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May
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answered $\log|x|\in\text{BMO}(\mathbb R^n)$
May
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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!
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accepted Is $\int_{\mathbb R} f(\sum_{k=1}^n\frac{1}{x-x_k})dx$ independent of $x_k$'s for certain $f$?
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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?
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Apr
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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.$