Questions tagged [inequalities]

for questions involving inequalities, upper and lower bounds.

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$L^p$ inequality for "positively correlated" random variables

Suppose that we have $m$ complex-valued random variables $\xi_1,\ldots,\xi_m$ and assume the following "positive correlation" property: for all non-negative integers $\alpha_1,\ldots,\...
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3 votes
1 answer
58 views

More on the inequality $f'(x)/(1-f(x)^2)-1/(1-x^2)\ge0$

A previous question was as follows: Assume that $f\colon[0,1]\to[0,1]$ is a diffeomorphism so that $(f''(x)/f'(x))'<0$ and that $f''(0)=0$. It seems to me that $$\frac{1-f(x)^2}{1-x^2}\le f'(x)$$ ...
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1 vote
1 answer
156 views

A condition on the inequality $f'(x)/(1-f(x)^2)-1/(1-x^2)\ge 0$

Assume that $f:[0,1]\to [0,1]$ is an diffeomorphism so that $(f''(x)/f'(x))'<0$ and that $f''(0)=0$. It seems to me that $$\frac{1-f(x)^2}{1-x^2}\le f'(x)$$ on $[0,1]$. But no proof so far. The ...
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Are these $L_2$-spectral radii approximations strictly increasing?

Suppose that $V$ is a finite dimensional complex Hilbert space. Let $L(V)$ denote the collection of all linear mappings from $V$ to $V$. Let $A_1,\dots,A_r:V\rightarrow V$ be linear operators. Then ...
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2 votes
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Is the Jaccard distance between continuous vectors a metric?

Define the Jaccard distance between two continuous vectors $a, b\in [0,1]^p$ as \begin{equation} J(a,b) = 1 - \frac{\|a\odot b\|_1}{\|a\odot b\|_1+\|a-b\|_1} \end{equation} where $\odot$ is the ...
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1 answer
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Is there any triangle inequality based distances comparison? [closed]

Is there any known approach or method to compare distances basing on the triangle inequality ? thanks.
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3 votes
2 answers
116 views

On finding an upper bound on the error of a sparse approximation

I posted this question on math.stackexchange earlier, but didn't see any response. So, I am posting it here, in case someone else has an answer. Original question: https://math.stackexchange.com/...
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2 votes
0 answers
114 views

Trying to prove an inequality

I am working on a problem and for that purpose, I need to prove the following inequality. Let $t\in [0,1]$ and set $$ z_0=1-4t(1-t)\sin^2(4x)\\ z_1=1-4z_0(1-z_0)\sin^2(3x) $$ I need to show that for ...
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  • 477
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307 views

Using Young's inequality to show elementary inequality?

Let $p, q\geq 2$, $s\geq p$ and $f,g$ be non-negative smooth enough functions. Then why does the following inequality hold: $$-f^{q-2}g^{s}|\nabla f|^{p}+f^{q-1}g^{s-1}|\nabla f|^{p-1}|\nabla g|\leq C(...
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How to prove that this inequality defines a convex sequence?

Let $k,n$ be given positive integers and let $a_n$ be defined as $$ a_{n}=\sqrt[n]{n!}+\sqrt[n-1]{(n-1)!}+\ldots+\sqrt[k]{k!}\qquad\forall k\le n. $$ Show that $$ a^2_{n}\ge a_{n+1}a_{n-1}.\label{1}\...
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  • 193
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A distribution $\pi \propto \exp(-f)$ satisfies log-Sobolev inequality, does $\exp(-af)$ also satisfy LSI?

Assume a distribution $\pi \propto e^{-f}$ satisfies log-Sobolev inequality (LSI) $$\forall \rho \in P(\mathbb{R}^n), \quad KL(\rho\| \pi) \le \frac{1}{2\lambda} I(\rho \| \pi)$$ with LSI constant $\...
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3 votes
1 answer
66 views

Hölder inequality between different Orlicz spaces

If we have a product of functions $fg$ with $f\in L^r$ and $g\in L^s$ for some $s,r>1$ satisfying $1/r+1/s=1$, then we know that $fg\in L^1$. But if $g$ is a little bit more than $L^s$, say $L^s \...
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Bound on triangle inequality with powers

Conjecture I believe the following inequality is true but I cannot prove it. Let $\alpha \in (0, 2)$ and $x, y \in \mathbb{R}$. Then, it holds that $$ \Big\vert \vert x + y \vert^\alpha - \vert x \...
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A question about an inequality for hafnians of some special matrices

Let $S$ by a complex symmetric $2m$ by $2m$ matrix. Let $\sigma$ be the $2m$ by $2m$ matrix which is the direct sum of $m$ copies of the following matrix: $$ \begin{pmatrix} 0 & 1 \\ 1 & 0 \...
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  • 3,594
2 votes
1 answer
92 views

Multivariate inequality of floor function

Define $$f(x,a) := (2x-a)\lfloor\frac{x}{a}\rfloor-a\lfloor\frac{x}{a}\rfloor^2.$$ It seems that $$f(x,a)+f(x,b)\geq 2f(x,c),\forall a,b \in [1,x],a+b=2c.$$ I have written a program that has checked ...
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2 votes
1 answer
117 views

Coefficients of certain Taylor series

For $t\in(-1,1)$, let $$f(t):=\left(\frac{1+t}{1-t}\right)^{(1-t)/2}+\left(\frac{1-t}{1+t}\right)^{(1+t)/2}$$ and $$g(t):=\frac1{f(t)}.$$ Note that the functions $f$ and $g$ are even. Question 1: Is ...
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1 vote
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106 views

examples of function difficult to prove to be $\geq0$?

I have often wondered whether there has ever come a point in your research, when you were confronted with an explicit real function $f(x_1,x_2,\ldots,x_n)$ and an explicitly defined compact set $S\...
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1 vote
1 answer
78 views

A question about mutual information

Let $A$ and $B$ be two, possibly dependent, random variables, and let $X$ be a random variable independent of $(A,B)$. For simplicity, let's concern ourselves with discrete random variables. Is the ...
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11 votes
2 answers
862 views

Decoupling inequality/counterexample

I am embarrassed to be stuck on this seemingly simple question. Suppose that $X,Y$ are mean-zero real-valued random variables and $\tilde X,\tilde Y$ are their "independent copies": $\tilde ...
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1 answer
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Proving maximum value of a determinant of $I - B$, where $B$ is nonnegative matrix

I have the following setting: Let $0 \leq r < 1$ and let $\{z_i\}_{i=1}^k$ be $k$ complex numbers such that $|z_i| \leq r$ for all $i$. Moreover, $r + \sum_{i=1}^k 2Re(z_i) \geq 0$ I am interested ...
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Show that for all $\delta$, we have $\|u\|^2_{\Gamma}\le c_\delta(\|u\|^2_{\omega(\delta)}+\|u\|_{\omega(\delta)}\|\nabla u\|_{\omega(\delta)})$

I am reading the article On existence of weak solutions of the Navier-Stokes equations in regions with moving boundaries from Fujita and Kato and at some point they use an argument I have some trouble ...
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  • 268
8 votes
1 answer
591 views

Prove an inequality related to sums of Legendre symbols

$\newcommand\Legendre{\genfrac(){}{}}$Let $p\equiv 1\pmod 4$ be a prime number, and $x_{i}\ge 0$ be such that $$x_{1}+x_{2}+\dotsb+x_{p}=1.$$ Show that $$\sum_{1\le i<j\le p}\Legendre{i-j}{p}x_{i}...
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  • 193
4 votes
2 answers
228 views

A convexity question

Let $Q=[0,1]\times[0,1]$ and let $a$ be a positive smooth function on $Q$. Does there exist a smooth positive function $u$ On $Q$ such that there holds $$ \frac{\partial^2}{\partial x_1^2}u <0 $$ ...
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  • 2,858
4 votes
1 answer
170 views

Nonnegativity locus of Schur polynomials

Let $a_1,\ldots,a_n \in \mathbb{C}$ be complex numbers that are the zeros of a real polynomial (meaning that the non-real ones come in complex conjugate pairs). Suppose that these numbers are such ...
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2 votes
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137 views

An oscillatory integral

Let $s>0, v\in \mathbb{R}^d, w\in \mathbb{R}, |w|\leq 1$. Pick a cut-off function $B(0,1)\prec \eta \prec B(0,2)$ and a large real number $N$. Do we have the following type of estimates? \begin{...
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4 votes
1 answer
112 views

How does the integral of pseudo Gaussian kernel on $(0,\infty)$ depend on its variance?

Let $a, b: \mathbb R_+ \to [0,1]$ be continuous functions. Let $k: \mathbb R_+\times\mathbb R \to [1,2]$ be $1-$Lipschitz. Set, for $0<s<t$ and $y>0$, $$A(s,t,y):=\int_s^t\frac{k(u,y)}{1+a(u)}...
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327 views

Is $\lim_{x\to\infty}\sum_{n=-\infty}^{\infty}\frac{x}{n^2+x^2}=\pi$?

It seems that $$\lim_{x\to\infty}\sum_{n=-\infty}^{\infty}\frac{x}{n^2+x^2}=\pi.$$ But I can't prove it. I cannot prove that the function is decreasing in $x$ either.
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2 votes
1 answer
84 views

How to connect the functions in spaces $H^1$ and $H_r$?

\begin{align*} L^2 (\mathbb{R}^3)& {}=\{ u : \int_{\mathbb{R}^3} \lvert u\rvert^2 dx<+\infty \}. \\ H^1(\mathbb{R}^3) & {}=\{ u\in L^2 (\mathbb{R}^3):\, \lvert\nabla u\rvert\in L^2(\mathbb{...
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2 votes
0 answers
134 views

Closeness of a rational approximation

What is $$p_*:=\inf\big\{p\in\mathbb R\colon\,\inf_{n\in\mathbb N}n^p\,\inf_{k\in\mathbb N} |2\sqrt{3n}-9\pi/4-k\pi|>0\big\},$$ where $\mathbb N:=\{1,2,\dots\}$? In other words, I would like to ...
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3 votes
1 answer
76 views

Tauberian lower bound for a series

Let $(a_n)_{n\in\mathbb{N}}$ be a sequence of positive number such that $\sum_n a_n < +\infty$ (i.e. $a_n \in \ell^1$) but $\sum_n r^n a_n = +\infty$ for every $r > 1$. Given $\sigma \in (0,1)$, ...
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9 votes
2 answers
269 views

Asymptotics of a quadratic recursion

Consider the sequence defined by \begin{align} c_0 &{}= 1 \\ c_n &{}= 2\,n\,c_{n-1}-\frac{1}{2}\sum_{m=1}^{n-1}c_m\,c_{n-m}. \end{align} How can you prove that it has the following asymptotics ...
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-1 votes
1 answer
111 views

What functions are equal to their symmetric decreasing rearrangement?

I am trying to understand the set $$\mathcal{A}=\{f:f\text{ non-negative, measurable}, \text{vanishes at infinity and } f(x)=f^*(x)\}$$ where $f:\mathbb{R}^n\to \mathbb{R}$ is vanishing at infinity if ...
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3 votes
0 answers
249 views

Inequalities involving traces of products of hermitian positive semidefinite matrices

$\DeclareMathOperator{\tr}{tr}$ Fix an integer $n \geq 2$. Let $A_1, \dotsc, A_n$ be hermitian positive semidefinite matrices, with each $A_i$ being $m$ by $m$. Consider the symmetric group $S_n$ on $...
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  • 3,594
0 votes
1 answer
56 views

Using Hölder's Inequality to prove the following equation [closed]

Here $B_r$ presents the open ball of radius r in $R^3$. So I hope to know how to prove the following inequality. $$ \int_{B_r} |u|\le Cr^{\frac{9}{5}}\left(\int_{B_r} |u|^2\right)^{\frac{1}{5}}\left(\...
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7 votes
2 answers
336 views

Prove or disprove that the power of positive term polynomial will be eventually single peak

This is a question that a classmate asked me three years ago. Let $P(x)=\sum_{i=0}^n a_ix^i$ be a polynomial such that each $a_i>0$. Prove or disprove that there exists a positive integer $r$ such ...
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  • 509
2 votes
2 answers
127 views

Lower bound for integrals like $\int_1^{t+1}e^{-\sqrt{s}}s^{-1}ds$

Let $$I(t) = \int_{1}^{t+1}\exp\left\{-c\frac{s^{1-\beta}}{1-\beta}\right\}s^{-2\beta}ds,$$ where $c$ is some positive constant and $\beta\in(0, 1)$. Since the integral $I(t)$ given above could not be ...
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0 votes
1 answer
65 views

Constant bound for the 1 dimensional Besicovitch covering theorem on real line

I recently looked through the proof of the Gagliardo–Nirenberg Interpolation Inequality, see proof and it says that for real line $R$, there exists a sequence of open intervals $\{I_k\}$, which covers ...
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2 votes
1 answer
59 views

A simple 1-dimensional inequality, maybe Poincaré inequality or Hölder inequality?

I'm reading a paper on the classical Gagliardo-Nirenberg interpolation inequality arXiv link and there is a inequality used $$ |v-\overline{v}|\le \left\Vert v' \right\Vert_{r,I} \ell^{1-\frac{1}{r}}, ...
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3 votes
1 answer
215 views

$\int_0^1 f(\sin(1/x)) \times g(\cos(1/x)) dx \leq \int_0^1 f(\sin(1/x)) dx \times \int_0^1 g(\cos(1/x))dx? $

I have noticed experimentally that the following question has a positive answer. Is it true that for all even and convex functions $f$, $g$: $$\int_0^1 f(\sin(1/x)) \times g(\cos(1/x)) dx \leq \int_0^...
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  • 3,013
4 votes
1 answer
157 views

Positivity of real functions in two variables

Assume that $f_0,f_1,f_2$ are polynomial functions of degree two in two variables. This means that the $f_i$ are linear combinations with real coefficients of $x^2,xy,x,y^2,y,1$. Consider the function ...
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3 votes
1 answer
115 views

Concentration inequality for Hilbert space valued random variables

I have read in a paper about the following result: Let $V$ be a separable Hilbert space and $(\Omega,A_{\Omega},P)$ a probability space. Suppose that $Y_1,Y_2,...$ is a sequence of independent $V$-...
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  • 85
3 votes
2 answers
175 views

An inequality involving binomial coefficients and the powers of two

I came across the following inequality, which should hold for any integer $k\geq 1$: $$\sum_{j=0}^{k-1}\frac{(-1)^{j}2^{k-1-j}\binom{k}{j}(k-j)}{2k+1-j}\leq \frac{1}{3}.$$ I have been struggling with ...
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  • 115
5 votes
1 answer
180 views

Dimension reduction for non-negativity of elementary symmetric polynomials

Fix integers $1 \leq k \leq n$ and suppose $\mathbf{x} \in \mathbb{R}^n$ is such that $e_j(x_1,x_2,\ldots,x_n) \geq 0$ for all $1 \leq j \leq k$, where $e_j$ is the $j$-th elementary symmetric ...
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2 votes
1 answer
108 views

A kind of Gagliardo-Nirenberg inequality proof

Could any one give a proof for this inequility here? I just know its some kind of Gagliardo-Nirenberg inequility, but where does the second term come from? Thx~ $$ \int_{B_r}|u|^q\le C\left(\int_{B_r}|...
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26 votes
0 answers
1k views

Is $\int_0^\infty{dx\over x^{x^{x^x}}}<\int_0^\infty{dx\over x^{x^{x^{x^{x^x}}}}}<\int_0^\infty{dx\over x^{x^{x^{x^{x^{x^{x^x}}}}}}}<\cdots$ true?

On MSE I asked whether each of $\int_0^\infty\frac{dx}{x^x},\int_0^\infty\frac{dx}{x^{x^{x^x}}},\int_0^\infty\frac{dx}{x^{x^{x^{x^{x^x}}}}},\cdots$ was less than $2$ and received answers on bounding ...
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3 votes
1 answer
199 views

Obtaining the "best possible" inequality by tuning hyper-parameters

I encountered the following problem in one of my research projects which can be encapsulated as follows. Let's say we have a set $\mathcal{C}$ of functions $f$ defined from $\mathbb R_+$ to $\mathbb R$...
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1 vote
1 answer
108 views

Extremizers of the Sobolev inequality

Background: I am reading the paper: Best constant in Sobolev inequality by Talenti (see here) and I am trying to understand the following step. On p. 365, the author is arguing that the solutions to ...
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  • 403
5 votes
1 answer
178 views

Variance of the norm of a random variable under finite-moment assumptions

There is the following exercise in Vershynin's book on High-Dimensional Probability. Exercise 3.1.6: Let $X = (X_1, \dots, X_n) \in \mathbb{R}^n$ be a random vector with independent coordinates $X_i$ ...
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1 vote
1 answer
232 views

A discrete version of Poincaré's inequality

Given a (bounded) sequence $\{q_n\}_{n\geq 0}$ such that $\lvert q_n\rvert \leq 1$ for all $n \geq 0$ and $\sum_{n\geq 0} q_n = 0$. We can impose the condition that $\sum_{n\geq 0} \lvert q_n\rvert \...
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  • 410
4 votes
1 answer
238 views

Probability of existence of $\lambda$ such that $\lambda a_i \geq b_i$, for i.i.d random variables $a_i$'s and $b_i$'s

Suppose $a_i$'s and $b_i$'s ($1\leq i\leq n$) are i.i.d Gaussian random variables. What's the probability that a $\lambda$ exists such that $\lambda a_i \geq b_i, ~\forall i$? Actually, an upper bound ...
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