelian group. The same is done for a real inner product space \(X\) with \( \dim X \geq 3 \) and the conditions \[ \begin{alignedat}{2} 2 \frac{f(2x + 2y)}{4} - f(x) - f(y) &\geq \phi(x, y),&\quad &x,y \in X,\\ \phi(x, -x) &\geq -\phi(x,x),&\quad &x \in X,\\ \phi(2x,2x) &\leq 4\phi(x,x),&\quad &x \in X,\\ \forall_{x \in X} \exists_{n_{0} \in \mathbb N} \forall_{n \geq n_{0}} f\biggl(\frac{x}{2^{n}}\biggr) &\geq 0, &&{}\\ \phi(x,y) &\geq 0, &\quad &x \perp y, x, y \in X . \end{alignedat} \]
Mathematics Subject Classification ID: 39B62
Mathematics Subject Classification ID: 39B52
zbMATH DE Number: 5156431
zbMATH Keywords: functional inequalities
zbMATH Keywords: quadratic and biadditive functionals
zbMATH Keywords: abelian group
zbMATH Keywords: inner product space
reviewed by: Andrzej Smajdor
MaRDI profile type: Publication
full work available at URL: j.jmaa.2006.10.033
OpenAlex ID: W1997846369
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: Characterization of quadratic mappings through a functional inequality
cites work: Q3218572
cites work: On orthogonally additive mappings
DOI: 10.1016/J.JMAA.2006.10.033
description: scientific article; zbMATH DE number 5156431