{"title":"初等函数的逆Holder不等式","authors":"A.O. Korenovskii","doi":"10.30970/ms.56.1.28-38","DOIUrl":null,"url":null,"abstract":"For a positive function $f$ on the interval $[0,1]$, the power mean of order $p\\in\\mathbb R$ is defined by \n\\smallskip\\centerline{$\\displaystyle\\|\\, f\\,\\|_p=\\left(\\int_0^1 f^p(x)\\,dx\\right)^{1/p}\\quad(p\\ne0),\\qquad\\|\\, f\\,\\|_0=\\exp\\left(\\int_0^1\\ln f(x)\\,dx\\right).$} \nAssume that $0<A<B$, $0<\\theta<1$ and consider the step function$g_{A<B,\\theta}=B\\cdot\\chi_{[0,\\theta)}+A\\cdot\\chi_{[\\theta,1]}$, where $\\chi_E$ is the characteristic function of the set $E$. \nLet $-\\infty<p<q<+\\infty$. The main result of this work consists in finding the term \n\\smallskip\\centerline{$\\displaystyleC_{p<q,A<B}=\\max\\limits_{0\\le\\theta\\le1}\\frac{\\|\\,g_{A<B,\\theta}\\,\\|_q}{\\|\\,g_{A<B,\\theta}\\,\\|_p}.$} \n\\smallskip For fixed $p<q$, we study the behaviour of $C_{p<q,A<B}$ and $\\theta_{p<q,A<B}$ with respect to $\\beta=B/A\\in(1,+\\infty)$.The cases $p=0$ or $q=0$ are considered separately. \nThe results of this work can be used in the study of the extremal properties of classes of functions, which satisfy the inverse H\\\"older inequality, e.g. the Muckenhoupt and Gehring ones. For functions from the Gurov-Reshetnyak classes, a similar problem has been investigated in~[4].","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2021-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"The reverse Holder inequality for an elementary function\",\"authors\":\"A.O. Korenovskii\",\"doi\":\"10.30970/ms.56.1.28-38\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"For a positive function $f$ on the interval $[0,1]$, the power mean of order $p\\\\in\\\\mathbb R$ is defined by \\n\\\\smallskip\\\\centerline{$\\\\displaystyle\\\\|\\\\, f\\\\,\\\\|_p=\\\\left(\\\\int_0^1 f^p(x)\\\\,dx\\\\right)^{1/p}\\\\quad(p\\\\ne0),\\\\qquad\\\\|\\\\, f\\\\,\\\\|_0=\\\\exp\\\\left(\\\\int_0^1\\\\ln f(x)\\\\,dx\\\\right).$} \\nAssume that $0<A<B$, $0<\\\\theta<1$ and consider the step function$g_{A<B,\\\\theta}=B\\\\cdot\\\\chi_{[0,\\\\theta)}+A\\\\cdot\\\\chi_{[\\\\theta,1]}$, where $\\\\chi_E$ is the characteristic function of the set $E$. \\nLet $-\\\\infty<p<q<+\\\\infty$. The main result of this work consists in finding the term \\n\\\\smallskip\\\\centerline{$\\\\displaystyleC_{p<q,A<B}=\\\\max\\\\limits_{0\\\\le\\\\theta\\\\le1}\\\\frac{\\\\|\\\\,g_{A<B,\\\\theta}\\\\,\\\\|_q}{\\\\|\\\\,g_{A<B,\\\\theta}\\\\,\\\\|_p}.$} \\n\\\\smallskip For fixed $p<q$, we study the behaviour of $C_{p<q,A<B}$ and $\\\\theta_{p<q,A<B}$ with respect to $\\\\beta=B/A\\\\in(1,+\\\\infty)$.The cases $p=0$ or $q=0$ are considered separately. \\nThe results of this work can be used in the study of the extremal properties of classes of functions, which satisfy the inverse H\\\\\\\"older inequality, e.g. the Muckenhoupt and Gehring ones. For functions from the Gurov-Reshetnyak classes, a similar problem has been investigated in~[4].\",\"PeriodicalId\":37555,\"journal\":{\"name\":\"Matematychni Studii\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2021-10-23\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Matematychni Studii\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.30970/ms.56.1.28-38\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"Mathematics\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Matematychni Studii","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.30970/ms.56.1.28-38","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"Mathematics","Score":null,"Total":0}
The reverse Holder inequality for an elementary function
For a positive function $f$ on the interval $[0,1]$, the power mean of order $p\in\mathbb R$ is defined by
\smallskip\centerline{$\displaystyle\|\, f\,\|_p=\left(\int_0^1 f^p(x)\,dx\right)^{1/p}\quad(p\ne0),\qquad\|\, f\,\|_0=\exp\left(\int_0^1\ln f(x)\,dx\right).$}
Assume that $0
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