Pub Date : 2024-04-09DOI: 10.1186/s13660-024-03135-z
Saeed Montazeri
Correction to: J. Inequal. Appl.2023, 161 (2023). https://doi.org/10.1186/s13660-023-03074-1
Following publication of the original article [1], the author reported an error in the affiliation. The revised affiliation is indicated hereafter.
The incorrect affiliation reads:
1Faculty of Mechanical Engineering, Shahid Rajaee Teacher Training University, Tehran, Iran
The correct affiliation should read:
1Independent researcher, Tehran, Iran
All the changes requested are implemented in this correction article.
Montazeri, S.: New refinements of the Cauchy–Bunyakovsky inequality. J. Inequal. Appl. 2023, 161 (2023). https://doi.org/10.1186/s13660-023-03074-1
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Independent researcher, Tehran, Iran
Saeed Montazeri
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Correction to：J. Inequal.Appl. 2023, 161 (2023)。https://doi.org/10.1186/s13660-023-03074-1Following 原文[1]发表时，作者报告的单位有误。以下是修改后的单位。错误的所属单位为：1Faculty of Mechanical Engineering, Shahid Rajaee Teacher Training University, Tehran, Iran正确的所属单位应为：1Independent researcher, Tehran, Iran所有要求的修改均在本更正文章中实现。Montazeri, S.: New refinements of the Cauchy-Bunyakovsky inequality.J. Inequal.Appl. 2023, 161 (2023)。https://doi.org/10.1186/s13660-023-03074-1Article MathSciNet Google Scholar 下载参考文献作者和工作单位独立研究员，伊朗德黑兰Saeed Montazeri作者Saeed Montazeri查看作者发表的文章您还可以在PubMed Google Scholar中搜索该作者通讯作者Saeed Montazeri的通讯。出版商注释Springer Nature对出版地图和机构隶属关系中的管辖权主张保持中立。开放获取本文采用知识共享署名 4.0 国际许可协议进行许可，该协议允许以任何媒介或格式使用、共享、改编、分发和复制本文，但须注明原作者和出处，提供知识共享许可协议链接，并说明是否进行了修改。本文中的图片或其他第三方材料均包含在文章的知识共享许可协议中，除非在材料的署名栏中另有说明。如果材料未包含在文章的知识共享许可协议中，且您打算使用的材料不符合法律规定或超出许可使用范围，您需要直接从版权所有者处获得许可。要查看该许可的副本，请访问 http://creativecommons.org/licenses/by/4.0/.Reprints and permissionsCite this articleMontazeri, S. Correction to：平等中的考奇-布尼亚科夫斯基的新完善.J Inequal Appl 2024, 54 (2024). https://doi.org/10.1186/s13660-024-03135-zDownload citationPublished: 09 April 2024DOI: https://doi.org/10.1186/s13660-024-03135-zShare this articleAnyone you share the following link with will be able to read this content:Get shareable linkSorry, a shareable link is not currently available for this article.Copy to clipboard Provided by the Springer Nature SharedIt content-sharing initiative
{"title":"Correction to: New refinements of the Cauchy–Bunyakovsky in equality","authors":"Saeed Montazeri","doi":"10.1186/s13660-024-03135-z","DOIUrl":"https://doi.org/10.1186/s13660-024-03135-z","url":null,"abstract":"<p>Correction to: <i>J. Inequal. Appl.</i> <b>2023</b>, 161 (2023). https://doi.org/10.1186/s13660-023-03074-1</p><p>Following publication of the original article [1], the author reported an error in the affiliation. The revised affiliation is indicated hereafter. </p><ul>\u0000<li>\u0000<p>The incorrect affiliation reads:</p>\u0000<p><sup>1</sup>Faculty of Mechanical Engineering, Shahid Rajaee Teacher Training University, Tehran, Iran</p>\u0000</li>\u0000<li>\u0000<p>The correct affiliation should read:</p>\u0000<p><sup>1</sup>Independent researcher, Tehran, Iran</p>\u0000</li>\u0000</ul><p> All the changes requested are implemented in this correction article.</p><ol data-track-component=\"outbound reference\"><li data-counter=\"1.\"><p> Montazeri, S.: New refinements of the Cauchy–Bunyakovsky inequality. J. Inequal. Appl. <b>2023</b>, 161 (2023). https://doi.org/10.1186/s13660-023-03074-1</p><p>Article MathSciNet Google Scholar </p></li></ol><p>Download references<svg aria-hidden=\"true\" focusable=\"false\" height=\"16\" role=\"img\" width=\"16\"><use xlink:href=\"#icon-eds-i-download-medium\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"></use></svg></p><h3>Authors and Affiliations</h3><ol><li><p>Independent researcher, Tehran, Iran</p><p>Saeed Montazeri</p></li></ol><span>Authors</span><ol><li><span>Saeed Montazeri</span>View author publications<p>You can also search for this author in <span>PubMed<span> </span>Google Scholar</span></p></li></ol><h3>Corresponding author</h3><p>Correspondence to Saeed Montazeri.</p><h3>Publisher’s Note</h3><p>Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</p><p><b>Open Access</b> This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.</p>\u0000<p>Reprints and permissions</p><img alt=\"Check for updates. Verify currency and authenticity via CrossMark\" height=\"81\" loading=\"lazy\" src=\"data:image/svg+xml;base64,PHN2ZyBoZWlnaHQ9IjgxIiB3aWR0aD0iNTciIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8yMDAwL3N2ZyI+PGcgZmlsbD0ibm9uZSIgZmlsbC1ydWxlPSJldmVub2RkIj48cGF0aCBkPSJtMTcuMzUgMzUuNDUgMjEuMy0xNC4ydi0xNy4wM2gtMjEuMyIgZmlsbD0iIzk4OTg5OCIvPjxwYXRoIGQ9Im0zOC42NSAzNS40NS0yMS4zLTE0LjJ2LTE3LjAzaDIxLjMiIGZpbGw9IiM3NDc0NzQiLz48cGF0aCBkPSJtMjggLjVjLTEyLjk4IDAtMjMuNSAxMC","PeriodicalId":16088,"journal":{"name":"Journal of Inequalities and Applications","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140575071","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-04-08DOI: 10.1186/s13660-024-03117-1
Teodor Bulboacă, Hanaa M. Zayed
In continuation of Zayed and Bulboacă work in (J. Inequal. Appl. 2022:158, 2022), this paper discusses the geometric characterization of the normalized form of the generalized Bessel function defined by $$begin{aligned} mathrm{V}_{rho,r}(z):=z+sum_{k=1}^{infty} frac{(-r)^{k}}{4^{k}(1)_{k}(rho )_{k}}z^{k+1}, quad zin mathbb{U}, end{aligned}$$ for $rho, rin mathbb{C}^{ast}:=mathbb{C}setminus {0}$ . Precisely, we will use a sharp estimate for the Pochhammer symbol, that is, $Gamma (a+n)/Gamma (a+1)>(a+alpha )^{n-1}$ , or equivalently $(a)_{n}>a(a+alpha )^{n-1}$ , that was firstly proved by Baricz and Ponnusamy for $nin mathbb{N}setminus {1,2}$ , $a>0$ and $alpha in [0,1.302775637ldots ]$ in (Integral Transforms Spec. Funct. 21(9):641–653, 2010), and then proved in our paper by another method to improve it using the partial derivatives and the two-variable functions’ extremum technique for $nin mathbb{N}setminus {1,2}$ , $a>0$ and $0leq alpha leq sqrt{2}$ , and used to investigate the orders of starlikeness and convexity. We provide the reader with some examples to illustrate the efficiency of our theory. Our results improve, complement, and generalize some well-known (nonsharp) estimates, as seen in the Concluding Remarks and Outlook section.
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We introduce a novel multiparameterized fractional multiplicative integral identity and utilize it to derive a range of inequalities for multiplicatively s-convex mappings in connection with different quadrature rules involving one, two, and three points. Our results cover both new findings and established ones, offering a holistic framework for comprehending these inequalities. To validate our outcomes, we provide an illustrative example with visual aids. Furthermore, we highlight the practical significance of our discoveries by applying them to special means of real numbers within the realm of multiplicative calculus.
我们引入了一个新颖的多参数化分数乘法积分标识，并利用它推导出一系列与涉及一点、两点和三点的不同正交规则相关的乘法 s 凸映射不等式。我们的成果既有新发现，也有已有成果，为理解这些不等式提供了一个整体框架。为了验证我们的成果，我们提供了一个带有直观教具的示例。此外，我们还将这些发现应用于乘法微积分领域中实数的特殊手段，从而强调了这些发现的实际意义。
{"title":"On the multiparameterized fractional multiplicative integral inequalities","authors":"Mohammed Bakheet Almatrafi, Wedad Saleh, Abdelghani Lakhdari, Fahd Jarad, Badreddine Meftah","doi":"10.1186/s13660-024-03127-z","DOIUrl":"https://doi.org/10.1186/s13660-024-03127-z","url":null,"abstract":"We introduce a novel multiparameterized fractional multiplicative integral identity and utilize it to derive a range of inequalities for multiplicatively s-convex mappings in connection with different quadrature rules involving one, two, and three points. Our results cover both new findings and established ones, offering a holistic framework for comprehending these inequalities. To validate our outcomes, we provide an illustrative example with visual aids. Furthermore, we highlight the practical significance of our discoveries by applying them to special means of real numbers within the realm of multiplicative calculus.","PeriodicalId":16088,"journal":{"name":"Journal of Inequalities and Applications","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140575072","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-04-04DOI: 10.1186/s13660-024-03131-3
Tahar Bouali, Rafik Guefaifia, Salah Boulaaras
This work deals with the existence and multiplicity of solutions for a class of variable-exponent equations involving the Kirchhoff term in variable-exponent Sobolev spaces according to some conditions, where we used the sub-supersolutions method combined with the mountain pass theory.
{"title":"Multiplicity of solutions for fractional (p ( z ) )-Kirchhoff-type equation","authors":"Tahar Bouali, Rafik Guefaifia, Salah Boulaaras","doi":"10.1186/s13660-024-03131-3","DOIUrl":"https://doi.org/10.1186/s13660-024-03131-3","url":null,"abstract":"This work deals with the existence and multiplicity of solutions for a class of variable-exponent equations involving the Kirchhoff term in variable-exponent Sobolev spaces according to some conditions, where we used the sub-supersolutions method combined with the mountain pass theory.","PeriodicalId":16088,"journal":{"name":"Journal of Inequalities and Applications","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140575075","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-04-03DOI: 10.1186/s13660-024-03116-2
Abbas Najati, Yavar Khedmati Yengejeh, Kandhasamy Tamilvanan, Masho Jima Kabeto
In this article, with simple and short proofs without applying fixed point theorems, some hyperstability results corresponding to the functional equations of Cauchy and Jensen are presented in 2-normed spaces. We also obtain some results on hyperstability for the general linear functional equation $f(ax+by)=Af(x)+Bf(y)+C$ .
{"title":"Hyperstability of Cauchy and Jensen functional equations in 2-normed spaces","authors":"Abbas Najati, Yavar Khedmati Yengejeh, Kandhasamy Tamilvanan, Masho Jima Kabeto","doi":"10.1186/s13660-024-03116-2","DOIUrl":"https://doi.org/10.1186/s13660-024-03116-2","url":null,"abstract":"In this article, with simple and short proofs without applying fixed point theorems, some hyperstability results corresponding to the functional equations of Cauchy and Jensen are presented in 2-normed spaces. We also obtain some results on hyperstability for the general linear functional equation $f(ax+by)=Af(x)+Bf(y)+C$ .","PeriodicalId":16088,"journal":{"name":"Journal of Inequalities and Applications","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140575073","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-04-03DOI: 10.1186/s13660-024-03086-5
Qun Wang, Aixia Qian
We study the following nonlinear mass supercritical Kirchhoff equation: $$ - biggl(a+b int _{mathbb{R}^{N}} vert nabla u vert ^{2} biggr) triangle u+ mu u=f(u) quad text{in } {mathbb{R}^{N}}, $$ where $a ,b,m>0$ are prescribed, and the normalized constrain $int _{mathbb{R}^{N}}|u|^{2},dx =m$ is satisfied in the case $1leq Nleq 3$ . The nonlinearity f is more general and satisfies weak mass supercritical conditions. Under some mild assumptions, we establish the existence of ground state when $1leq Nleq 3$ and obtain infinitely many radial solutions when $2leq Nleq 3$ by constructing a particular bounded Palais–Smale sequence.
我们研究了以下非线性质量超临界基尔霍夫方程：$$ - biggl(a+b int _{mathbb{R}^{N}} vert nabla u vert ^{2} biggr) triangle u+ mu u=f(u) quad text{in }。{mathbb{R}^{N}}, $$ 其中$a,b,m>0$是规定的，并且归一化约束$int _mathbb{R}^{N}}|u|^{2},dx =m$在$1leq Nleq 3$的情况下是满足的。非线性 f 更为一般，满足弱质量超临界条件。在一些温和的假设条件下，我们确定了当 $1leq Nleq 3$ 时基态的存在，并通过构造一个特殊的有界 Palais-Smale 序列得到了当 $2leq Nleq 3$ 时的无穷多个径向解。
{"title":"Ground state normalized solutions to the Kirchhoff equation with general nonlinearities: mass supercritical case","authors":"Qun Wang, Aixia Qian","doi":"10.1186/s13660-024-03086-5","DOIUrl":"https://doi.org/10.1186/s13660-024-03086-5","url":null,"abstract":"We study the following nonlinear mass supercritical Kirchhoff equation: $$ - biggl(a+b int _{mathbb{R}^{N}} vert nabla u vert ^{2} biggr) triangle u+ mu u=f(u) quad text{in } {mathbb{R}^{N}}, $$ where $a ,b,m>0$ are prescribed, and the normalized constrain $int _{mathbb{R}^{N}}|u|^{2},dx =m$ is satisfied in the case $1leq Nleq 3$ . The nonlinearity f is more general and satisfies weak mass supercritical conditions. Under some mild assumptions, we establish the existence of ground state when $1leq Nleq 3$ and obtain infinitely many radial solutions when $2leq Nleq 3$ by constructing a particular bounded Palais–Smale sequence.","PeriodicalId":16088,"journal":{"name":"Journal of Inequalities and Applications","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140575276","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-04-02DOI: 10.1186/s13660-024-03130-4
Fatih Hezenci, Hüseyin Budak
In this paper, we prove an equality for twice-differentiable convex functions involving the conformable fractional integrals. Moreover, several Bullen-type inequalities are established for twice-differentiable functions. More precisely, conformable fractional integrals are used to derive such inequalities. Furthermore, sundry significant inequalities are obtained by taking advantage of the convexity, Hölder inequality, and power-mean inequality. Finally, we provide our results by using special cases of obtained theorems.
{"title":"Bullen-type inequalities for twice-differentiable functions by using conformable fractional integrals","authors":"Fatih Hezenci, Hüseyin Budak","doi":"10.1186/s13660-024-03130-4","DOIUrl":"https://doi.org/10.1186/s13660-024-03130-4","url":null,"abstract":"In this paper, we prove an equality for twice-differentiable convex functions involving the conformable fractional integrals. Moreover, several Bullen-type inequalities are established for twice-differentiable functions. More precisely, conformable fractional integrals are used to derive such inequalities. Furthermore, sundry significant inequalities are obtained by taking advantage of the convexity, Hölder inequality, and power-mean inequality. Finally, we provide our results by using special cases of obtained theorems.","PeriodicalId":16088,"journal":{"name":"Journal of Inequalities and Applications","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140575170","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-04-02DOI: 10.1186/s13660-024-03114-4
Hari Mohan Srivastava, Pishtiwan Othman Sabir, Sevtap Sümer Eker, Abbas Kareem Wanas, Pshtiwan Othman Mohammed, Nejmeddine Chorfi, Dumitru Baleanu
The Ruscheweyh derivative operator is used in this paper to introduce and investigate interesting general subclasses of the function class $Sigma_{m}$ of m-fold symmetric bi-univalent analytic functions. Estimates of the initial Taylor-Maclaurin coefficients $vert a_{m+1} vert $ and $vert a_{2 m+1} vert $ are obtained for functions of the subclasses introduced in this study, and the consequences of the results are discussed. Additionally, the Fekete-Szegö inequalities for these classes are investigated. The results presented could generalize and improve some recent and earlier works. In some cases, our estimates are better than the existing coefficient bounds. Furthermore, within the engineering domain, the utilization of the Ruscheweyh derivative operator can encompass a broad spectrum of engineering applications, including the robotic manipulation control, optimizing optical systems, antenna array signal processing, image compression, and control system filter design. It emphasizes the potential for innovative solutions that can significantly enhance the reliability and effectiveness of engineering applications.
{"title":"Some m-fold symmetric bi-univalent function classes and their associated Taylor-Maclaurin coefficient bounds","authors":"Hari Mohan Srivastava, Pishtiwan Othman Sabir, Sevtap Sümer Eker, Abbas Kareem Wanas, Pshtiwan Othman Mohammed, Nejmeddine Chorfi, Dumitru Baleanu","doi":"10.1186/s13660-024-03114-4","DOIUrl":"https://doi.org/10.1186/s13660-024-03114-4","url":null,"abstract":"The Ruscheweyh derivative operator is used in this paper to introduce and investigate interesting general subclasses of the function class $Sigma_{m}$ of m-fold symmetric bi-univalent analytic functions. Estimates of the initial Taylor-Maclaurin coefficients $vert a_{m+1} vert $ and $vert a_{2 m+1} vert $ are obtained for functions of the subclasses introduced in this study, and the consequences of the results are discussed. Additionally, the Fekete-Szegö inequalities for these classes are investigated. The results presented could generalize and improve some recent and earlier works. In some cases, our estimates are better than the existing coefficient bounds. Furthermore, within the engineering domain, the utilization of the Ruscheweyh derivative operator can encompass a broad spectrum of engineering applications, including the robotic manipulation control, optimizing optical systems, antenna array signal processing, image compression, and control system filter design. It emphasizes the potential for innovative solutions that can significantly enhance the reliability and effectiveness of engineering applications.","PeriodicalId":16088,"journal":{"name":"Journal of Inequalities and Applications","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140574974","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-04-02DOI: 10.1186/s13660-024-03111-7
K. M. Shadimetov, J. R. Davronov
The discrete analog of the differential operator plays a significant role in constructing interpolation, quadrature, and cubature formulas. In this work, we consider a discrete analog $D_{m}(hbeta )$ of the differential operator $frac{d^{2m}}{dx^{2m}}+1$ designed specifically for even natural numbers m. The operator’s effectiveness in constructing an optimal quadrature formula in the $L_{2}^{(2,0)}(0,1)$ space is demonstrated. The errors of the optimal quadrature formula in the $W_{2}^{(2,1)}(0,1)$ space and in the $L_{2}^{(2,0)}(0,1)$ space are compared numerically. The numerical results indicate that the optimal quadrature formula constructed in this work has a smaller error than the one constructed in the $W_{2}^{(2,1)}(0,1)$ space.
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Pub Date : 2024-04-02DOI: 10.1186/s13660-024-03129-x
Yen-Chang Huang
The classical Cauchy surface area formula states that the surface area of the boundary $partial K=Sigma $ of any n-dimensional convex body in the n-dimensional Euclidean space $mathbb{R}^{n}$ can be obtained by the average of the projected areas of Σ along all directions in $mathbb{S}^{n-1}$ . In this note, we generalize the formula to the boundary of arbitrary n-dimensional submanifold in $mathbb{R}^{n}$ by introducing a natural notion of projected areas along any direction in $mathbb{S}^{n-1}$ . This surface area formula derived from the new notion coincides with not only the result of the Crofton formula but also with that of De Jong (Math. Semesterber. 60(1):81–83, 2013) by using a tubular neighborhood. We also define the projected r-volumes of Σ onto any r-dimensional subspaces and obtain a recursive formula for mean projected r-volumes of Σ.
经典的 Cauchy 表面积公式指出，n 维欧几里得空间 $mathbb{R}^{n}$ 中任意 n 维凸体的边界 $partial K=Sigma $ 的表面积可以通过 Σ 沿 $mathbb{S}^{n-1}$ 中所有方向的投影面积的平均值得到。在本注释中，我们通过引入沿 $mathbb{S}^{n-1}$ 中任意方向的投影面积的自然概念，将该公式推广到 $mathbb{R}^{n}$ 中任意 n 维子曲面的边界。这个由新概念导出的表面积公式不仅与 Crofton 公式的结果相吻合，而且与 De Jong (Math. Semesterber.Semesterber.60(1):81-83,2013）使用管状邻域的结果相吻合。我们还定义了Σ在任意r维子空间上的投影r卷，并得到了Σ的平均投影r卷的递推公式。
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