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Let $n$ be a given even positive integer. We have the following integral \begin{eqnarray} &&\int_0^1\cdots\int_0^1\prod\limits_{i=1}^n\prod\limits_{j=1}^n(x_i-y_j)dx_1\cdots dx_ndy_1\cdots dy_n\\ &=&\int_0^1\cdots\int_0^1\left(\int_0^1\prod\limits_{j=1}^n(x-y_j)dx\right)^ndy_1\cdots dy_n\\ &>&0. \end{eqnarray} Let's consider a similar integral where $n$ is also an even positive integer: $$A_n=\int_0^1\cdots\int_0^1\int_0^{2\pi}\cdots\int_0^{2\pi}\prod\limits_{i=1}^n\prod\limits_{j=1}^n\left(x_i-y_j+i\cos(\alpha_i-\beta_j)\right)dx_1\cdots dx_ndy_1\cdots dy_nd\alpha_1\cdots d\alpha_nd\beta_1\cdots d\beta_n.$$ It is easy to see that $A_n$ is a real number for every $n$. My question is whether $A_n$ is positive or not.

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  • $\begingroup$ Did you try any numerical calculations? $\endgroup$
    – Wojowu
    Jun 19, 2017 at 19:44
  • $\begingroup$ For $n=2$ Maple answers the integral over $\alpha$s and $\beta$s equals $$2\, \left( 8\,{\pi}^{3}A_{{1,1}}A_{{1,2}}A_{{2,1}}A_{{2,2}}+4\,{\pi}^{ 3} \right) \pi, $$ where $A_{i,j}:=x_i-y_j .$ $\endgroup$
    – user64494
    Jun 19, 2017 at 20:12
  • $\begingroup$ Is this a question at the research level? I find it rather an exercise from George Pólya and Gábor Szegő, Aufgaben und Lehrsätze aus der Analysis, 1st edn. 1925.[11] ("Problems and theorems in analysis“). Springer, Berlin 1975 (with Gábor Szegő). Have you asked that in math.stackexchange.com ? $\endgroup$
    – user64494
    Jun 19, 2017 at 20:18
  • $\begingroup$ user64494: Thank you for your attention! This question is not an exercise of any book for me. You said that you found it rather an exercise in the book "Problems and theorems in analysis". Would you please tell me where I can find this exercise?I really want to know more about this question! $\endgroup$
    – user173856
    Jun 20, 2017 at 15:18
  • $\begingroup$ the book mentioned by @user64494 is here, but I don't see this excercise there. $\endgroup$ Jun 23, 2017 at 11:02

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