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 Oct 22 comment A Problem on Improper Integrals @ParamanandSingh: I wrote my answer in this way because I thought you would prefer a hint rather than a full answer. Is it better if I add all the details to my answer? Oct 22 comment For a closed plane curve, showing some inequalities. @JeongNam-ho: You are welcome. Oct 22 revised A Problem on Improper Integrals added 150 characters in body Oct 22 comment For a closed plane curve, showing some inequalities. @JeongNam-ho: You don't have to hear that. You can prove it by yourself, as I said in the first comment. Oct 22 answered A Problem on Improper Integrals Oct 22 comment For a closed plane curve, showing some inequalities. @JeongNam-ho: Yes, $\langle,\rangle$ denotes inner product. I thought it was a conventional notation, so I didn't mention it. Sorry about the confusion. For the remark, I think it's not very hard to prove, but it need quite a few words to write down all the details. Oct 22 comment For a closed plane curve, showing some inequalities. @JeongNam-ho: It's just the linearity of integral. To see this more clearly, you may write write $v=(v_1,v_2)$, $\gamma'(t)=(x'(t),y'(t))$, and evaluate both sides of the equality. Oct 22 revised For a closed plane curve, showing some inequalities. added 1495 characters in body Oct 22 answered For a closed plane curve, showing some inequalities. Oct 21 comment Computing a limit almost surely using the strong law of large numbers @Shanks: You are welcome! :) Oct 21 revised Computing a limit almost surely using the strong law of large numbers added 17 characters in body Oct 21 answered Computing a limit almost surely using the strong law of large numbers Oct 21 revised convergence of total variation measure added 512 characters in body Oct 21 answered convergence of total variation measure Oct 20 comment Could someone explain chirality from a group theory point of view? You are welcome and thank you for your understanding. Oct 20 revised Order of Double Coset deleted 175 characters in body Oct 20 answered Order of Double Coset Oct 20 comment Is $x_1^{\alpha_1} + \dotsb + x_n^{\alpha_n}\geq x_1^{h/n}\dotsb x_n^{h/n}$ an example of power means? Compare $\lambda_1y_1+\cdots+\lambda_ny_n\ge \left(y_1^{\lambda_1}\cdots y_n^{\lambda_n}\right)^r$ with the inequality in my last comment. Then we can find the following correspondence: $\frac{h}{n\alpha_i}\leftrightarrow \lambda_i$ and $x_i^{\alpha_i}\leftrightarrow y_i$. Oct 20 comment Is $x_1^{\alpha_1} + \dotsb + x_n^{\alpha_n}\geq x_1^{h/n}\dotsb x_n^{h/n}$ an example of power means? It might be noticed that the inequality in your second paragraph can be replaced by a more precise one, $\frac{h}{n}(\frac{x_1^{\alpha_1}}{\alpha_1} + \dotsb + \frac{x_n^{\alpha_n}}{\alpha_n})\geq x_1^{h/n}\dotsb x_n^{h/n}$, which is equivalent to the inequality between weighted means. Oct 19 comment Let $a_{i} \in\mathbb{R}$ ($i=1,2,\dots,n$), and $f(x)=\sum_{i=0}^{n}a_{i}x^i$ such that if $|x|\leqslant 1$, then $|f(x)|\leqslant 1$. Prove that: Thank you. If I cannot figure it out in a couple of days, I will consider to post an question later.