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 Mar 26 awarded Popular Question Dec 13 awarded Good Question Oct 26 awarded Yearling Oct 23 awarded Enlightened Oct 23 awarded Nice Answer Jun 14 awarded Enlightened Jun 14 awarded Nice Answer May 7 answered $\log|x|\in\text{BMO}(\mathbb R^n)$ May 2 comment Is $\int_{\mathbb R} f(\sum_{k=1}^n\frac{1}{x-x_k})dx$ independent of $x_k$'s for certain $f$? I have noticed that Theorem 3 in sos440's answer to the original post about this topic has solved the problem in my last comment. Thank you! May 2 accepted Is $\int_{\mathbb R} f(\sum_{k=1}^n\frac{1}{x-x_k})dx$ independent of $x_k$'s for certain $f$? May 2 comment Is $\int_{\mathbb R} f(\sum_{k=1}^n\frac{1}{x-x_k})dx$ independent of $x_k$'s for certain $f$? Dear David Speyer, thank you for your excellent answer! I still have one question. The displayed equality in your answer also seems to hold for some $f$ that fails to be holomorphic somewhere, say $f(z)=\frac{1}{1+|z|^\alpha}$ for real $\alpha>2$. Could such $f$ be covered as a simple corollary of your result? May 1 awarded Enlightened Apr 30 awarded Nice Answer Apr 29 comment Is $\int_{\mathbb R} f(\sum_{k=1}^n\frac{1}{x-x_k})dx$ independent of $x_k$'s for certain $f$? @DavidSpeyer: Thank you for your comments. I agree with your observation. I think in fact the statement for general positive real $a_k$ can be deduced from the statement for integers. First, by change of variables for $x$, the statement holds for rational $a_k$; second, use rationals to approximate reals. Apr 29 comment Is $\int_{\mathbb R} f(\sum_{k=1}^n\frac{1}{x-x_k})dx$ independent of $x_k$'s for certain $f$? @SamratMukhopadhyay: Thank you, but I have no idea how to apply argument principle for this problem. Apr 29 comment On the inequality $\int_{-\infty}^{+\infty}\frac{(p'(x))^2}{(p'(x))^2+(p(x))^2}\,dx \le n^{3/2}\pi.$ @DavidSpeyer: I have posted the problem. Please refer to the linked question to this post. Apr 29 asked Is $\int_{\mathbb R} f(\sum_{k=1}^n\frac{1}{x-x_k})dx$ independent of $x_k$'s for certain $f$? Apr 29 comment On the inequality $\int_{-\infty}^{+\infty}\frac{(p'(x))^2}{(p'(x))^2+(p(x))^2}\,dx \le n^{3/2}\pi.$ @DavidSpeyer: You are welcome. Yes, I also got stuck in showing the independence of $x_k$. It seems to work for many $f$'s; for example $f(t)=\frac{t^{2m}}{1+t^{2n}}$, $1\le m\le n$ are integers. Would you mind if I post a new question for this problem? Apr 29 comment On the inequality $\int_{-\infty}^{+\infty}\frac{(p'(x))^2}{(p'(x))^2+(p(x))^2}\,dx \le n^{3/2}\pi.$ @DavidSpeyer: The statement in your last comment is equivalent to $\int f(\sum_{k=1}^n\frac{1}{x-x_k})dx=n\cdot\int f(\frac{1}{x})dx$ for $f(t)=\frac{t^2}{1+t^2}$. It seems that the same equality holds for more general $f$, but I am completely clueless about the reason. Apr 28 answered On the inequality $\int_{-\infty}^{+\infty}\frac{(p'(x))^2}{(p'(x))^2+(p(x))^2}\,dx \le n^{3/2}\pi.$