Questions tagged [inequalities]
for questions involving inequalities, upper and lower bounds.
1,659
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Hardy's inequality can be improved?
Hardy's inequality in $L^{p}$-spaces, taking the form [3]
$${\displaystyle \left\|{\frac {f}{|x|}}\right\|_{L^{p}(R^{n})}\leq {\frac {p}{n-p}}\|\nabla f\|_{L^{p}(R^{n})},2\leq n,1\leq p<n,}$$
where
...
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An inequality for a concave function $f(x)=x^{p/2}$
Assume that $p\in(1,2]$, $a,b\ge 1$, $b\le -\frac{1}{2} \left(\cos\frac{\pi }{p}+\sec\frac{\pi }{p}\right)$, and $t\in[0,\pi]$. How to prove this inequality $$\left(\frac{a+\cos t}{b+\cos\frac{\pi }{...
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Monotone likelihood ratio of a kernel based on $\log(\cosh(x))$
Let $f(x) = \log(\cosh(x))$, and define the kernel density:
$$p_r(\phi;\theta) = \Big(f\big(r\cos(\phi-\theta)\big) - f\big(r\cos(\phi+\theta)\big) \Big)\hspace{0.5pt} \frac{\sin(2\phi)}{\sin(2\theta)}...
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What conclusions can I derive from this family of trace inequalities?
Problem. Let $n_1,\ldots,n_s,m_1,\ldots,m_s\ge 0$ be nonnegative integers and set $m := \sum_{i=1}^s m_i$ and $n := \sum_{i=1}^s n_i$. Let $\oplus$ be an operation on matrices which stacks them in a ...
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Monotone likelihood ratio of densities based on power function
Given $p,\phi,\theta \in \mathbb{R}$ such that $p>2$ and $0 \le \phi,\theta\le \pi/2$ define the density function:
$$f(\phi;\theta) =
\mbox{$\Large\frac{1}{p B\big(\hspace{-1pt}\frac{3}{2},\frac{p+...
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Reference request: Inequalities involving convex sets and Gaussian variables stated in a paper by Talagrand
I'm looking for references for two facts that are stated without proof in the paper:
Talagrand, M., Are all sets of
positive measure essentially convex?, Lindenstrauss, J. (ed.) et al.,
Geometric ...
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Sign Regularity of a Density Kernel with Convexity Properties
(Asking a final time in a new question because the previous version had insufficient conditions, as pointed out in the answer there.)
Define the densities:
$$p(\phi;\theta,r) = \Big(f\big(r\cos(\phi-\...
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Min problem on integers
Let $n$ be any integer greater than $2^{10^6}$. Given any $s\le (\log_2 n)/1000$ integers $1=q_1\le q_2\le \cdots q_{s-1}\le q_s=n$. Prove that
$$\min_\ell\left(\sum_{i=1}^\ell q_i\right)\left(\sum_{i=...
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Non-negativity of a complicated function
Show that $f(x)\ge 0$ for $0\le x \le 1$, where:
$$f(x) = \arccos(x)^2 -8x(5x^2-2) \sqrt{1-x^2}\arccos(x)+36 x^8-112 x^6+93 x^4-17 x^2$$
The endpoints are $f(0)=\pi^2/4$ and $f(1)=0$. Plotting ...
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Validity of Hölder inequality for the homogeneous Besov spaces $\dot{B}^0_{1,2}(\mathbb{R}^n)$ and $\dot{B}^0_{2,2}(\mathbb{R}^n)=L^2(\mathbb{R}^n)$
I am looking at Corollary 1. in p.244-245 of the book
"Sobolev Spaces of Fractional Order,
Nemytskij Operators,
and Nonlinear
Partial Differential Equations" (1996) by Thomas Runst
Winfried ...
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Hardy type inequality with singular weights
Recall that Hardy's inequality involving distance from the boundary of a convex set $\Omega \subsetneq \mathbb{R}^n ; n \geq 1$, asserts that
$$
\int_{\Omega}|\nabla u|^p \, d x \geq\left(\frac{p-1}{p}...
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Is this inequality in two variables true?
It it true that for all $p\in(0,1/3]$ and all real $t$ we have
$$4
\ln(1-p +p\cosh t)
\ln\frac{1+\sqrt{1-2p}}{1-\sqrt{1-2p}}
\le t^2 (1+c p) \sqrt{1-2p} ,$$
where $c:=2\sqrt{3}\, \ln(2+\sqrt{3})-3$?
...
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Estimating the integral $\int_{\epsilon}^1 \Bigl\lvert \int_0^x \frac{f(y)}{\lvert x-y\rvert^{1/2}} dy\Bigr\rvert^2 dx$ for $L^2$ function $f(y)$?
I guess the chances are slim but still curious about the integral in the title.
Let $f : [0, \infty) \to \mathbb{R}$ be a locally "square-integrable" function on $[0,\infty)$.
Then, for any $...
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Decomposition of a vector in parts with bounded $\ell^1$ and $\ell^2$ norms
I ended up needing the following statement, which seems true but to which I couldn't find a proof nor a counterexample. Let $q \in [4/3, 2]$ and $\lambda \in \mathbb{R}^n$ with $\|\lambda\|_q = 1$. Is ...
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Comparing the slackness of Jensen's inequality for some coupled random variables
Let $f:\mathbb{R} \to \mathbb{R}$ be convex and $X,Y$ be random variables with a coupling such that $\mathbb{E}[Y\mid X=x] = x$. A straightforward application of Jensen's inequality gives that $\...
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Sobolev spaces H^{k}, k an integer
I have two questions:
is it true that we have equivalence between these two norms?
$$\lVert u\rVert_{H^{k}}^{2}\sim \lVert u\rVert_{L^{2}}^{2}+\sum_{\lvert \alpha\rvert=k} \lVert D^{\alpha}u\rVert_{...
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Sharpened versions of the Fisher information inequality?
Let $X, Y$ be two independent random vectors in $\mathbb{R}^d$. In the paper [1, see ineq. (11)], the following convolution inequality is proven for the Fisher information:
$$
I(X + Y) \preceq \Big(I(...
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$L^\infty$ bound of $x^m \psi_n(x)$ where $\psi_n$ is a Hermite function and $m,n \in \mathbb{N}$ - extension from Cramer's inequality
For each $n \in \mathbb{N}$, the Hermite function $\psi_n : \mathbb{R} \to \mathbb{R}$ is a Schwartz function defined by
\begin{equation}
\psi_n(x):=(-1)^n(2^n n!\sqrt{\pi})^{-1/2} e^{x^2/2} \frac{d^n}...
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On the validity of a certain Grönwall-type inequality
Assume that $u~ \colon \mathbb{R}_+ \to [-M,M]$ is a bounded continuously differentiable function such that $u(0) = 0$ and $$u(t) \leq \int_0^t \lambda(s)~u(s)~\mathrm{d}s + C \label{1}\tag{1}$$ where ...
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Finding bounds for a "smooth" mapping $u : [0,T] \to \mathcal{S}(\mathbb{R})$ in terms of Schwartz seminorms
Let $u(t,x) : [0,T] \times \mathbb{R} \to \mathbb{R}$ be a smooth function with the following properties:
Partial derivatives of $u(t,x)$ with respect to $t$ up to any order are Schwartz functions of ...
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An inequality involving 2 variables
How to prove that the following expression is positive for $u\ge 0$ and $q\in(0,\pi/3)$. $$\frac{2 \sqrt{2} ((2+u) (1+\cos(q)))^{3/4}}{3^{3/4}}-\frac{2^{3/4} (1+\sec(q))}{\sqrt{3} \sec(q)^{3/4}}-\left(...
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Poincaré inequality in dimension one with mixed boundary condition
Suppose that $u$ is $C^1$ in $[0,\pi/2]$ and $u(0)=u’(\pi/2)=0$. I want to derive the following Poincaré inequality
$$
\int_0^{\pi/2} u(x)^2\,dx \leq \int_0^{\pi/2} u’(x)^2\,dx.
$$
Since $u(0)=0$, we ...
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Inverse of Sobolev interpolation inequality : $\lVert u \rVert_2 \lVert \Delta u \rVert_2 \leq C\lVert \nabla u \rVert_2^2$?
If $u : \mathbb{T}^3 \to \mathbb{R}$ is a smooth function on the $3$-dimensional torus $\mathbb{T}^3$, I wonder it is possible to reverse the Sobolev interpolation inequality in the sense that
\begin{...
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Bound a function with parameter involving logarithm
In my master project I encounter the following function
$$f_\varepsilon(x) = \ln\left(\frac{x^{1 + a} + \varepsilon^\beta}{\lambda(x)^2 + \varepsilon^2}\right)$$
for $a$ close to zero, $\beta \in (1, ...
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Two-Sided Bounds on Binomial Sum
I came across this partial sum which I cannot find reasonable bounds on; I feel this must be known in the literature, but I do not know where to look. Here is the problem:
Let $s\in (0,1)$ and ...
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Log Sobolev inequality for log concave perturbations of uniform measure
Suppose $\Omega$ is a convex bounded open set of $\mathbb{R}^n$ (I would be happy with just $\Omega$ as the $n$-dimensional cube). Let $\mu$ be the uniform measure on $\Omega$ and consider the ...
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A geometric proof that there are infinitely many primes?
Let $e_d$ be the $d$-th standard-basis vector in the Hilbert space $H=l_2(\mathbb{N})$.
Let $h(n) = J_2(n)$ be the second Jordan totient function, defined by:
$$J_2(n) = n^2 \prod_{p|n}(1-1/p^2)$$
...
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Extending Gromov's inequality
In 1981 Gromov proved that all Riemannian metrics on the complex projective space $\mathbb CP^n$ satisfy the bound
$$\DeclareMathOperator{stsys}{stsys} \DeclareMathOperator{vol}{vol}
\frac{\stsys_2^n}{...
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Bound on an integral representing a difference of two relative entropies
Let $ f : [0,1] \to \mathbb{R} $ be a function satisfying: 1.) $ |f(x)| \leqslant a $ for some $ a < 1 $, and 2.) $ \int_0^1 f(x) {\mathrm d}x = 0 $. I would like to know whether the following ...
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A proof of the Gagliardo-Nirenberg interpolation inequality using Jensen's inequality
A brief look at the statement of Gagliardo-Nirenberg interpolation inequality would suggest that there should exist a proof by a clever use of Jensen's inequality. In other words, there should be a ...
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Gaussian Poincare inequality in $1$ dimensions together with localization issue
Let $d\mu$ be a Gaussian measure on $\mathbb{R}$ with the center $a \in \mathbb{R}$ and variance $1$.
Let $B(a,r) \subset \mathbb{R}$ be the interval $[a-r,a+r]$.
Then, for any smooth mapping $f : \...
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Bounding the variance of a truncated Gaussian random variable
Suppose $X_1, X_2, X_3 \sim N(0, 1)$ are three independent standard normal random variables. I am interested in showing that:
$$\text{Var}[X_2\mid X_2 \geq X_1 - a, X_1 \leq X_3 + b] < 1,$$
where ...
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Prime gap conjecture $ \pi_{2a}(n+(6a+4)^3)+(6a+4)^3 > \pi_{4a}(n)$ counterexamples?
Consider prime constellations $p,p+2s$ where both $p,p+2s$ are prime.
For instance for $s=1$ we get the twin primes.
We define the counting function $\pi_{2s}(n)$ to count the number of such pairs $p,...
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$f(n) = \frac{n^2 + n + 4}{2}$, $g(f(n)) = f(g(n))$ such that $g(n)$ is an integer
Let $n$ be a strict positive integer and let's define an integer sequence $f(n)$ :
$$f(n) = \frac{n^2 + n + 4}{2}$$
so
$$
\begin{split}
f (\Bbb N)& \triangleq {3,5,8,12,17,23,30,38,47,\ldots}\\
f(...
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A counterexample showing $BV_p \neq AC_p$
I am trying to work through a supposedly simple counterexample given in papers by Love and Gehring regarding a $p$-power generalization of bounded variation and absolute continuity.
Let $p > 1$. ...
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A potential new norm for matrices and Horn's inequalities
I am investigating a function defined in terms of the singular values of matrices. Initially, I simplified the problem by focusing on the eigenvalues of $2 \times 2$ Hermitian, positive-definite ...
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How big can a wedge of $n$ 2-forms in $\mathbb{R}^{2n}$ be?
$\def\RR{\mathbb{R}}$Let $\omega$ be a $2$-form on $\RR^{2n}$, where $\RR^{2n}$ has the usual Euclidean norm. The comass of $\omega$ is defined to be $\max_{|u|, |v| \leq 1} \omega(u,v)$. Here is the ...
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139
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Log Sobolev inequality uniform in parameters
Fix a positive integer $N$. For $\theta \in [0,2\pi]$, set $\sigma_k(\theta) :=(\cos(k\theta),\sin(k\theta)) \in S^1$ for each integer $1\leq k\leq N$. Now for vectors $x_1,\ldots,x_N\in \mathbb{R}^2$,...
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214
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How to estimate an integral by the variation and upper bound of the integrand?
Suppose that $f$ is a continuous function on $\mathbb{R}$. I want to estimate the definite integral
$$ I:= \int_{0}^a [f(x)-f(0)]dx $$
by the upper bound $M = \sup_{x\in[0,a]}|f(x)|$ and the variation ...
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A one-sided/monotone version of min/max-stable distributions -- does this have a name?
In a couple of papers I am working on (in random graph theory) I have encountered the following property of certain probability distributions, which I will describe shortly, and I am wondering if this ...
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463
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Cauchy-Schwarz-like inequality with a power $p$ term
We set :
$\phi_1, \phi_2 : \mathbb{R} \to \mathbb{R}^M$ with compact support
$\theta \in \mathbb{R}^M$ non-zero, $\theta_{p-1} = (\operatorname{sign}(\theta_i)|\theta_i|^{p-1})_{1 \leq i \leq M} \in \...
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votes
2
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299
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An inequality problem for certain positive-definite matrices
Suppose that $G=(G_{ij})$ is an $n\times n$ positive-definite symmetric matrix with the diagonal entries all equal $1$ and all off-diagonal entries $<0$. Let $a$ be a column $n\times1$ matrix such ...
- 118k
0
votes
1
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131
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Deducing norm concentration from MGF bounds
Suppose that $X$ is a centered, $\mathbf{R}^d$-valued random variable such that for all $t \in \mathbf{R}^d$, there holds the bound $$\log \mathbf{E} \left[ \exp \langle t, X \rangle \right] \leqslant ...
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votes
1
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210
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An inequality for certain positive-definite matrices
Suppose that $G=(G_{ij})$ is an $n\times n$ positive-definite symmetric matrix with the diagonal entries all equal $1$ and all off-diagonal entries $\le0$. Let $a$ be a column $n\times1$ matrix with ...
- 118k
4
votes
1
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452
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An inequality for certain positive-semidefinite matrices
Suppose that $G=(G_{ij})$ is a positive-semidefinite symmetric matrix with the diagonal entries all equal $1$ and all off-diagonal entries $\le0$. Does it then necessarily follow that
$$\sum_{i,j}(G^5)...
- 118k
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Interpolation of scalars
For $a,b$ and $\alpha_i, \beta_i $ where $ i \in \{ 1,2 \} $, are non-negative real numbers, is it possible to find a constant $C$ such that
$$(\alpha_1 a + \beta_1 b) ^{(1-\theta)} (\alpha_2 a + \...
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3
votes
1
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292
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Gronwall lemma for a $2$-dimensional system of linear differential inequalities
Let $$z(t)=\begin{pmatrix}x(t) \\ y(t)\end{pmatrix}~,~ A=\begin{pmatrix}0 & -\beta \\ \alpha & -(\alpha + \beta)\end{pmatrix}$$ satisfies the following system of linear differential ...
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On $[0,1]$ with the Lesbegue measure, is it possible to have $\lVert \cdot \rVert^{1/n}_p \leq \lVert \cdot \rVert_q$ for $p>q$ and $n$ large?
The question is as in the above.
In all literature, I only find that on $[0,1]$ with the Lebesgue measure, $\lVert \cdot \rVert_q \leq \lVert \cdot \rVert_p$ for $p>q$.
(I deleted the last question ...
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Proof of the inequality $\frac{y}{x}-1-\log\left(\frac{y}{x}\right)\geq \frac{1}{2}\frac{(x-y)^2}{x}$ when $x,y \in (0,1]$
I am trying to prove the following inequality:
$$\frac{y}{x}-1-\log\left(\frac{y}{x}\right)\geq \frac{1}{2}\frac{(x-y)^2}{x} \quad \forall x,y \in (0,1]$$
The statement looks simple enough that it may ...
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0
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Question on a mixed-norm estimate
I am currently reading the paper Global existence and scattering for rough solutions to generalized nonlinear Schrödinger equations on $\mathbb{R}$ by Colliander, Holmer, Visan, Zhang. In this article,...
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