Comparison of means generated by two functions and a measure
type: Item
instance of: scholarly article
title: Comparison of means generated by two functions and a measure
zbMATH Open document ID: 1148.26036
author: László Losonczi
author: Zsolt Páles
published in: Journal of Mathematical Analysis and Applications
publication date: 2008-06-24T00:00:00Z
review text: Let \(I\subset{\mathbb R}\) be an open interval, \(f,g:I\to{\mathbb R}\) be continuous, \(g\)~positive and \(f/g\) strictly monotone. Let \(\mu\) be a~probability Borel measure on \([0,1]\). Then \[ M_{f,g;\mu}(x,y):=\biggl(\frac{f}{g}\biggr)^{-1} \left( \frac{\int_0^1f\bigl(tx+(1-t)y\bigr)\,d\mu(t)} {\int_0^1g\bigl(tx+(1-t)y\bigr)\,d\mu(t)} \right),\qquad x,y\in I \] defines a~class of means containing many well known means: Bajraktarević, quasiarithmetic, Cauchy, Lagrangian, Gini, Stolarsky etc. The comparison problem is studied for these means. Conditions for \((f,g)\), \((h,k)\) and measures \(\mu\), \(\nu\) are found that the inequality \(M_{f,g;\mu}(x,y)\leq M_{h,k;\nu}(x,y)\) (\(x,y\in I\)) holds. Necessary conditions and sufficient conditions are studied.
reviewed by: Szymon Wąsowicz
Mathematics Subject Classification ID: 26E60
Mathematics Subject Classification ID: 26D15
zbMATH DE Number: 5292294
zbMATH Keywords: generalized integral means
zbMATH Keywords: generalized Gini means
zbMATH Keywords: comparison problem
MaRDI profile type: Publication
full work available at URL: j.jmaa.2008.04.004
OpenAlex ID: W1963949165
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DOI: 10.1016/J.JMAA.2008.04.004
description: scientific article; zbMATH DE number 5292294