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$\newcommand\tH{\tilde H}$This problem is one of real algebraic geometry, which can be solved purely algorithmically. In Mathematica, such algorithms are implemented by Reduce and similar commands. He …

$\newcommand{\de}{\delta} \newcommand{\De}{\Delta} \newcommand{\ep}{\varepsilon} \newcommand{\ga}{\gamma} \newcommand{\Ga}{\Gamma} \newcommand{\la}{\lambda} \newcommand{\Si}{\Sigma} \newcommand{\thh}{ …

$\newcommand{\de}{\delta} \newcommand{\De}{\Delta} \newcommand{\ep}{\varepsilon} \newcommand{\ga}{\gamma} \newcommand{\Ga}{\Gamma} \newcommand{\la}{\lambda} \newcommand{\Si}{\Sigma} \newcommand{\thh}{ …

This inequality is false in general, by homogeneity considerations. Indeed, it can be rewritten as $L(x)\ge R(x)$, where $L(x):=\left(\chi(0)\chi(2)-\chi(1)^{2}\right)\left(\chi(4)\chi(2)-\chi(3)^{2}\ …

A counterexample is given by the following conditions: $n=185$, $$a_k=\sum_{j=k}^n b_j,\quad b_j:=\frac{c_j}{j+1}, \quad c_j:=\frac{1000}7\,1(j=47)+\frac{302}7\,1(j=185).$$ Indeed, then $a_0\ge\cdots …

$\newcommand{\al}{\alpha} \newcommand{\de}{\delta} \newcommand{\De}{\Delta} \newcommand{\ep}{\varepsilon} \newcommand{\ga}{\gamma} \newcommand{\Ga}{\Gamma} \newcommand{\la}{\lambda} \newcommand{\Si}{\ …

Indeed, disjoint tiny smooth spikes, of small heights and even much-much smaller widths, will do. Let $K\in C^\infty(\mathbb R)$ be such that $K\ge0$, $K(x)=0$ if $|x|>1/2$, and $a:=K(1/3)-K(0)\ne0 …

$\newcommand{\ga}{\gamma} $ Of course, without further assumptions on the $e_t$'s, no good bound can be given. However, looking at your examples, it appears that you are primarily interested in situat …

This is to complement the answer by Carlo Beenakker by showing that $$I(t)\le\pi^4 te^{-\pi t}\tag{1}\label{1}$$ for real $t\ge0$, where $I(t)$ is the integral in question. Indeed, according to Carlo …

$\newcommand\th\theta\newcommand\R{\mathbb R}$This is false for any real $p>2$ (actually, this is false for any real $p>1$ such that $p\ne2$). Indeed, if this were true, then, by continuity, we could …

By the main result of the paper Exact Rosenthal-type bounds, we have $$E|X-np|^r\le c^r E|\Pi_\lambda-\lambda|^r $$ for real $r\in(2,\infty)\setminus(3,4)\setminus(4,5)$, where real $c>0$ and $\lambd …

Perhaps, the following re-arrangement of the argument will help remove the mystery of the choice of $g(x,y)=\frac{\partial}{\partial x} \ln f(x,y)=\frac{f'_x(x,y)}{f(x,y)}$, where $f:=f_{X,Y}$ and ${} …

This conjecture does not hold even in the case $\gamma=\gamma'=1/2$. Indeed, consider the values of $\ell,t,k$ such that $$|\ell-n/2|\ll\sqrt n,\ |t-n/2|\ll\sqrt n,\ |k-t/2|\ll\sqrt n,$$ where $A\l …

It is not enough to assume that $Y=C_l$ is decreasing in $l$. Indeed, if your inequality were true for all such $Y$, then its non-strict version would be true for all constant $Y$, say for $Y=1$ for a …

$\newcommand{\de}{\delta} \newcommand{\si}{\sigma} \newcommand{\ep}{\varepsilon}$ Let us present the exact lower bound on $EXY$ in terms of $\mu_1:=\mu_X$, $\mu_2:=\mu_Y$, $\si_1:=\si_X$, $\si_2:=\si_ …