{"title":"关于加权拟对称均值的Hardy型不等式","authors":"Zsolt P'ales, P. Pasteczka","doi":"10.7153/mia-2020-23-75","DOIUrl":null,"url":null,"abstract":"Using recent results concerning the homogenization and the Hardy property of weighted means, we establish sharp Hardy constants for concave and monotone weighted quasideviation means and for a few particular subclasses of this broad family. More precisely, for a mean $\\mathscr{D}$ like above and a sequence $(\\lambda_n)$ of positive weights such that $\\lambda_n/(\\lambda_1+\\dots+\\lambda_n)$ is nondecreasing, we determine the smallest number $H \\in (1,+\\infty]$ such that \n$$ \n\\sum_{n=1}^\\infty \n\\lambda_n \\mathscr{D}\\big((x_1,\\dots,x_n),(\\lambda_1,\\dots,\\lambda_n)\\big) \\le H \\cdot \\sum_{n=1}^\\infty \\lambda_n x_n \\text{ for all }x \\in \\ell_1(\\lambda). \n$$ It turns out that $H$ depends only on the limit of the sequence $(\\lambda_n/(\\lambda_1+\\dots+\\lambda_n))$ and the behaviour of the mean $\\mathscr{D}$ near zero.","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":"77 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2019-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"5","resultStr":"{\"title\":\"On Hardy type inequalities for weighted quasideviation means\",\"authors\":\"Zsolt P'ales, P. Pasteczka\",\"doi\":\"10.7153/mia-2020-23-75\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Using recent results concerning the homogenization and the Hardy property of weighted means, we establish sharp Hardy constants for concave and monotone weighted quasideviation means and for a few particular subclasses of this broad family. More precisely, for a mean $\\\\mathscr{D}$ like above and a sequence $(\\\\lambda_n)$ of positive weights such that $\\\\lambda_n/(\\\\lambda_1+\\\\dots+\\\\lambda_n)$ is nondecreasing, we determine the smallest number $H \\\\in (1,+\\\\infty]$ such that \\n$$ \\n\\\\sum_{n=1}^\\\\infty \\n\\\\lambda_n \\\\mathscr{D}\\\\big((x_1,\\\\dots,x_n),(\\\\lambda_1,\\\\dots,\\\\lambda_n)\\\\big) \\\\le H \\\\cdot \\\\sum_{n=1}^\\\\infty \\\\lambda_n x_n \\\\text{ for all }x \\\\in \\\\ell_1(\\\\lambda). \\n$$ It turns out that $H$ depends only on the limit of the sequence $(\\\\lambda_n/(\\\\lambda_1+\\\\dots+\\\\lambda_n))$ and the behaviour of the mean $\\\\mathscr{D}$ near zero.\",\"PeriodicalId\":8451,\"journal\":{\"name\":\"arXiv: Classical Analysis and ODEs\",\"volume\":\"77 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2019-10-14\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"5\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv: Classical Analysis and ODEs\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.7153/mia-2020-23-75\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv: Classical Analysis and ODEs","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.7153/mia-2020-23-75","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 5
摘要
利用最近关于加权均值的齐次化和Hardy性质的结果,我们建立了凹拟合均值和单调拟合均值的尖锐Hardy常数,以及这个大族的几个特殊子类。更准确地说,对于像上面这样的平均值$\mathscr{D}$和一个正权重的序列$(\lambda_n)$,使得$\lambda_n/(\lambda_1+\dots+\lambda_n)$是非递减的,我们确定最小的数$H \in (1,+\infty]$,使得$$ \sum_{n=1}^\infty \lambda_n \mathscr{D}\big((x_1,\dots,x_n),(\lambda_1,\dots,\lambda_n)\big) \le H \cdot \sum_{n=1}^\infty \lambda_n x_n \text{ for all }x \in \ell_1(\lambda). $$事实证明,$H$仅取决于序列$(\lambda_n/(\lambda_1+\dots+\lambda_n))$的极限和平均值$\mathscr{D}$接近零的行为。
On Hardy type inequalities for weighted quasideviation means
Using recent results concerning the homogenization and the Hardy property of weighted means, we establish sharp Hardy constants for concave and monotone weighted quasideviation means and for a few particular subclasses of this broad family. More precisely, for a mean $\mathscr{D}$ like above and a sequence $(\lambda_n)$ of positive weights such that $\lambda_n/(\lambda_1+\dots+\lambda_n)$ is nondecreasing, we determine the smallest number $H \in (1,+\infty]$ such that
$$
\sum_{n=1}^\infty
\lambda_n \mathscr{D}\big((x_1,\dots,x_n),(\lambda_1,\dots,\lambda_n)\big) \le H \cdot \sum_{n=1}^\infty \lambda_n x_n \text{ for all }x \in \ell_1(\lambda).
$$ It turns out that $H$ depends only on the limit of the sequence $(\lambda_n/(\lambda_1+\dots+\lambda_n))$ and the behaviour of the mean $\mathscr{D}$ near zero.