Questions tagged [inequalities]

for questions involving inequalities, upper and lower bounds.

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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-...
Navin K.'s user avatar
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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 ...
Elena Yudovina's user avatar
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2k views

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)...
math110's user avatar
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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 \...
Shivam Patel's user avatar
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271 views

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)...
user51604's user avatar
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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 ...
gandalfthegreat's user avatar
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487 views

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. $(\int_{B_{4}}e^{4u}dx)^{\frac{1}{4}} \leq S(\int_{\partial B_{4}}e^{3u}d\xi)^{\frac{1}{3}}$, ...
user13289's user avatar
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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 $\...
Guy Blanc's user avatar
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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 ...
Ryan Hendricks's user avatar
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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(...
mick's user avatar
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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)\...
Sergei's user avatar
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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{$r_{n,k,j}:=\frac{x_{n,2j}}{y_{n,k,j}\;y_{n,k+1,j}},\quad$ $...
Iosif Pinelis's user avatar
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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 \...
LL 3.14's user avatar
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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\...
Tom Solberg's user avatar
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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 ...
Steve's user avatar
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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 ...
Margaret Kail's user avatar
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131 views

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. ...
Iosif Pinelis's user avatar
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218 views

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 $\...
JoshuaZ's user avatar
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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\...
neitherother's user avatar
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165 views

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\...
combNightmare's user avatar
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123 views

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 ...
gouliguo's user avatar
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267 views

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 $ ...
Frederik Ravn Klausen's user avatar
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1 answer
706 views

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 ...
0xbadf00d's user avatar
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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 ...
Ivan Meir's user avatar
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404 views

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}\...
Artemy's user avatar
  • 650
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402 views

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 ...
D_809's user avatar
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260 views

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,...
J.Mayol's user avatar
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0 answers
143 views

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\...
Iosif Pinelis's user avatar
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0 answers
198 views

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 ...
user113925's user avatar
4 votes
0 answers
165 views

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, \...
Victor Reis's user avatar
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0 answers
200 views

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 \...
Steve's user avatar
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238 views

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,...
Anthony Carapetis's user avatar
4 votes
0 answers
631 views

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)...
Joe's user avatar
  • 759
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0 answers
343 views

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 <...
Narek Margaryan's user avatar
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127 views

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}\...
Henry.L's user avatar
  • 7,951
4 votes
0 answers
200 views

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 ...
nsoum's user avatar
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456 views

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 ...
Johnny T.'s user avatar
  • 3,547
4 votes
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626 views

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?
Aryeh Kontorovich's user avatar
4 votes
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195 views

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} ...
Iosif Pinelis's user avatar
4 votes
0 answers
202 views

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 $i$. ...
Iosif Pinelis's user avatar
4 votes
0 answers
173 views

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 ...
Destiny freedom's user avatar
4 votes
0 answers
172 views

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 ("...
Neal's user avatar
  • 826
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0 answers
143 views

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{...
Boggie Georgiev's user avatar
4 votes
0 answers
1k views

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, $...
Alexander's user avatar
4 votes
0 answers
202 views

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,\...
YotamH's user avatar
  • 41
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0 answers
183 views

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 ...
Wolfgang's user avatar
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4 votes
0 answers
461 views

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,\...
Alex R.'s user avatar
  • 4,902
4 votes
0 answers
246 views

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 ...
Jernej's user avatar
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3 votes
0 answers
315 views

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 ...
mathoverflowUser's user avatar
3 votes
0 answers
78 views

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. ...
Luís Ferreira's user avatar

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