Mohamed Amine Ighachane, Fuad Kittaneh, Zakaria Taki
{"title":"通过杨氏不等式对一些经典不等式的新完善","authors":"Mohamed Amine Ighachane, Fuad Kittaneh, Zakaria Taki","doi":"10.1007/s43036-024-00347-4","DOIUrl":null,"url":null,"abstract":"<div><p>The main objective of this paper is to use a new refinement of Young’s inequality to obtain two new scalar inequalities. As an application, we derive several new improvements of some well-known inequalities, which include the generalized mixed Schwarz inequality, numerical radius inequalities, Jensen inequalities and others. For example, for every <span>\\(T,S \\in {\\mathcal {B(H)}}\\)</span>, <span>\\(\\alpha \\in (0,1)\\)</span> and <span>\\(x, y \\in {\\mathcal {H}}\\)</span>, we prove that </p><div><div><span>$$\\begin{aligned}{} & {} \\left( 1+ L(\\alpha )\\log ^2\\left( \\frac{|\\langle TS x, y\\rangle | }{r(S)\\Vert f(|T|) x\\Vert \\left\\| g\\left( \\left| T^*\\right| \\right) y\\right\\| }\\right) \\right) |\\langle TSx, y\\rangle | \\\\{} & {} \\quad \\le r(S)\\Vert f(|T|) x\\Vert \\left\\| g\\left( \\left| T^*\\right| \\right) y\\right\\| , \\end{aligned}$$</span></div></div><p>where <i>L</i> is a positive 1-periodic function and <i>r</i>(<i>S</i>) is the spectral radius of <i>S</i>, which gives an improvement of the well-known generalized mixed Schwarz inequality: </p><div><div><span>$$\\begin{aligned} \\left| \\langle TSx,y \\rangle \\right| \\le r(S)\\Vert f(|T|) x\\Vert \\left\\| g\\left( \\left| T^*\\right| \\right) y\\right\\| , \\end{aligned}$$</span></div></div><p>where <span>\\(|T| S=S^*|T|\\)</span> and <i>f</i>, <i>g</i> are non-negative continuous functions defined on <span>\\([0, \\infty )\\)</span> satisfying that <span>\\(f(t) g(t)=t\\,(t \\ge 0)\\)</span>.</p></div>","PeriodicalId":44371,"journal":{"name":"Advances in Operator Theory","volume":"9 3","pages":""},"PeriodicalIF":0.8000,"publicationDate":"2024-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"New refinements of some classical inequalities via Young’s inequality\",\"authors\":\"Mohamed Amine Ighachane, Fuad Kittaneh, Zakaria Taki\",\"doi\":\"10.1007/s43036-024-00347-4\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>The main objective of this paper is to use a new refinement of Young’s inequality to obtain two new scalar inequalities. As an application, we derive several new improvements of some well-known inequalities, which include the generalized mixed Schwarz inequality, numerical radius inequalities, Jensen inequalities and others. For example, for every <span>\\\\(T,S \\\\in {\\\\mathcal {B(H)}}\\\\)</span>, <span>\\\\(\\\\alpha \\\\in (0,1)\\\\)</span> and <span>\\\\(x, y \\\\in {\\\\mathcal {H}}\\\\)</span>, we prove that </p><div><div><span>$$\\\\begin{aligned}{} & {} \\\\left( 1+ L(\\\\alpha )\\\\log ^2\\\\left( \\\\frac{|\\\\langle TS x, y\\\\rangle | }{r(S)\\\\Vert f(|T|) x\\\\Vert \\\\left\\\\| g\\\\left( \\\\left| T^*\\\\right| \\\\right) y\\\\right\\\\| }\\\\right) \\\\right) |\\\\langle TSx, y\\\\rangle | \\\\\\\\{} & {} \\\\quad \\\\le r(S)\\\\Vert f(|T|) x\\\\Vert \\\\left\\\\| g\\\\left( \\\\left| T^*\\\\right| \\\\right) y\\\\right\\\\| , \\\\end{aligned}$$</span></div></div><p>where <i>L</i> is a positive 1-periodic function and <i>r</i>(<i>S</i>) is the spectral radius of <i>S</i>, which gives an improvement of the well-known generalized mixed Schwarz inequality: </p><div><div><span>$$\\\\begin{aligned} \\\\left| \\\\langle TSx,y \\\\rangle \\\\right| \\\\le r(S)\\\\Vert f(|T|) x\\\\Vert \\\\left\\\\| g\\\\left( \\\\left| T^*\\\\right| \\\\right) y\\\\right\\\\| , \\\\end{aligned}$$</span></div></div><p>where <span>\\\\(|T| S=S^*|T|\\\\)</span> and <i>f</i>, <i>g</i> are non-negative continuous functions defined on <span>\\\\([0, \\\\infty )\\\\)</span> satisfying that <span>\\\\(f(t) g(t)=t\\\\,(t \\\\ge 0)\\\\)</span>.</p></div>\",\"PeriodicalId\":44371,\"journal\":{\"name\":\"Advances in Operator Theory\",\"volume\":\"9 3\",\"pages\":\"\"},\"PeriodicalIF\":0.8000,\"publicationDate\":\"2024-05-18\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Advances in Operator Theory\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s43036-024-00347-4\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Advances in Operator Theory","FirstCategoryId":"1085","ListUrlMain":"https://link.springer.com/article/10.1007/s43036-024-00347-4","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
本文的主要目的是利用杨氏不等式的新改进,得到两个新的标量不等式。作为应用,我们推导了一些著名不等式的新改进,其中包括广义混合施瓦茨不等式、数值半径不等式、詹森不等式等。例如,对于每一个(T,S 在{B(H)}}中),(alpha 在(0,1)中)和(x, y 在{H}}中),我们证明$$begin{aligned}{} & {}.\Left( 1+ L(α )log ^2\left( \frac{|langle TS x, y\rangle | }{r(S)\Vert f(|T|) x\Vert \left\| g\left( \left| T^*\right| \right) y\right| }\right) \right) |langle TSx, y\rangle | \{} & {}\quad \le r(S)\Vert f(|T|) x\Vert \left\| g\left( \left| T^*\right| \right) y\right\| , \end{aligned}$$其中 L 是正的 1-periodic 函数,r(S) 是 S 的光谱半径,这给出了著名的广义混合 Schwarz 不等式的改进:$$\begin{aligned}$$其中 L 是正的 1-periodic 函数,r(S) 是 S 的光谱半径。\left | \langle TSx,y \rangle \right| \le r(S)\Vert f(|T|) x\Vert \left | g\left( \left| T^*\right| \right) y\right| 、\end{aligned}$where \(|T| S=S^*|T|\) and f, g are non-negative continuous functions defined on \([0, \infty )\) satisfying that \(f(t) g(t)=t\,(t \ge 0)\).
New refinements of some classical inequalities via Young’s inequality
The main objective of this paper is to use a new refinement of Young’s inequality to obtain two new scalar inequalities. As an application, we derive several new improvements of some well-known inequalities, which include the generalized mixed Schwarz inequality, numerical radius inequalities, Jensen inequalities and others. For example, for every \(T,S \in {\mathcal {B(H)}}\), \(\alpha \in (0,1)\) and \(x, y \in {\mathcal {H}}\), we prove that
where L is a positive 1-periodic function and r(S) is the spectral radius of S, which gives an improvement of the well-known generalized mixed Schwarz inequality: