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This identity is now established, since An integral identity cited in the above post is now proved.

This is to detail Carlo Beenakker's assertion about the poles of the integrand. Suppose that $t=x+iy$ is such a pole, where $x$ and $y$ are real. Then $$1-y=e^{-uy}\cos ux,\quad x=e^{-uy}\sin ux.$$ Su …

As in Conrad's comment, let $$f(z):=\prod_1^\infty\Big(1-\frac{z^2}{z_n^2}\Big).$$ Then for complex $z\ne0$ $$|f(z)|=\prod_1^\infty\Big|1-\frac{z^2}{z_n^2}\Big| \le\prod_1^\infty\Big(1+\frac{|z|^2}{z_ …

$\newcommand\ol\overline$For $x=(x_1,\dots,x_n)$, $$f_{A,B,v}(x)=\sum_{i,j,k,l,m}A_{ij}x_j\ol{B_{ik}}\ol{x_k}v_l\ol{x_l}\,\ol{v_m}x_m =\sum_{j,k,l,m}c_{j,k,l,m}x_j\ol{x_k}\,\ol{x_l}x_m,$$ where $\ol{ …

This is to complement the inequality $$|(1+it)^n-1|\ge b_1(n,t):=|e^{int}-1|,\tag{10}\label{10}$$ proved by Terry Tao for real $t$ and natural $n$, by the following inequality: $$|(1+it)^n-1|\ge b_2(n …

Because Alexandre's solution uses a version of the Phragmén–-Lindelöf principle (PLP) that requires some extra effort to find or prove, I would like to present the following modification of his answer …

$\newcommand{\ep}{\epsilon} \newcommand{\ga}{\gamma} \newcommand{\Ga}{\Gamma} \newcommand{\la}{\lambda} \newcommand{\Si}{\Sigma} \newcommand{\thh}{\theta} \newcommand{\R}{\mathbb{R}} \newcommand{\E}{\ …

The answer is no. E.g., let $n=3$ and $z_j=e^{i(j-1)2\pi/3}$ for $j=1,2,3$. Then $f(z)>|z|+15/100>|z|$ if $|z|\le1$. This counterexample generalizes to any $n\ge3$. Indeed, take any $n\ge3$ and let $ …

Using the formula for the sum of the first $n$ terms of a geometric series, we have $$|f(x)|^2=\frac{\sin^2(Nx/2)}{\sin^2(x/2)}$$ and hence for $t\downarrow 0$ $$g(t)=2\int_0^t |f(x)|^2\,dx =2\int_0^t …

The key is the following integral representation: \begin{equation} \begin{aligned} s_n(z)&:= \sum_{j>n} \frac{z^j e^{-j/n}}j \\ &=\sum_{j>n} z^j e^{-j/n} \int_0^\infty du\,e^{-ju} \\ & …

Not exactly your setting, with $a_n=1$ (and i.i.d. $a_0,\dots,a_{n-1}$?), but closely related settings were considered e.g. by Götze and Zaporozhets and Ibragimov and Zaporozhets. See also further ref …

$$ \begin{aligned} \left|\sqrt{f(x)}-\sqrt{f(y)}\right|&=\frac{\bigl|f(x)-f(y)\bigr|}{\sqrt{f(x)}+\sqrt{f(y)}} \\ &\le\min\left(\sqrt{f(x)}+\sqrt{f(y)},\frac{C|x-y|^\alpha}{\sqrt{f(x)}+\sqrt{f(y)}}\r …

$\newcommand\ol\overline$As pointed out by AlgebraicsAnonymous, it is not a problem to find all even-order moments of $Y$. However, Mathematica cannot find $EY$ even when $n=1$, $a_0=1$, and $a_1=2$. …

For $n=1$, such simple (non-tautological) conditions do not exist, because real-analytic functions can mimic any $C^1$ functions in terms of their monotonicity patterns and the limit value at $\infty- …

$\newcommand\Ga\Gamma \newcommand{\R}{\mathbb{R}} \newcommand{\de}{\delta} \newcommand{\ga}{\gamma} \newcommand{\Si}{\Sigma}$ We have to show that for $a:=-\ln(2^{1/b-1/2}/v)\in\R$ and $b:=\beta\in(1, …