The mapping \(\delta_{x,y}\) in normed linear spaces and refinements of the Cauchy-Schwarz inequality
type: Item
instance of: scholarly article
title: The mapping \(\delta_{x,y}\) in normed linear spaces and refinements of the Cauchy-Schwarz inequality
zbMATH Open document ID: 0958.46009
DOI: 10.1016/S0362-546X(98)00274-0
author: J. J. Koliha
author: Silvestru Sever Dragomir
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2000-07-09T00:00:00Z
review text: Let \((X,\|\cdot\|)\) be a real normed linear space. The authors introduce the lower and upper semi-inner product by \[ (y,x)_i= \lim_{t\to 0^+}{\|x+ ty\|^2-\|x\|^2\over 2t} \quad\text{and}\quad (y,x)_s= \lim_{t\to 0^+}{\|x+ ty\|^2-\|x\|^2\over 2t}, \] respectively. Studying some geometrical properties of the function \(\delta_{x,y}: \mathbb{R}\to\mathbb{R}\) defined by \(\delta_{x,y}(t)= 2\|x+ ty\|-\|x+2ty\|\) for fixed \(x,y\in X\) they obtain certain refinements of the Cauchy-Schwarz inequality and give some applications to \(L^p(\Omega)\) spaces for \(p> 1\) and spaces \(\ell^1(\mathbb{R})\), \(\ell^2(\mathbb{R})\) and \(c_0(\mathbb{R})\).
Mathematics Subject Classification ID: 46C15
zbMATH DE Number: 1472254
zbMATH Keywords: lower and upper semi-inner product
zbMATH Keywords: Cauchy-Schwarz inequality
reviewed by: Ludvík Janoš
MaRDI profile type: Publication
full work available at URL: s0362-546x(98)00274-0
OpenAlex ID: W2058795933
description: scientific article; zbMATH DE number 1472254