Questions tagged [inequalities]
for questions involving inequalities, upper and lower bounds.
1,659
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Upper bound for complex integral
I am interested in obtaining a good upper bound for the absolute value of the following integral
$$
\left| \int_{0}^{\pi/3} e^{-itn} \left( 1-e^{it} \right)^{k} dt \right|,
$$
when $n>k>0$ are ...
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2
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Approximation of the Gaussian Mills' ratio
Let $R(t) = \frac{1 - \Phi(t)}{\varphi(t)}$ where $\Phi, \varphi$ represent the CDF and PDF of the standard Normal distribution, respectively.
I am interested in approximations of the function for $t &...
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1
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1
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Inequalities involving entropy: quantum discord and mutual information
My question is inspired by the following paper of Olivier and Żurek but for this question to be self-contained I will recall all the necessary definitions: for a quantum state $\rho$ we define the ...
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1
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Optimal constant comparing $f(1/2)$ and $\|f\|_2$ when $f$ is $t$-Hölder?
Suppose that $f \colon [0, 1] \to \mathbb{R}$ is $k$ times continuously differentiable and Holder in the sense that for some
$t = k + \beta$, where $\beta \in (0, 1]$ and $k$ is a nonnegative integer ...
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2
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Is the passage in argument of existence solution of PDE correct?
The passage below is from a fixed point argument in Wang's paper enter link description here pg 566. At the end of the $I_1$ calculation, it somehow makes the following estimate
$$C\|\phi\|_X (e^{-|x|^...
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2
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165
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Relating function value to $L^2$ norm in Holder space
Suppose that $f \colon [0, 1] \to \mathbb{R}$ is $k$ times continuously differentiable and Holder in the sense that for some
$t = k + \beta$, where $\beta \in (0, 1]$ and $k$ is a nonnegative integer ...
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1
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394
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Inequality of inclusion-exclusion term
This question was initially posted on math.stackexchange.com but did not receive any answers for half a week.
While analyzing the properties of an algorithm I am working on (I'm a computer scientist), ...
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1
answer
165
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What is the minimum and the maximum perimeter of a triangle with area $x$ that can be inscribed in a circle?
This question was posted in MSE but is still open hence posting in MO.
The area of the largest triangle that can be inscribed in a circle of raidus $1$ is $\displaystyle \frac{3 \sqrt{3}}{4}$ for a ...
- 4,638
1
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1
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Kolmogorov inequality for Bernoulli random variables
This question is also asked on math stackexchange. The question is about one inequality which shows in Kolmogorov's paper (inequality (3.1)) but is not proved. The inequality says that, if we assume $...
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Is there a sharper Golden–Thompson inequality?
For any two Hermitian matrices $A$ and $B$, the Golden–Thompson inequality
$$\mathrm{Tr} (e^A e^B) \geq \mathrm{Tr} \, e^{A + B}$$
holds, and it is known to be a strict inequality whenever $[A, B] \...
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3
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1
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Derive distributional inequalities from pointwise estimates
My question is how to prove the following claim:
Suppose that $E$ is an algebraic set in $\mathbb{R}^n (n\ge3)$ with dimension $\le n-2$, and $u$ is locally Lipschitz continuous
on $\mathbb{R}^n$. If ...
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1
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750
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Find a special integer coefficients polynomial which takes small absolute value on [0,4]
The question is easy to state: Is there a non-constant $f\in\mathbb{Z}[x]$ such that for all $x\in [0,4]$, we have $|f(x)|\leq 1$? I do not know where to find a useful reference for it.
I did a few ...
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Minimizing $P=\frac{\sqrt{5a+8bc}}{8a+5bc}+\frac{\sqrt{5b+8ca}}{8b+5ca}+\frac{\sqrt{5c+8ab}}{8c+5ab}.$ [closed]
Olympiad inequality. Let $a,b,c\ge 0: ab+bc+ca=1.$ Find the minimal value $P$ of $$f:=\frac{\sqrt{5a+8bc}}{8a+5bc}+\frac{\sqrt{5b+8ca}}{8b+5ca}+\frac{\sqrt{5c+8ab}}{8c+5ab}.$$
Note: Often Stack ...
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Ratio of the constants of the Marcinkiewicz–Zygmund inequality for p=1
The Marcinkiewicz–Zygmund inequality states that
$$
{\displaystyle A_{p}E\left(\left(\sum _{i=1}^{n}\left\vert X_{i}\right\vert ^{2}\right)_{}^{p/2}\right)\leq E\left(\left\vert \sum _{i=1}^{n}X_{i}\...
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Weighted version of the Gagliardo Nirenberg inequality
I'm searching for a weighted Gagliardo-Nirenberg inequality similar as in https://arxiv.org/pdf/1307.1363.pdf where the weight is a power of the last component. Is there an inequality of the form
$$
\...
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1
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A question about the prime counting function
I was playing around with the prime counting function and came across something that seemed correct to me, maybe it's already been proven but I don't know so I decided to ask here.
maybe a stupid ...
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300
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Recover unknown vector through shifted argmax queries
$\DeclareMathOperator*{\argmax}{arg\,max}$
I am interested in finding an efficient algorithm for the following problem:
Let $x \in [0,1]^n$ be some vector, with $x_n = 1$.
We want to recover $x$, ...
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201
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Erdős–Rényi random graphs — reproducing 2 inequalities
In Erdős and Renyi's 1959 paper On random graphs I
, I'm trying to reproduce, starting from Eq.\eqref{1} in their paper, the two inequalities that appear in Eq.\eqref{2}.
Eq.\eqref{1} is:
$$
P \le \...
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0
answers
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How to prove this weighted sum inequality with non-increasing sequences?
Problem
I have two non-increasing sequences, $X = (x_1, x_3, x_5, \ldots, x_{n-1})$ and $Y = (y_1, y_3, y_5, \ldots, y_{n -1})$, $n$ is an even integer. I want to prove this inequality:
$$
\sum_{i=1}^{...
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Maximizing a sum minus its maximal summand
This is a followup to a question that appeared on m.SE:
Maximize $\displaystyle f(\pi)=\left(\sum_{i=1}^{n}{i\pi_i}\right)-\max_{1\le i\le n}{(i\pi_i)}$ over permutations $\pi\in S_n$.
The problem ...
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Explicit upper and lower bounds for a certain support function
Let $a_1 \geq a_2 \geq \cdots \geq a_n \geq 0$ be a sequence of nonincreasing nonnegative real numbers. Define the set, for $t > 1$,
$$
B_t = \Big\{b \in \mathbb{R}^n : b_i \geq 0, \sum_i b_i^2 \...
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How to get this inequality in Santambrogio's book about optimal transport?
Let $\hat{\varrho}, \tilde{\varrho}$ be probability density functions on $\mathbb R^d$ where $\tilde{\varrho} \in L^{\infty} (\mathbb R^d)$. For $\varepsilon \in [0, 1]$, we define $\varrho_{\...
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Inequality for extremal values of product of Hermitian matrices
I am looking for a reference to verify the following inequality, where $X$ and $Y$ are Hermitian positive semidefinite matrices:
$$
\lambda_n(X^{1/2}YX^{1/2}) = \lambda_n(XY) \leq \lambda_n(X)\...
6
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1
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A Poincaré-like inequality
Is it true that for some real $K>0$ and all real $u\in C_0^\infty((0,1))$ we have
$$\int_0^1 (u'(x)^2+u(x)^2)\,dx\,\int_0^1 u(x)^2\,dx
\le K\Big(\int_0^1 x\,u'(x)^2\,dx\Big)^2\text{ ?}$$
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Most elementary proof showing that exponential growth wins against polynomial growth
This question is motivated by teaching : I would like to see a completely elementary proof showing for example that for all natural integers $k$ we have eventually $2^n>n^k$.
All proofs I know rely ...
1
vote
0
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59
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A possible upper bound for a function that satisfies a singular integral inequality
I am currently working on an analysis problem in fractional calculus and after some work I have encountered the following inequality:
$$
|v(s)|~\leq \epsilon+\beta \int_{0}^{1}|s-x|^{\alpha-1}\left(
|...
0
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1
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For which $p>p^*$ does the inequality $\cos^2(−π/4+π/p)>1/2+π/p^2$ hold?
I have the inequality $\cos^2(−\frac{\pi}{4}+\frac{\pi}{p})>\frac{1}{2}+\frac{π}{p^2}$ and have found that it holds for $p>p^*$, where $p^*$ is some positive number (around ~2.8). I'm looking ...
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Friedrich's second inequality for functions with zero average
Friedrich's second inequality (or Maxwell Estimates or Gaffney’s inequality in the literature) is referred as follows: for all $\mathbf{u} \in H^1(\Omega)^2$ satisfying either $\mathbf{n} \cdot \...
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Does "perpendicular phase incoherence" satisfy the triangle inequality?
I asked this question at https://math.stackexchange.com/q/4783968/222867, but even after a 200-point bounty, no solution was provided, only some thoughts regarding possible directions. So I'm now ...
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Bounds tighter than the additive Chernoff
Additive Chernoff
Suppose $X_1, \ldots, X_n$ are i.i.d. random variables, taking values in $\{0,1\}$. Let $p=\mathrm{E}\left[X_i\right]$ and $\varepsilon>0$.
\begin{gather*}
\operatorname{Pr}\left(\...
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3
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Jensen-like inequality for random matrix: $\Bbb E[\det X^2]\ge\det\Bbb E[X^2]$
Let $X\in M_n(\Bbb R)$ be a random matrix with iid elements following a continuous distribution.
What are the necessary and sufficient conditions for $$\Bbb E[\det X^2]\ge\det\Bbb E[X^2]$$ to hold? Is ...
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Is it possible obtain an identity of the type $\dfrac{|x-y|}{t} + \dfrac{|y-z|}{s} \approx a(t,s) | y - b(t,x,s,z)| + c(t,s)|x-z|$?
using the Euclidean inner product, and by completing squares it is possible to prove that
$$\dfrac{|x-y|^2}{t} + \dfrac{|y-z|^2}{s} = \dfrac{t+s}{ts} \left| y - \dfrac{sx+tz}{s+t} \right|^2 + \dfrac{|...
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estimate involving Gaussian data
Let $x\in \mathbb{R}^{d}$, $d\geq 1$ and $p>\frac{2}{d}$. For any $R>0$ and any $0<s<\frac{dp}{2}-1$
\begin{align}
&\int_{\mathbb{R}}\int_{|x|>R}\left(\frac{1}{1+t^2} \right)^{\frac{...
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Bounding a sum of reciprocals of square-free integers
(Cross-posted from MSE, as the question did not get any clear answer)
Fix positive integers $k$ and $n$. Let $N_1,\dots,N_r$ be all the integers less than or equal to $n$ that are squarefree and have ...
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108
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Integral inner product with exponential function
Suppose on some unknown interval $[0, I]$ we have non-negative functions $f, g : [0, I] \rightarrow \mathbb{R}^{\geq 0}$.
If we know that
\begin{aligned}
\int_0^I f & = c \\
\int_0^I e^f & = e^...
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votes
1
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361
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On norm of the Sobolev space $H^2(\Omega)$, $\Omega \subset \mathbb{R}^n; n \geq 2$
Let the Sobolev space $H^2(\Omega)$ be defined with the norm $\|u\|_{H^2(\Omega)}=\Big(\sum_{|\alpha|\leq 2})\|D^{\alpha}u\|^2_{L^2(\Omega)}\Big)^\frac{1}{2}$.
I have found in several research ...
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Do these cousins of permanents satisfy the following inequality?
Let $H$ denote an $n$ by $n$ hermitian positive semidefinite matrix. Let $G$ and $K$ be two subgroups of the symmetric group $\Sigma_n$. Define
$$ f_{G, K}(H) = \sum_{(\sigma, \tau) \in G \times K} \...
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Sub-Gaussian random variables and convex ordering
Suppose that $X$ is a $1$-sub-Gaussian real-valued random variable, i.e. for all $t \in \mathbf{R}$, it holds that $\log \mathbf{E} \exp \left( t X \right) \leqslant \frac{1}{2} t^2 $.
Does there ...
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1
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Possibility of bounding one functional by another functional (under certain constraints)
Suppose that we consider a class of $L^2(\mathbb{R}_+)$ functions $h$ such that $h$ can be expressed as a difference of two cumulative distribution functions $F$ and $G$ (whose corresponding densities ...
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0
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Help with understanding a proof of existence of solutions
In El Fatini and Boukanjime "Stochastic analysis of a two delayed epidemic model incorporating Lévy processes with a general non-linear transmission" paper, can someone give a detailed ...
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0
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Can we bound the squared Gaussian curvature of genus three triply periodic minimal surfaces?
Assume that $\mathcal{M}$ is a balanced triply periodic minimal surface of genus 3, embedded in a flat torus $T^3=\mathbb{R}^3/\Lambda$ for a lattice $\Lambda$ with volume 1. I want to understand the ...
11
votes
1
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Understanding the application of two inequalities?
I am reading the paper "The long-time behaviour of a stochastic SIR epidemic model with distributed delay and multidimensional Levy jumps" by Driss Kiouach and Yassine Sabbar.
I have two ...
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votes
1
answer
117
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Simplified upper bounds for moment-generating function of symmetrised random variable
Let $X$ be a nonnegative random variable such that $\mathbf{E} \left[ \exp X \right] < \infty$. For $\theta \leqslant 1$, an appropriate application of Jensen's inequality, yields that
\begin{align}...
- 706
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1
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219
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Equivalence between the $L^2$ norm and the $L^2$ norm of Laplace transform
It is well-known that the Laplace transform, defined by $$\mathcal{L} \colon f(x) \in L^2(\mathbb{R}_+) \to \hat{f}(\xi) \in L^2(\mathbb{R}_+)$$ via $$\hat{f}(\xi) = \int_{\mathbb{R}_+} f(x)\,\mathrm{...
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votes
4
answers
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views
Understanding the condition $\frac{1}{p} + \frac{1}{q} = 1$ in the estimate $xy \le \frac{1}{p}x^p + \frac{1}{q}y^q$
I just read a proof of Holder's inequality in measure theory, which boils down to the following inequality:
$$xy \le \frac{1}{p}x^p + \frac{1}{q}y^q$$
where $x,y\ge 0$ and $\frac{1}{p} + \frac{1}{q} = ...
- 7,161
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1
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107
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$L^1$ error between indicator function and smoothed out version
For a large parameter $r>0$, consider the indicator function $1_{[-r,r]}$ and its convolution with the (normalized) Gaussian $\frac{1}{\sqrt{\pi}}e^{-x^2}$, that is,
$$f_r(x) = \frac{1}{\sqrt{\pi}}\...
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2
votes
1
answer
186
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A question about equivalence of weighted Sobolev space norm in S. Benzoni-Gavage and D. Serre's book
This question may not be at the research level, but it has really bothered me for a long time. The following space is used for handling initial boundary value problem for first order hyperbolic ...
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2
answers
2k
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One specific inequality
How to prove this inequality $$\left(a+\frac{1}{2} \left(a b-\sqrt{a^2-1} \sqrt{b^2-1}\right)\right)^{3/4}-\frac{\sqrt{3} \cos\left[\frac{3 (\pi -t)}{4}\right]}{2 \left(\frac{1}{2}+b\right)^{1/4}}-(a+\...
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0
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Inequality concerning the imaginary parts of a recurrent sequence, Laplacian eigenvectors
Let $u=(u_1,\dots,x_n)\in\mathbb{C}^n$ be a sequence that satisfies the cyclic recurrence
$$
\lambda+1 =a_{i-1}\frac {u_{i-1}}{u_i} + (1-a_{i+1})\frac{ u_{i+1} }{u_i }
$$
with $a_i \in (0,1)$ and $\...
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2
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Is there any bilinear Poincaré/Sobolev inequality?
Is the following, I call it bilinear Poincaré inequality, true?
Let $\Omega$ be an open bounded set in $\mathbf R^n\DeclareMathOperator{\dL}{d\!}$. There exists $C > 0$ such that for any $u, v \in ...
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