<p>The goal of this paper is to refine the classical Cauchy-Schwarz inequality, which is then utilized to obtain various refinements of some well-known numerical radius inequalities. Orlicz type numerical radius inequalities have been given. In particular, it is shown that if <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1911_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="74" /> </InlineMediaObject> <EquationSource Format="TEX">$A\in \mathcal {B(H)}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>∈</mo> <mrow> <mi mathvariant="script">B</mi> <mo stretchy="false">(</mo> <mi mathvariant="script">H</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, then <Equation ID="Equ39"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1911_Article_Equ39.gif" Format="GIF" Height="78" Rendition="HTML" Resolution="72" Type="Linedraw" Width="481" /> </MediaObject> <EquationSource Format="TEX">$\begin{aligned} w^{4p}(A)&\le \frac{w^{2p}(|A|^{2}+i|A^{}|^{2})}{2^{p+2}}+\frac{1}{4}w^{2p}(A^{2}) + \frac{w^{p}(A^{2}) (\Vert |A|^{2}+|A^{}|^{2}\Vert )^p}{2^{p+1}} \\&\le \frac{1}{2}\left\Vert |A|^{4p}+|A^{*}|^{4p}\right\Vert . \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msup> <mi>w</mi> <mrow> <mn>4</mn> <mi>p</mi> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mo>≤</mo> <mfrac> <mrow> <msup> <mi>w</mi> <mrow> <mn>2</mn> <mi>p</mi> </mrow> </msup> <msup> <mrow> <mo stretchy="false">(</mo> <mo stretchy="false">|</mo> <mi>A</mi> <mo stretchy="false">|</mo> </mrow> <mn>2</mn> </msup> <mrow> <mo>+</mo> <mi>i</mi> <mo stretchy="false">|</mo> </mrow> <mmultiscripts> <mi>A</mi> <mrow /> <mrow> <mrow /> <mo>∗</mo> </mrow> </mmultiscripts> <mrow> <msup> <mo stretchy="false">|</mo> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> <msup> <mn>2</mn> <mrow> <mi>p</mi> <mo>+</mo> <mn>2</mn> </mrow> </msup> </mfrac> <mo>+</mo> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> <msup> <mi>w</mi> <mrow> <mn>2</mn> <mi>p</mi> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mi>A</mi> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mfrac> <mrow> <msup> <mi>w</mi> <mi>p</mi> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mi>A</mi> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> <msup> <mrow> <mo stretchy="false">(</mo> <mo stretchy="false">‖</mo> <mo stretchy="false">|</mo> <mi>A</mi> <mo stretchy="false">|</mo> </mrow> <mn>2</mn> </msup> <mrow> <mo>+</mo> <mo stretchy="false">|</mo> </mrow> <mmultiscripts> <mi>A</mi> <mrow /> <mrow> <mrow /> <mo>∗</mo> </mrow> </mmultiscripts> <msup> <mrow> <mo stretchy="false">|</mo> </mrow> <mn>2</mn> </msup> <msup> <mrow> <mo stretchy="false">‖</mo> <mo stretchy="false">)</mo> </mrow> <mi>p</mi> </msup> </mrow> <msup> <mn>2</mn> <mrow> <mi>p</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="right"> <mrow /> </mtd> <mtd columnalign="left"> <mrow> <mo>≤</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mfenced close="∥" open="∥"> <msup> <mrow> <mo stretchy="false">|</mo> <mi>A</mi> <mo stretchy="false">|</mo> </mrow> <mrow> <mn>4</mn> <mi>p</mi> </mrow> </msup> <mo>+</mo> <mmultiscripts> <mrow> <mo stretchy="false">|</mo> <mmultiscripts> <mi>A</mi> <mrow /> <mrow> <mrow /> <mo>∗</mo> </mrow> </mmultiscripts> <mo stretchy="false">|</mo> </mrow> <mrow /> <mrow> <mn>4</mn> <mi>p</mi> </mrow> </mmultiscripts> </mfenced> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation></p>

Various numerical radius inequalities concerning the refined Cauchy-Schwarz inequality

期刊论文
发表日期：

2025年6月25日

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Fuad Kittaneh,
Vuk Stojiljković

摘要

The goal of this paper is to refine the classical Cauchy-Schwarz inequality, which is then utilized to obtain various refinements of some well-known numerical radius inequalities. Orlicz type numerical radius inequalities have been given. In particular, it is shown that if $A\in \mathcal {B(H)}$ $A \in B (H)$ , then $\begin{aligned} w^{4p}(A)&\le \frac{w^{2p}(|A|^{2}+i|A^{*}|^{2})}{2^{p+2}}+\frac{1}{4}w^{2p}(A^{2}) + \frac{w^{p}(A^{2}) (\Vert |A|^{2}+|A^{*}|^{2}\Vert )^p}{2^{p+1}} \\&\le \frac{1}{2}\left\Vert |A|^{4p}+|A^{*}|^{4p}\right\Vert . \end{aligned}$ $\begin{matrix} w^{4 p} (A) & \leq \frac{w^{2 p} {(| A |}^{2} + i | A^{*} |^{2})}{2^{p + 2}} + \frac{1}{4} w^{2 p} (A^{2}) + \frac{w^{p} (A^{2}) {(‖ | A |}^{2} + | A^{*} |^{2} {‖)}^{p}}{2^{p + 1}} \\ \leq \frac{1}{2} ∥{| A |}^{4 p} + {| A^{*} |}^{4 p}∥ . \end{matrix}$