Isoperimetric inequalities in surround system and space science

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• 作者：JiaJin Wen ; Jun Yuan ; ShanHe Wu
• 关键词：p ; power mean ; centered 2 ; surround system ; l ; central region ; λ ; gravity function
• 刊名：Journal of Inequalities and Applications
• 出版年：2016
• 出版时间：December 2016
• 年：2016
• 卷：2016
• 期：1
• 全文大小：1,725 KB
• 刊物主题：Analysis; Applications of Mathematics; Mathematics, general;
• 出版者：Springer International Publishing
• ISSN：1029-242X

文摘

By means of the algebraic, analysis, convex geometry, computer, and inequality theories we establish the following isoperimetric inequality in the centered 2-surround system \(S^{(2)} \{P,\varGamma ,l \}\): $$\begin{aligned}& \biggl(\frac{1}{|\varGamma |} \oint_{\varGamma }\bar{r}_{P}^{p} \biggr)^{1/p}\leqslant\frac{|\varGamma |}{4\pi}\sin\frac{l\pi}{|\varGamma |} \biggl[ \csc \frac{l\pi}{|\varGamma |}+\cot^{2} \frac{l\pi}{|\varGamma |} \ln \biggl(\tan \frac{l\pi}{|\varGamma |}+\sec\frac{l\pi}{|\varGamma |} \biggr) \biggr], \\& \quad \forall p\leqslant -2. \end{aligned}$$ As an application of the inequality in space science, we obtain the best lower bounds of the mean λ-gravity norm \(\overline{\Vert {\mathbf{F}}_{\lambda} ( \varGamma ,P )\Vert }\) as follows: $$\overline{\bigl\Vert {\mathbf{F}}_{\lambda} ( \varGamma ,P )\bigr\Vert } \triangleq\frac{1}{|\varGamma |} \oint_{\varGamma }\frac{1}{\|A-P\|^{\lambda }}\geqslant \biggl(\frac{2\pi}{|\varGamma |} \biggr)^{\lambda},\quad \forall \lambda\geqslant2. $$ Keywords p-power mean centered 2-surround system l-central region λ-gravity function MSC 26D15 26E60 51K05 52A40 1 IntroductionThe gravity is an essential attribute of any physical matter. Therefore, the study of gravity has great theoretical significance and extensive application value.