# Questions tagged [inequalities]

for questions involving inequalities, upper and lower bounds.

1,370
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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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### 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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### 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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### 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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36
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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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### 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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### 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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### 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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### 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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### 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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67
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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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### 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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1
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117
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### 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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106
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### 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
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78
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### 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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862
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### 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
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83
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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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78
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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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591
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### 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}...

4
votes

2
answers

228
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### 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
$$ \frac{\partial^2}{\partial x_1^2}u <0 $$
...

4
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170
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### 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 ...

2
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### 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{...

4
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1
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### 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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### 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.

2
votes

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### 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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### 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 ...

3
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1
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76
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### 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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### 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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votes

1
answer

111
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### 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 ...

3
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### 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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1
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### 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(\...

7
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2
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336
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### 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 ...

2
votes

2
answers

127
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### 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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1
answer

65
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### 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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### 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}}, ...

3
votes

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215
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### $\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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### 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 ...

3
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115
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### 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 $V$-...

3
votes

2
answers

175
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### 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 ...

5
votes

1
answer

180
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### 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 ...

2
votes

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108
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### 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}|...

26
votes

0
answers

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### 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 ...

3
votes

1
answer

199
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### 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 $\mathbb R$...

1
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1
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108
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### 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 ...

5
votes

1
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178
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### 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 $X_i$ ...

1
vote

1
answer

232
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### 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 \...

4
votes

1
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238
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### 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 ...