# Questions tagged [inequalities]

for questions involving inequalities.

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### 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( \...

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**1**answer

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 $\mathbb{E}[X] \leq 0$?
...

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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}...

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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 ...

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**1**answer

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 $\...

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**1**answer

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 $$\lim_{t\nearrow T}u(t)=+\infty.$$
...

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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(...

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**1**answer

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\|...

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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{...

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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}^{...

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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 ...

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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 ...

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**1**answer

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 ...

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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
$f^{(0)} = f$, ...

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**1**answer

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,
\...

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**1**answer

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 $x>0$,
...

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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}(...

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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}...

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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\...

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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 ...

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### 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}\...

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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 $$a_{1}a_{2}\cdots a_{n}=1$$
...

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**1**answer

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 $\Sigma \in \mathbb{R}^{d \times d}$ ...

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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(\...

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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 $\...

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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, $f(t,x)$ ...

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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|=...

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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)\...

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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+ ...

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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 ...

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**1**answer

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)}) \...

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**3**answers

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_{...

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**4**answers

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 ...

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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]...

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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 ...

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**1**answer

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 $x\to+\infty$?

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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....

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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 ...

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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....

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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 $x=(x_1,\dots,x_n),y=(y_1,\dots,y_n)$ ...

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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 $...

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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)?

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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 ...

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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 ...

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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). $\sum_{i=1}^K a_i, \sum_{i=1}^K b_i, \sum_{i=1}^K c_i \ge 1/2 + \epsilon$;
(...

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**1**answer

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\...

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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 ...

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**3**answers

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 $P$ ...

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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 ...

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**2**answers

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 ...