Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [inequalities]

for questions involving inequalities.

1
vote
0answers
42 views

Can the Sobolev Inequalities be derived from the Weighted Hardy Inequality, or vice versa?

Here, for the weighted version of the Hardy Inequality, I refer to Muckenhoupt's formulation in Theorem 1 of 1 Sobolev Inequality: $$C_d \int_{\mathbb{R}^d} \vert \nabla \phi \vert^2 \geq \left( \...
1
vote
1answer
45 views

Randomly scaled random variables

Consider two possibly correlated scalar random variables $N$ and $X$. It is known that $1\leq N \leq N_{\max}$. Given that $\mathbb{E}[NX]\leq 0$, does it always hold that $\mathbb{E}[X] \leq 0$? ...
5
votes
1answer
119 views

Bounds on the L^1 norm of a discrete Fourier spectrum

I am dealing with a function $f$ of the form \begin{equation} f(t):=\sum_{k=1}^Na_ke^{\mathrm{i}\phi_k t} \end{equation} and I have a promise that \begin{equation} 0\leq f(t)\leq C\;\;\;\text{for all}...
0
votes
0answers
43 views

Energy estimates involving test functions for weak solutions of PDE problems

I was reading an article on Arxiv.org about Navier-Stokes system ([Breit]) and I stumble on this sentence on the second page: "A weak (in the PDE sense) solution satisfying the energy inequality ...
4
votes
1answer
131 views

Сoincidence of discrete random variables

Let $\xi, \eta$ be a discrete random values and $\mathbb E| ξ |$, $\mathbb E | η | < +\infty$, and any value of these values ​​are accepted with a non-zero probability. How to prove that from $\...
2
votes
1answer
79 views

The blow-up rate of a nonlinear oscillator

(Related to this Math.SE question.) For $p>1$, let $u$ be a solution to $$\tag{1}\frac{d^2 u}{dt^2} + u = |u|^{p-1}u$$ that blows up at $T>0$, that is $$\lim_{t\nearrow T}u(t)=+\infty.$$ ...
3
votes
0answers
116 views

Proving that the triangle inequality holds for a metric on $\mathbb{C}$ [closed]

Following problem had post mathstack three years ago,and until now no one solve it.so I ask here,Thanks for you help. let $$f(x,y)=\dfrac{|x-y|}{\sqrt{|x|^2+1}+\sqrt{|y|^2+1}}$$ show that $$f(x,y)+f(...
2
votes
1answer
65 views

Hoeffding's inequality for Hilbert space valued random elements

Suppose that $\mathbb H$ is a separable Hilbert space and $X_1,\ldots,X_n$ are independent zero mean $\mathbb H$-valued random elements such that $\|X_i\|\le s$ for each $1\le i\le n$, where $\|\cdot\|...
2
votes
1answer
184 views

Complicated bound after using Stirling's approximation

I have this inequality $$\frac{1}{a}\exp\bigl\{-\frac{4}{h^2}\bigr\} \geq \frac{1}{f}$$ where $$ a \leq \Bigl(\pi^{d/2}\Gamma(\frac{1}{2}d+1)^{-1} + 1\Bigr) \left(\frac{h^{d+1}}{2} \Gamma \left(\frac{...
1
vote
2answers
196 views

Inequality involving three functions

I have the follwing inequality, which I am not sure if it is correct or not. $$\int_{0}^{h} \int_{0}^{h} \max(u,v) f(u) f(v) du dv \geq \int_{0}^{h} \int_{0}^{h} \min(u,v) du dv \int_{0}^{h}\int_{0}^{...
2
votes
0answers
53 views

Proof of a technical fact in the book of Schapire and Freund on boosting

Disclaimer: I asked this question on math.stackexchange.com two weeks ago but it has not been answered yet so I figured that I might as well try to also post it here. I am currently looking at ...
1
vote
0answers
46 views

Concrete Hanson-Wright inequality?

I'm working on a paper that requires bounding $$\Pr\left[|\vec x^\top Q \vec y| >= t\right]$$ where $Q$ is a matrix (happens to be symmetric) and $\vec x,\vec y$ are iid real mean-zero subgaussian ...
1
vote
1answer
55 views

Is $H_0^1$ a redundant assumption in the 2D Agmon inequality?

The Wikipedia article on Agomon's inequality states the following: Let $u\in H^2(\Omega)\cap H^1_0(\Omega)$ where $\Omega\subset\mathbb{R}^2$. Then Agmon's inequality in 2D states that there ...
16
votes
2answers
623 views

“Insanely increasing” $C^\infty$ function with upper bound

Let $C^\infty$ denote the collection of functions $f:\mathbb{R}\to\mathbb{R}$ such that for every positive integer $n$, the $n$-th derivative of $f$ exists. For $f\in C^\infty$ we set $f^{(0)} = f$, ...
-1
votes
1answer
38 views

Characterisation of a superset of the simplex

Does there exist a nice description of the following set: \begin{equation} A:=\left\lbrace x\in\mathbb{R}^{n}\ \colon\ 0< x_{i}-\bar{x}+\frac{1}{n}< 1\ \text{for} \ i=1,\dots,n\right\rbrace, \...
2
votes
1answer
164 views

Simple but entangled inequalities

Do there exist functions $F,G$ on $[0,1]$ with $0\le F,G< 1$, such that for all $x, y\in [0,1]$ with $x+y\le 1$, the following hold? 1) $G(x)\le x$, 2) $G(1)<1$, 3) $F(x)>0$ if $x>0$, ...
3
votes
2answers
139 views

Is the covariance of squares always bounded from below by two times the covariance?

I came across the following inequality in one of my calculations ($X,Y$ are centered random variables): $$\operatorname{E}(X^2Y^2)-\operatorname{E}(X^2)\operatorname{E}(Y^2) \geq 2 \operatorname{E}(...
3
votes
1answer
261 views

A moment inequality

Let $\chi(s)=\int_{0}^{1}x(t)^{s}f(t)dt$, where $x(t)$ and $f(t)$ are real valued continuous functions for $t\in[0,1]$, and $f(t)\geq0$. Is it possible to show that $\left(\chi(0)\chi(2)-\chi(1)^{2}...
4
votes
1answer
259 views

Expectation of exponential of a function of independent Rademacher r.v.'s involving the error function

Let $Z,Z'\in\{-1,1\}^n$ be two independent vectors of i.i.d. Rademacher r.v.'s, where $1\leq n \leq d$ are two integers ($d\gg 1$). I am trying to get an upper bound on $$ \mathbb{E}_{ZZ'}\left[ \exp\...
0
votes
0answers
129 views

Find all placement methods are made so that $S=\sum\limits_{1\leq i<j\leq n}|P_iP_j| ^2$ takes the maximum value

A few days ago, China held the Mathematics Olympiad. In this competition, all the students did not do the last question. But what I'm more interested in right now is, what's the background to this ...
16
votes
3answers
1k views

I conjecture inequalities $\sum_{k=1}^{n}\{kx\}\le\frac{n}{2}x$

I conjecture the following inequality: For $x > 1$, and $n$ a positive integer, $$\sum_{k=1}^{n}\{kx\}\le\dfrac{n}{2}x.$$ For $n=1$, the inequality becomes $$\{x\}\le\dfrac{x}{2}\...
8
votes
2answers
294 views

Find the maximum of the value $c(n)$ (similar to Hardy's inequality)

This problem has been posted on Math.SE for seven days, without a solution. Let $n\ge 2$ be a given positive integer, and $a_{1},a_{2},\cdots,a_{n}>0$, such that $$a_{1}a_{2}\cdots a_{n}=1$$ ...
4
votes
1answer
100 views

Norm/trace of product inequality involving skew symmetric matrices

I wonder if the following inequality involving skew symmetric matrices is true: Suppose that $B,C \in \mathbb{R}^{d \times d}$ are skew-symmetric matrices, and $\Sigma \in \mathbb{R}^{d \times d}$ ...
1
vote
1answer
40 views

Monotonicity given an implicit function containing a Measure integral

The following question seems simple but I am not sure how to handle it correctly because of the integral with respect to a measure. I would be very thankful for any reply.Cheers! Knowing that $$f(\...
0
votes
0answers
31 views

Specific Bounds on Divergence operator in Sobolev setting

Given that $\vec{u}=(u_1,u_2)\in H^1(\Omega)\times H^1(\Omega)$ where $\Omega\subseteq\mathbb{R}^n$ is open and bounded with Lipschitz boundary, does there exist specific inequalities which links $\...
0
votes
0answers
81 views

Example of a function satisfying certain conditions on its derivatives

I am searching for examples of a non-negative function $f \in C^1((0,1];C^\infty _b(\mathbb{R}^n))$ (where $C^\infty _b(\mathbb{R}^n)$ is set of all smooth functions with bounded derivatives, $f(t,x)$ ...
9
votes
2answers
259 views

Maximum of a quantity for two normal orthogonal vectors in $\mathbb{R}^n$

Let's define for every pair of vectors $u,v\in\mathbb{R}^n$, a quantity as follows: $$f(u,v) = \sum_{1\leq i,j\leq n}|u_iu_j-v_iv_j|.$$ I want to find: $$M(n)= \max \{f(u,v): u,v\in \mathbb{R}^n, |u|=...
5
votes
0answers
212 views

An integral trigonometric inequality

Problem 1. Suppose that $\xi>0$ and $\sin(2\xi)<0$. Let $$b_\nu=(N-v+1)\tfrac{\pi}{\xi}\quad\mbox{for}\quad\nu=1,\dots,N:=\big[\tfrac{\xi}{\pi}\big].$$ Prove that $$\mathrm{sgn}(\sin \xi)\...
3
votes
2answers
220 views

Is it possible to find the maximum value of a sum of absolute differences?

Let $a_1$, $a_2$, …, $a_n$ and $b_1$, $b_2$, …, $b_n$ be $2n$ strictly positive integers not greater than $M$, with $M$ a given positive integer, such that $$a_1+ a_2+ \dotsb+ a_n=b_1+ b_2+ \dotsb+ ...
1
vote
1answer
120 views

Upper bound of the fraction of gamma functions

Is there a simple upper bound of the following fraction of gamma functions for any $a,b\geq1/2$: $$\left(\frac{\Gamma(a+b)}{a\Gamma(a)\Gamma(b)}\right)^{1/a}$$ An upper bound in the following form is ...
4
votes
1answer
171 views

Intuitive proof of Golden-Thompson inequality

Sutter et al. [1] in their paper "Multivariate Trace Inequalities" give an intuitive proof of the following Golden-Thompson inequality: For any hermitian matrices $A,B$: $$ \text{tr}(\exp{(A+B)}) \...
8
votes
3answers
280 views

Matrix determinant inequality proof without using information theory

Let $A$ be a $k \times n$ orthogonal matrix; i.e., $AA^T = I_{k \times k}$. For $1 \leq j \leq n$, let the squared norm of the $j$-th column of $A$ be denoted by $\alpha_j^2$; i.e., $$\sum_{i=1}^k a_{...
6
votes
4answers
282 views

Improvement of Chernoff bound in Binomial case

We know from Chernoff bound $P\bigg(X \leq (\frac{1}{2}-\epsilon)N\bigg)\leq e^{-2\epsilon^2 N}$ where $X$ follows Binomial($N, \frac{1}{2}$). If I take $N=1000, \epsilon=0.01$, the upper bound is ...
9
votes
1answer
195 views

The discrete Hardy-Littlewood-Sobolev inequality

Let $p>1$, $q>1$, $0<\lambda<1$ be such that $\frac{1}{p}+\frac{1}{q}+\lambda=2$. Suppose that $(a_{k})\in \ell^{p}(\mathbb{Z})$ and $(b_{k})\in \ell^{q}(\mathbb{Z})$. It is known ([1,2,3]...
3
votes
1answer
152 views

it's convex sequence inequality

A sequence $a_0,a_1,\dots,a_n$ of real numbers is called concave if $a_{0}=0$, and for each $0<i<n$, we have $a_i\geq\dfrac{a_{i-1}+a_{i+1}}{2}$. Find the largest $c(n)$ such that for every ...
0
votes
1answer
33 views

Right tail decay of F distribution [closed]

Suppose $X\sim F(a,b)$. Is there any sharp upper bound of the following probability with large $x$? $$\mathbb{P}(X\geq x)$$ what is the order of the above probability as $x\to+\infty$?
24
votes
4answers
987 views

show this nice and hard inequality with $ \prod_{i=1}^{n}|x_{i}-y_{i}|<e^{\frac{n}{2}}$

I saw the following results in a book. The author said it was not difficult to prove how I felt it was difficult to prove, so I asked here. The result comes from a book that has no electronic version....
1
vote
2answers
119 views

lower bound the probability of at least L collisions

Lets say we get a list $M$ containing $|M|=\sqrt{L\cdot N}$ randomly and independtly drawn elements from a set of size $N$. And lets denote the $i$-th element of the list $M$ by $M[i]$. If we now ask ...
16
votes
1answer
1k views

A conjecture of Littlewood

The following is a conjecture due to Littlewood. For any set of distinct non-zero integers $n_1,\ldots,n_k$ the inequality $$\int_0^{2\pi}|1+e^{in_1x}+\cdots+e^{in_kx}| \, dx\geq C\log k$$ holds....
15
votes
4answers
751 views

What is the minimum of this quantity on $S^{n-2}\times S^{n-2}$?

My question is to find the minimum of the following expression: $$A(x,y) = \sum_{1\leq i<j\leq n} |x_i-x_j|\ |y_i-y_j|,$$ over the set of pairs of real vectors $x=(x_1,\dots,x_n),y=(y_1,\dots,y_n)$ ...
4
votes
1answer
180 views

An inequality of T. Carleman

I'm looking for the name and some references for the proof of the inequality below. I founded that is due to T. Carleman but no reference was given. Let $f(z)$ be an analytic function on a subdomain $...
0
votes
2answers
245 views

Strange inequality with $\zeta(5)$ [closed]

$$\frac{\pi^2}{1+\exp(-1/\pi^2)}<\sum\limits_{k=1}^{\infty}\frac{5}{k^5}<\frac{\pi^2}{1+\exp(-\pi/31)}$$ How can I prove it (not only with computation)?
3
votes
1answer
113 views

Log concavity of the maximum of dependent Gaussians

Let $Z_1,\dots,Z_n$ be dependent Gaussian random variables. Is it true that $X=\max\{Z_1,\dots,Z_n\}$ has a log-concave distribution function? This is true for the independent case, but is it true in ...
3
votes
1answer
143 views

Why does the assumption $|U_t| \le \frac1{p_{\min}}$ work in this paper?

I am reading a 2009 paper right now "Importance Weighted Active Learning" and on page 5, there is a theorem that uses the inequality $|U_t| \leq \frac{1}{p_{\min}}$. I am not sure how the paper found ...
1
vote
1answer
61 views

additive discrepancy under a multiplicative constraint

Consider four sequences of numbers, $0 \le a_i, b_i, c_i, d_i \le 1$, suppose they satisfy the following constraints: (1). $\sum_{i=1}^K a_i, \sum_{i=1}^K b_i, \sum_{i=1}^K c_i \ge 1/2 + \epsilon$; (...
1
vote
1answer
77 views

How to solve such integer program problem?

Consider a $3$-tuple $(a,b,s)$ with $a,b\in\mathbb{Z}_+,s\in\mathbb{Q}_+$. Denote $ab-s$ by $\Delta$. Let $A$ be a positive number. What are the values of $A$ such that for any $(a,b,s)$ with $\Delta\...
1
vote
1answer
95 views

Any inequalities / estimates for a lower bound of the $L^2$ inner product of a quantity and its derivative?

For some numerical analysis of a fluid, I am wondering if there is any inequality that provides a lower bound (in the $L^2$ norm), for the $L^2$ inner product of a quantity with its derivative. In ...
10
votes
3answers
533 views

Entropy and total variation distance

Let $X$, $Y$ be discrete random variables taking values within the same set of $N$ elements. Let the total variation distance $|P-Q|$ (which is half the $L_1$ distance between the distributions of $P$ ...
5
votes
2answers
401 views

On the upper bound of $\sum_{i=1}^{n}x^m_{i}$ subject to the conditions $\sum_{i=1}^{n}x_{i}=0$ and $\sum_{i=1}^{n}x^2_{i}=n$

The following question has been posted on mathematics stackexchange: inequalities problem, perhaps arising from a question on expectations. Let $x_{1},x_{2},\cdots,x_{n}$ are real numbers, and such ...
2
votes
2answers
119 views

How to estimate a recursive inequality with an upper bound

The below is a simplification of part of a proof I'm working on, in numerical analysis. It is similar to a paper that I studied some months ago, for which I got some advice here on MathOverflow. I ...