Questions tagged [inequalities]
for questions involving inequalities, upper and lower bounds.
411
questions with no upvoted or accepted answers
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An inequality concerning non-negative integer matrices with constant row and column sums
[I posted this question on math.stackexchange a few weeks back, but no luck there so far: https://math.stackexchange.com/questions/1095659/an-inequality-concerning-non-negative-integer-matrices-with-...
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325
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Maximal inequalities for square of partial sums
Let $S_n = \sum_{i \leq n} X_i$ be the partial sums of a nice sequence of random variables $X_i$. In my application, $X_i$ is a functional of a finite-state, irreducible, aperiodic Markov chain, so ...
5
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2k
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A stronger Cauchy-Schwarz inequality for traces of compression matrices
Assume that $A$ and $B$ are contractions, so
$I-AA^T$ and $I-BB^T$ are positive-definite matrices. Let
$C=(I-AB^T)^{-1}(I-AA^{T})(I-BA^{T})^{-1}$, and show that:
$$Tr\left(\frac{1}{1-AA^T}\right)...
5
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136
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On the comparison of Egyptian fractions of two kinds
I posted the question on MSE here but it did not get any answer.
Consider $$S(n)=\left\{(a_1 ,a_2,a_3, \dots, a_n)\mid a_1\le a_2\le\cdots\le a_n, \; \sum_{r=1}^{n}\frac{1}{a_r} = 1\right\} \subset \...
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271
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Carleman estimates on monotonicity formulas
I am trying to derive a monotonicity formula for a certain Dirichlet critical point (or even maybe a minimizer) of an energy of the type, say for simplicity, an energy of the from
$$\int_{B_r} (Ae(u)...
5
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337
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Maximizing Renyi entropy for a certain channel
The channel under consideration is $T = A + B$, where $A$ and $B$ take on values in $\{0, 1\}$ according to a probability mass function. Let (joint) random vector $(A_1, A_2,\ldots, A_n)$ be denoted ...
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487
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Any similar inequality in literature?
I got the following inequality:
$B_{4}$ is the unit ball in $R^{4}$, $\partial B_{4}$ is its boundary.
$(\int_{B_{4}}e^{4u}dx)^{\frac{1}{4}} \leq S(\int_{\partial B_{4}}e^{3u}d\xi)^{\frac{1}{3}}$,
...
4
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137
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Comparing the slackness of Jensen's inequality for some coupled random variables
Let $f:\mathbb{R} \to \mathbb{R}$ be convex and $X,Y$ be random variables with a coupling such that $\mathbb{E}[Y\mid X=x] = x$. A straightforward application of Jensen's inequality gives that $\...
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247
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A proof of the Gagliardo-Nirenberg interpolation inequality using Jensen's inequality
A brief look at the statement of Gagliardo-Nirenberg interpolation inequality would suggest that there should exist a proof by a clever use of Jensen's inequality. In other words, there should be a ...
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$f(n) = \frac{n^2 + n + 4}{2}$, $g(f(n)) = f(g(n))$ such that $g(n)$ is an integer
Let $n$ be a strict positive integer and let's define an integer sequence $f(n)$ :
$$f(n) = \frac{n^2 + n + 4}{2}$$
so
$$
\begin{split}
f (\Bbb N)& \triangleq {3,5,8,12,17,23,30,38,47,\ldots}\\
f(...
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199
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Integral inequality of Polya
In the book Math Problems AMM (1957), Problem 230, there is the next inequality of D. Polya:
let $a,b>0$, $0\leq x \leq a $,
$f(x)$ --- being not a linear function, and $f(0)=0$, $f(a)=b$, $f(x)\...
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123
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A "counterbalancing" trigonometric sum inequality
Is it true that
$$s_{n,k}:=\sum_{j=1}^{n-1} r_{n,k,j}
<0$$
for all natural $n\ge2$ and all natural $k\in\{1,\dots,n-1\}$,
where
$$\text{$r_{n,k,j}:=\frac{x_{n,2j}}{y_{n,k,j}\;y_{n,k+1,j}},\quad$
$...
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141
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Approximation by gaussian mollification in Sobolev spaces
I have been trying to find a good estimate on the constant in the inequality (in dimension $d=3$ to simplify)
$$\label{0}\tag{0}
\|(1-e^{t\Delta})f\|_{L^{5/3}(\Bbb R^3)} \leq C_0\,t^{2/5}\, \|\nabla \...
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122
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Log of a truncated binomial
Let $X$ follow a binomial distribution with $n$ trials and success probability $p$, and let $0\leq k\leq n$. Are there any natural approximations or bounds for the ratio $$\frac{\boldsymbol{E}\log\...
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164
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Questions on a cone defined by elementary symmetric functions
Let $1\le k\le n$ be given integers. Define the following cone
$$\Gamma_k=\{\lambda\in\mathbb{R}^n| S_j(\lambda)>0, j=1, ..., k\},$$
where $S_j(\lambda)$ is the $j$th elementary symmetric function ...
4
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94
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Is this conjecture about the binomial and beta distributions true?
Let $X$ follow a binomial distribution with parameters $n$ and $p$, and also fix $k$ such that $1<k<n$. Define
$$a = \mathbb{E}(X-k)^+$$
and
$$b = \mathbb{E}\log\binom{X}{(X-k)^+}$$
where the ...
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An inequality for three iid random variables with a log-concave density
It was previously shown that
$$H\ge cG,\tag{1}$$
where $c:=1/14334$,
$$G:=E|X-Y|,\quad H:=E|X-Y|-\tfrac12\,E|X+Y-2Z|,$$
and $X,Y,Z$ are independent random variables with the same log-concave density.
...
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218
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An inequality involving a polynomial and its first and second derivative
Given a real polynomial $P(x)$ all whose roots are real, it is not hard to show that
$$P(x)P''(x) \leq P'(x)^2 \, \, \, \, (1).$$
Proof sketch: Assume that $P(x) = \prod_{i=1}^n (x-r_i)$. Look at $\...
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405
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Simmons' inequality on binomial random variables
Fix two positive integers $n,m$ such that $n>2m$ and $n\ge 3$, and let $X\sim \text{Bin}(n,\frac{m}{n})$ be a binomial random variable. For each $i\in \{1,\ldots,m\}$, set
$\alpha_i = \mathbb{P}(X\...
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165
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Pedestrian proof of Gaussian chaos for order-two polynomial?
Let $\ell \geqslant 1$. Let us consider $(g_n)_{n \in \mathbb{N}}$ identically distributed independent real gaussian variables and real number $(a_{n_1,\dots n_{\ell}})_{(n_1, \dots, n_{\ell}s)\in\...
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An inequality about metric spaces
I started studying this article(《$L^2$ CURVATURE BOUNDS ON MANIFOLDS WITH BOUNDED RICCI CURVATURE》) about 3 months ago: arxiv.org/abs/1605.05583
In this article, there is a seemingly simple assertion ...
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267
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Examples of measures that satisfy FKG, but not the FKG lattice condition
Let a percolation measure be a measure on $\{0,1\}^n$. We have a natural partial order on $\{0,1\}^n$ given by comparing all coordinates. An event $A$ is called increasing if for all $ \omega \in A $ ...
4
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answer
706
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Bound the operator norm of the Fréchet derivative of a Lipschitz function in this setting
I want to find a bound for the operator norm of the Fréchet derivative of a Lipschitz continuous function in the following setting:
Let
$E$ be a $\mathbb R$-Banach space;
$v:E\to[1,\infty)$ be ...
4
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97
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Closed curves with minimal total curvature in the unit circle
Chakerian proved in this paper that a closed curve of length L in the unit ball in $\mathbb{R}^n$ has total curvature at least L.
In this later paper Chakerian gave a simpler proof and noted that ...
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404
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Inequalities for trace/eigenvalues of product of multiple 2x2 matrices
Consider the matrix product $\prod_i^n A_i$,
where each $A_i$ is a $2\times2$ matrix having the form $A_i = \left( \begin{smallmatrix} \lambda + \alpha_i & -\beta_i \\ 1 & 0\end{smallmatrix}\...
4
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402
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A symmetrization-majorization inequality for i.i.d. zero mean random variables
Let $k\geq 2$ and
$$f(x_1,\ldots,x_k)=\Bigl(\prod_{i\leq k}(1+x_i)+\prod_{i\leq k}(1-x_i)\Bigr)\log\Bigl(\prod_{i\leq k}(1+x_i)+\prod_{i\leq k}(1-x_i)\Bigr)-k(1-x_1 x_2)\log(1-x_1 x_2).$$
If random ...
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260
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An inequality in harmonic analysis with the BMO flavour
I am asking myself this question (which seems to be a natural generalization of Remark 4.4 of these lecture notes).
Question. Let $I_s, s \in \mathcal{S}$ be a collection of intervals included in
$[0,...
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A Pythagorian inequality characterization of inner-product spaces
Let $(X,\|\cdot\|)$ be a real normed space. For any points $A$ and $B$ in $X$, let $AB:=\|A-B\|$. Suppose that for any points $A$ and $B$ in $X$ and any straight line $\ell\subseteq X$ such that $B\...
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198
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Strange inequality relating Binomial pmf and cdf
I'm encountering a strange inequality I need to prove, relating the Binomial pmf and cdf.
Suppose we have $n$ coin flips, and fix an arbitrary $k \le n/2$ heads. Suppose further that we have some ...
4
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165
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Sum of eigenvalues is nonpositive
Let $A$ be a symmetric positive semidefinite $n \times n$ matrix. How can I show that the sum of the largest $n-k+1$ eigenvalues of $A - k\cdot \textrm{diag}(A)$ is nonpositive, for any $k \in \{1, \...
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200
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Sharp constant for inequality with convex functions
This is a follow up to this question, where the optimal constant was left open.
Let $P \subset \mathbb{R}^n$ be bounded, convex, and open. Let
\begin{equation}
\mathcal{H} := \{f : P \rightarrow \...
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238
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Space-time Poincaré inequality for solution of parabolic equation
If $u : \mathbb R^n \to \mathbb R$ is a smooth enough function then on any Euclidean $n$-ball $B_R$ of radius $R$ we have the very well-known Poincaré inequality
$$ \int_{B_R} |u - \bar u|^2 \le C(R,...
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Problem with an integral equation taken from a paper
I am reading a paper (the 2015 paper by A. Falkowski and L. Slominski Stochastic Differential Equation with Constraints Driven by Processes with Bounded $p-$variation, page 353, proof of the Lemma 3.1)...
4
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343
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Fractional integral inequality (Hardy-Littlewood-Sobolev)
I am investigating the following integral
\begin{equation}
I^*(x) = \int_{\mathbb{R}} \frac{f(y) \ln |y-x| }{|y - x|^{\mu}} \, dy
\end{equation}
where $f \in L_p(\mathbb{R})$, $ 1 < p < q <...
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Is there an example that both Berry-Essen bound and DKW bound are attained?
The Berry-Essen bound stated that
$$\sup _{{x\in {\mathbb R}}}\left|\widehat{F_{n}(x)}-\Phi (x)\right|\leq C_{0}\cdot \psi _{0}$$
where $\psi _{0}(n)={\Big (}{\textstyle \sum \limits _{{i=1}}^{n}\...
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200
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Minkowski determinant inequality for the Fuglede-Kadison determinant
For positive-semidefinite matrices $A, B$ in $M_n(\mathbb{C})$, the Minkowski determinant theorem tells us that $\det(A+B)^{\frac{1}{n}} \ge \det(A)^{\frac{1}{n}} + \det(B)^{\frac{1}{n}}$. For a ...
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456
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An inequality from Green-Tao "The primes contain arbitrary long arithmetic progressions"
I have been reading "The primes contain arbitrary long arithmetic progressions" by Green and Tao (The version on the ArXiv: https://arxiv.org/pdf/math/0404188v6.pdf) going through the details and ...
4
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626
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Weyl-type inequality for non-Hermitian matrices?
What is the weakest known condition under which a Weyl-type eigenvalue perturbation inequality holds? Does some analogue hold for normal matrices, for example?
4
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195
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Dynamics of an inequality
The dynamics $D\ni(r_i,r_{i+1})\mapsto(r_{i+1},r_{i+2})\in D$ on the set $D:=\{(x,y)\in\mathbb{R}^2\colon x>0,y>x^2/2\}$ is given by the recurrence
$$r_{i+2}=\frac{r_{i+1}^2}2+\frac1{r_{i+1}^3}
...
4
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202
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Dimension reduction for low-order moments of Rademacher-weighted sums of vectors
Let $x_1,\dots,x_n$ be vectors in a Euclidean space $H$. Let $\varepsilon_1,\dots,\varepsilon_n$ be independent Rademacher random variables (r.v.'s), so that $P(\varepsilon_i=\pm1)=1/2$ for all $i$.
...
4
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173
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Do more generalizations of Schur's inequality exist?
I meet this following problem
If $$n\ge 3,\sum_{i=1}^{n}\left(\prod_{j\neq i}(a_{i}-a_{j})\right)\ge 0$$
where $a_{i}$ are real numbers.
when $n=3$, it is Schur's inequality
so which $n$ such ...
4
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172
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Finding an explicit constant in finite element error estimates
Background: In a finite element approximation to the solution of a linear PDEs, estimates on the order of convergence of the approximation to the solution rely on a theorem of Bramble and Hilbert ("...
4
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143
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A "gradient" weak Harnack inequality for quasilinear elliptic equations
Suppose we are in the following loosely described setting:
we have a non-negative supersolution $h$ of the following elliptic equation:
\begin{equation}
\Delta h + \|\nabla h\|^2 + f(x) \geq 0
\end{...
4
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1k
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Total variation and Hellinger distance inequality between truncated Gaussians
We know that the total variation distance, $d_{TV}(P,Q) = \frac{1}{2}\left|\left|P-Q\right|\right|_1$, between any two distributions $P$ and $Q$ is lower bounded by their squared Hellinger distance, $...
4
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0
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202
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Optimization problem involving Multivariate Normal
I use $\phi(t)$ to describe the standard normal distribution density and $\Phi(t)$ as the normal distribution CDF and would like to prove that for all
$n\geq3$, the function:
$$h(\mu_{1},\ldots,\...
4
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0
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183
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A challenging non homogenous fractional inequality
I have posted this question on Stackexchange but it has received no answer so far. It is a challenging generalization of several difficult inequalities, where none of the usual methods used in ...
4
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0
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461
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System of Equations Upper Bound
I asked a related question on math.stackexchange here but would now like to obtain a better bound. This question comes from a graph theory problem. I'll restate the new question here:
For $i=1,2,\...
4
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246
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Applying the amplification trick + probabilistic method on connected graphs
First of all, I apologise if my question is not very specific. I am not really looking for a concrete answer but rather some hints and pointers what could be used/applied in this situation. A concrete ...
3
votes
0
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315
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Some specific case of the abc-conjecture related to the Mason-Stothers proof of Serge Lang for polynomials?
I stumbled upon a version of the Mason-Stothers theorem, which can be modified to make a similar statement for natural numbers, which I think I can prove here. It is not the abc-conjecture, but it has ...
3
votes
0
answers
78
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Finite dimensional distribution of a stochastic process Lipschitz on every relatively compact set
Let $X_t$ be a Markovian Itô diffusion process, defined by an SDE
\begin{equation}
dX_t = \mu(X_t)\,dt + \sigma(X_t)\,dW_t\,.
\end{equation}
Let $f(x,t|x_0,0)$ denote its transition density function. ...