The inequalities tag has no wiki summary.

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115 views

### Placing Bounds on Correlation/Covariance Through Correlation with an Intermediate Variable

I am trying to make the most of computations that have already been performed in previous steps of an algorithm. Throughout this problem statement I am only mentioning correlation, but I think it is ...

**5**

votes

**1**answer

285 views

### A question on the Mahler conjecture

In its asymmetric version, the Mahler conjecture states that if $K \subset \mathbb{R^n}$ is a convex body containing the origin as an interior point and
$$
K^* := \{y \in \mathbb{R}^n : \langle y, x ...

**7**

votes

**1**answer

471 views

### A spectral inequality for positive-definite matrices

Question. Given a positive-definite $n \times n$ matrix $A = (a_{ij})$ with eigenvalues
$$
\lambda_1 \leq \cdots \leq \lambda_n ,
$$
is there a sharp upper bound for the product $\lambda_2 \cdots ...

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votes

**0**answers

53 views

### Minimizing/Maximizing the tail of the convex combinations of Chi Squared i.i.d random variables

Consider $N$ i.i.d random variables, $X_{1}, X_{2}, \ldots, X_{N}$ , that are chi-squared of degree $K \geq 2$. Also consider the following 3 vectors:
\begin{eqnarray*}
\bar{a} &=& ...

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votes

**1**answer

186 views

### Inequality of Partial Taylor Series

Hi,
For a given $\theta < 1$, and $N$ a positive integer, I am trying to find an $x > 0$ (preferably the smallest such $x$) such that the following inequality holds:
$$\sum_{k=0}^{N} ...

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**0**answers

337 views

### Noncommutative arithmetic mean geometric mean inequality and symmetric polynomials

While analyzing convergence speed of stochastic-gradient methods for convex optimization problems, Recht et al (2011) posed a tantalizing conjecture. It seems quite tricky, so after having struggled a ...

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votes

**1**answer

247 views

### Chernoff-Hoeffding bound for complex values

Consider the Chernoff-Hoeffding bound, stated as follows: Let $X_1, \dots, X_K$ be i.i.d. real-valued random variables with expectation value
$\mu$ and satisfying $|X_i| \le b$.
Let $\epsilon > 0$. ...

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**1**answer

881 views

### What goes wrong for the Sobolev embeddings at $k=n/p$?

For $u\in W^{k,p}(U)$, where $U\subseteq\mathbb{R}^n$ is open and bounded with $C^1$-boundary, we have the celebrated Sobolev inequalities:
If $k < n/p$ then $u\in L^q(U)$ for $q$ satisfying ...

**1**

vote

**1**answer

352 views

### for what arguments the function reaches maximum?

Hi,
What is the maximum of the following function?:
$f(x_i,w_i)=\frac { \sum w_i}{ \sum \frac {w_i}{x_i} } - \frac{ 1 - \prod \left ( 1 - w_{i}\right )}{ 1 - \prod \left ( 1 - \frac{w_{i}}{ ...

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**2**answers

482 views

### The fraction of the sphere a fixed distance from a subspace

The following problem has a beautiful geometric interpretation in terms of the proportion of points on the Euclidean sphere in $\mathbb{R}^d$ that lie at least a certain distance away from a ...

**12**

votes

**5**answers

782 views

### Understanding Gibbs's inequality

Short version
Gibbs's inequality is a simple inequality for real numbers, usually
understood information-theoretically. In the jargon, it states that
for two probability measures on a finite set, ...

**2**

votes

**1**answer

266 views

### concentration inequality for averages of dependent random variables

Let $X \in R^n$ be a random vector such that
$$P(|X_i| > \epsilon) \le e^{-\epsilon^2}$$
What is a tight bound on
$$P(\sum_{i=1}^n |X_i| > \epsilon)$$
and on
$$P(\max_{1\le i\le n} |X_i| ...

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votes

**1**answer

243 views

### An interpolation inequality.

For all $s>0$ define for $\epsilon\in(0,1)$ the function:
\begin{equation}
g(\epsilon)=\sum_{k=0}^{\infty}(1+k)^s(\sqrt{1-\epsilon})^k.
\end{equation}
Prove that $\exists C>0$ and $\phi(s)$ such ...

**2**

votes

**1**answer

447 views

### Combinatorial Inequality

Consider a set of $2^n-1$ non negative integers $S= ${$ a_{i,j}|1\le i\le n; 1\le j\le 2^{i-1} $} such that:
\begin{align}{}
1.\ \ &a_{i,j}\le 2^{n+1-i} \\\\
2.\ \ &a_{i,j}\le a_{i-1,j} \\\\
...

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**1**answer

342 views

### Inequality on Trigonometric polynomials

My question comes from trying to understand a technical step in this paper by Bourgain.
Let $R,L$ be positive integers and let $f(x)=\sum_{|n|\leq RL}a_ne^{2\pi inx}$ be a trigonometric polynomial. ...

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**0**answers

90 views

### Multivariate polynomial with positive coefficients

This question was originally asked at stack exchange (http://math.stackexchange.com/questions/292922/multivariate-polynomial-with-all-coefficients-positive), but did not receive any feedback for more ...

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**2**answers

289 views

### Spherical Bessel functions

I wish to show that
$|j_n(x)| < \frac{1}{\sqrt{x}}$
for $n=0,1,2,\ldots$ and $x>0$, where $j_n$ is the spherical Bessel function of the first kind.
Experimenting with Matlab I am sure that ...

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votes

**2**answers

340 views

### Efficient algorithm finding 'a' solution of system of linear inequalities

I'm working on rational number field $\mathbb{Q}$.
Is there an efficient algorithm finding a solution of system of linear inequalities?
In many computer algebra systems like Sage or Maple,
there ...

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votes

**1**answer

217 views

### another diameter-perimeter-area inequality

Recently I learnt that $$
\inf\frac{diam(C)(per(C)-2diam(C))}{area(C)}=0$$ where the infimum is taken over all plane convex bodies $C$ (say, with non-zero area). In other words, there is no ...

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**2**answers

262 views

### a diameter-perimeter-area inequality for convex figures

Is the following inequality known? I believe it's true, but I could find no reference.
For any convex body $C$ in the plane
we have
$$\left(4-\frac{8}{\pi}\right)area(C)\leq
> ...

**2**

votes

**1**answer

253 views

### on an inequality of Brezis-Lieb

In their 1983 JFA paper Brezis and Lieb have shown, among many other things, a PoincarÃ©-type inequality: in the case of a harmonic function $f$ on a bounded domain $\Omega$, their inequality ((3.14) ...

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**2**answers

397 views

### What is $A+A^T$ when $A$ is row-stochastic ?

This is motivated by this MO question.
If $A\in{\bf M}_n({\mathbb R})$ is row-stochastic (entrywise non-negative, and $\sum_j a_{ij}=1$ for all $i$), then $M:=A+A^T$ is
symmetric,
entrywise ...

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votes

**2**answers

114 views

### a monotone relation for s-numbers

Assume $A, B$ are self-ajoint compact operators. Is it true that $\|A+iB\|\le \|2A+iB\|$? Do we have a stronger inequality $\prod_{k=1}^ns_k(A+iB)\le \prod_{k=1}^ns_k(2A+iB)$ or even stronger one ...

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**1**answer

132 views

### Convex functions: bounding the difference

Suppose you are given a convex function $f: R^d \rightarrow R$. Let us say you are given $x,x' \in R^d$ and $x_1, x_2, \ldots, x_n \in R^d$ such that
$$\sum_{i=1}^n (x_i - x') = x - x'.$$
Is it ...

**0**

votes

**1**answer

378 views

### How to solve this optimization with the orthogonal constraint?

Problem
Supposing that $A$ is a symmetric real matrix and $\{\mathbf{w}\_i\}_{i=1}^n$ is any orthogonal basis on $\mathbb{R}^n$ such that $W^\top W=WW^\top=\mathbf{I}_n$ where ...

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**1**answer

203 views

### A Wirtinger-like inequality involving two functions

Let $f(t)$ and $g(t)$ be periodic functions on $t\in[0,2\pi]$. By using the Fourier series of the two functions, we can easily prove the inequality
$$\left|\int_0^{2\pi}f(t)g'(t)dt\right|=
...

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**0**answers

137 views

### Upper bound on integrals of Legendre polynomials

Hi,
If $P_n(x) $ is unnormalized shifted Legendre polynomial, and $g_{n,m}(x) = \int_0^x P_n(x_1)x_1^m dx_1, n>m $ then what is the upper bound $ |g_{n,m} (x)|_{max} , x\in (0,1) $ as a function ...

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**2**answers

330 views

### On a inequality [closed]

I hope the following kind of inequality holds: let $a_i,b_i\in R$ with $b_i>0$, $\sum _{i=1}^mt_i=1$ with $t_i>0$, then
...

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**1**answer

177 views

### a simple probability inequality

For independent Rademacher random variables $\epsilon_i, i=1,2, \cdots, n$, i.e. $P(\epsilon_i=-1)=P(\epsilon_i=1)=\frac{1}{2}$, do we have
$$max_{0\le a_i\le b, ...

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**2**answers

259 views

### Inequality involving the weak second moment

I want to ask the following probability inequality:
Is it true that for any random variable $X\ge 0$, we have
$$
\sup_{t>0}(t\mathbb E(X\mathbf 1_{X\ge t}))
\le
2\sup_{t>0}(t^2 \mathbb P(X ...

**3**

votes

**0**answers

301 views

### A curious inequality

Let $r_k>0$ for $k = 1,\ldots, n$, let $\alpha_k, \beta_k\in \mathbb{R}$ be given such that
$|\alpha_k|\le \beta_k\le \frac{\pi}{2}$. Suppose further that
...

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**0**answers

166 views

### A Multiplicative version of McDiarmid's Inequality like the one of Chernoff-Hoeffding Bounds

McDiarmid's Inequality basically says the following:
Let $X_1, X_2, X_3, \ldots, X_n$ denote independent random variables and $f$ is a function of $n$ real arguments. If changing the value of the ...

**1**

vote

**1**answer

150 views

### An upper bound on a simple sum

Hi,
I am trying to put a bound on a sum. Given $\omega=\exp(2\pi i/3)$ and $n$ positive real numbers
$ 0=\tau_0 < \tau_1 < \tau_2 < ...\tau_{n-1}< \tau_n=1 $
such that
...

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votes

**1**answer

272 views

### Trajectorial version of Doob's $L^2$ inequality

In the paper http://www.mat.univie.ac.at/~schachermayer/pubs/preprnts/prpr0154.pdf
you can find a trajectorial version of Doob's inequality. It is given by:
...

**0**

votes

**1**answer

238 views

### Inequalities Involving Wedge Product (Reference Request)

Hello,
I'm looking for references on various inequalities involving the wedge product and exterior forms. The only references I could hunt down are references on Hadamard-Schwarz Inequality. The ...

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**2**answers

518 views

### Quadratic Farkas' Lemma?

The Farkas Lemma says that if a system of linear inequalities implies
yet another linear inequality, then this last inequality can be obtained by
taking a positive linear combination of the ...

**1**

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**2**answers

267 views

### A Gronwall-type inequality.

I want to derive a Gronwall-type inequality from the inequality below. Here all the functions are nonnegative, continuous and if you need some assumptions you may use that.
$$ f^2(t) \leqslant g^2(t) ...

**4**

votes

**1**answer

300 views

### Upper bound on joint Renyi entropy

Renyi entropy of a random pair $(X,Y)$ with probability distribution $p_{X,Y}$ is defined by
\begin{equation}
H_\alpha(X,Y) = \frac{1}{1-\alpha}\log\sum_{x,y} p_{X,Y}(x,y)^\alpha.
\end{equation}
...

**11**

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**1**answer

610 views

### A delicate elementary inequality

The following "piecewise-quadratic" inequality emerged in a joint work of Rom
Pinchasi and myself. The inequality is surprisingly delicate, and all our
attempts to simplify it made it false. By the ...

**4**

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**0**answers

403 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 ...

**5**

votes

**4**answers

663 views

### An inequality involving sums of powers

I asked this question at Stack Exchange but received no answer. The origins of the question are unclear, as I came across it rummaging through old notebooks from highschool, in one of which it was ...

**1**

vote

**1**answer

228 views

### Does this variable have an upper bound?

Let $x$ be a positive scalar variable whose time derivative satisfies
$$|\dot{x}(t)|\leq \exp \left\{\left(-\int_{0}^{t}\frac{1}{x(\tau)} \mathrm{d} \tau \right)\right\},$$
where $|\cdot|$ denotes the ...

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votes

**1**answer

396 views

### Trace matrix inequality

Hello all,
I come across the following problem.
Is it true that for a positive definite matrix $X^{n\times n}$, the following holds
$\text{trace}(X^{-1})\geq\text{trace}([\text{diag}(X)]^{-1})$,
...

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votes

**1**answer

415 views

### Hadamard-like inequalites for positive definite symmetric matrices

Let $S$ be any positive semi-definite symmetric matrix (Hermitian psd matrices work as well). The Hadamard inequality is that
$$\det S\le\prod_{i=1}^n s_{ii}.$$
My question is whether there are some ...

**0**

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**1**answer

83 views

### Continuous variant of KFG Inequality

Context
Let $t_1, .., t_n$ be jointly independent boolean random variables.
Let $X, Y$ be monotone functions (i.e. $\forall i: t_i \geq t'_i$ implies $X(t_1, ... t_i ..., t_n) \geq X(t'_1, ... t'_i, ...

**1**

vote

**1**answer

112 views

### Approximating Moment of Sum of RVs

Given
$X_i$ are independent random variables.
$|X_i| < 1$
$E[X_i] = 0$
$X = \sum_i^n X_i$
$var(X)=\sigma$
Prove:
$$ E(X^p)^{1/p} = O(\sqrt{p}\sigma +p)$$ for all even p
Things I've tried:
...

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votes

**2**answers

134 views

### Are there better upper bounds on the rank of the commutant of a fusion module than the global dimension?

Suppose I have a fusion category $\mathcal{C}$ and an indecomposable module category $\mathcal{M}$ over it. The commutant $\mathcal{C}_\mathcal{M}^*$ is the category of module endofunctors, and gives ...

**1**

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**1**answer

161 views

### Unitary matrix and matrix inequality [closed]

Dear all,
Suppose U and V are unitary matrix, A and B are positive definite,
Does:
$UAU^{-1} < VBV^{-1}$
implies $A< B$
and vice versa?

**0**

votes

**1**answer

206 views

### Inequality with even powers of trigonometric functions

For $m>0$,
$0 < n\leqslant m+1$ ($m,n\in \mathbb{Z} $) , and $0 < a < 1$ , prove that
$$2^{n}\cdot \left( a^{n}\cos ^{2m}\dfrac {\pi a} {2}+\left( 1-a\right) ^{n}\sin ^{2m}\dfrac {\pi ...

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votes

**3**answers

290 views

### Asymptotic behaviour/upper bound for $\int_0^{\infty} \exp(-c x^a+K x^b)dx$ for $a>b>0$ as $K\rightarrow \infty$?

What is the asymptotic behaviour or an upper bound for $\int_0^{\infty} \exp(-c x^a+K x^b) \, dx$, for $a>b>0,$ as $K\rightarrow \infty$?
Or any good reference for tools to tackle this question?
...