Bounds for Hardy type differences
type: Item
instance of: scholarly article
title: Bounds for Hardy type differences
zbMATH Open document ID: 1225.26040
author: Neven Elezović
author: K. Krulić-Himmelreich
author: J. Pečarić
publication date: 2011-06-24T00:00:00Z
review text: The authors revisit the classical Hardy inequality and the Pólya-Knopp inequality. The paper contains necessary basic facts about convex, log-convex as well as exponentially convex functions. Let \(A_{k}\) be an integral operator defined by \[ A_k f\left( x \right) = \frac{1}{K\left( x \right)}\int_{\Omega _2} {k \left( {x,y} \right) f\left( y \right) d \mu_{2} \left( y \right),} \] where \(k: \Omega_{1} \times \Omega_{2} \rightarrow \mathbb R\) is a general nonnegative kernel, \((\Omega_{1}, \Sigma_{1}, \mu_{1}), (\Omega_{2}, \Sigma_{2}, \mu_{2})\) are measure spaces with \(\sigma \)-finite measures and \[ K\left( x \right) = \int_{\Omega _2 } {k\left( {x,y} \right)d\mu _2 \left( y \right),} \quad x \in \Omega_1. \] Guided by the above result, improvements and reversals of new weighted Hardy-type inequalities with integral operators are stated and proved. A new Cauchy-type mean is introduced and a monotonicity property of this mean is proved. The paper is useful for researchers in the field of this type of inequalities.
reviewed by: V. Lokesha
Mathematics Subject Classification ID: 26D15
Mathematics Subject Classification ID: 26B25
zbMATH DE Number: 5912885
zbMATH Keywords: inequalities
zbMATH Keywords: Cauchy inequality
zbMATH Keywords: Hilbert inequality
zbMATH Keywords: Hardy-type inequalities
MaRDI profile type: Publication
full work available at URL: s10114-011-9707-5
OpenAlex ID: W2095278186
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DOI: 10.1007/S10114-011-9707-5
published in: Acta Mathematica Sinica, English Series
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description: scientific article; zbMATH DE number 5912885