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2
votes
1answer
153 views

trigonometric sum and inequalities

let $x\in\mathbb{R}-\mathbb{Z}$ and $e(x)=e^{2\pi ix}$. If we have this sum $$\left|\overset{q}{\underset{h=1}{\sum}^{*}}e\left(h\, x\right)\underset{\underset{p\equiv h\,\textrm{mod}\, q}{p\leq N}}{\...
4
votes
2answers
282 views

Inequality for an integral involving $ \exp $, $ \sin $ and $ \cos $

Let $ t > 0 $ and $ k \in \{ 0,1,2,\ldots \} $. Does the following inequality hold? $$ \int_{k + 1/2}^{k + 3/2} \frac{x \sin(2 \pi x)}{1 + 2 e^{2 \pi t} \cos(2 \pi x) + e^{4 \pi t}} ...
4
votes
2answers
272 views

Operator norm versus Hlawka inequality

Let $E$ be a finite dimensional normed vector space. If $E$ is $\ell^1$-embeddable, then the norm satisfies Hlawka inequality $${\bf(H)}\qquad\|x+y\|+\|y+z\|+\|z+x\|\le\|x\|+\|y\|+\|z\|+\|x+y+z\|,\...
4
votes
0answers
395 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, $...
10
votes
1answer
405 views

Generalized Hlawka inequality

Let $E$ be a vector space over the real (the complex case is interesting too). We consider functions $f:E\rightarrow\mathbb R$ which satisfy the homogeneity property $$f(\lambda x)=|\lambda|\,f(x).$$ ...
6
votes
1answer
84 views

Summability of ratios of moments a weight

Recently, I encounter the following problem: Let $w$ be a probability density on $[0,1]$. Let mk be the $k$-th moment, i.e., $$m_k=\int_0^1t^kw(t)dt.$$ Under what condition can we have $$\sum_{k=0}^\...
2
votes
0answers
88 views

Hlawka inequality for Lorentz quadratic form

Let $K$ be a convex cone in ${\mathbb R}^n$. A continuous function $f:K\rightarrow\mathbb R$ satisfies a Hlawka inequality if $$f(0)+f(x+y)+f(y+z)+f(z+x)\le f(x)+f(y)+f(z)+f(x+y+z),\qquad\forall x,y,z\...
1
vote
2answers
129 views

investigating positivity/negativity of a function [closed]

I'm investigating if and how the positivity or negativity of a multivariable function can be proved. Consider $y_{1},y_{2},y_{3}\in\mathbb{R}$ and the following function $$f\left(y_{1},y_{2},y_{3}\...
15
votes
1answer
446 views

Hlawka inequality for determinants of positive definite matrices

It is mentioned here that if $A, B, C\in M_{n}(\mathbb C)$ are positive semidefinite, then $$\det (A+B+C)+\det C\ge \det (A+C)+\det (B+C)$$ (quoted from this article) and the special case ($C=\bf 0$) $...
3
votes
1answer
155 views

Conditional Form of Rosenthal's Inequality

Rosenthal's Inequality as stated in the book "Martingale Limit Theory and Its Application" by Hall and Heyde states the following: If $\{S_i, \mathcal{F}_i, 1\leq i \leq n\}$ is a martingale and $2\...
4
votes
3answers
537 views

Generalized Hardy-Littlewood-Sobolev Inequality

The Hardy-Littlewood-Sobolev Inequality says that $$\text{for $p,q,r\in (1,+\infty)$ such that }\quad 1-\frac1p+1-\frac1q=1-\frac1r,\tag {$\sharp$} $$ $$ \exists C, \forall u\in L^p(\mathbb R^n),\...
3
votes
1answer
673 views

Number of solutions in a sum of squares Diophantine equation

Let $n$ be an integer. I'm interested in upper bounding the number of integer solutions of \begin{equation*} x_1^2+x_2^2+\ldots+x_p^2=y_1^2+y_2^2+\ldots+y_p^2, \end{equation*} where for all $i\in\{1,...
3
votes
1answer
181 views

Hardy-type inequality for point boundary

Let $f$ be in $W^{2,p}(\mathbb{R}^n)$ for $n\geq 3$ and $p>n/2$, with $f=0$ at the origin. I want to show that the integral $$\int_{B(0,r)} (f |x|^{-2})^p dV <\infty$$ for some small $r>0$. A ...
2
votes
2answers
364 views

Does this simple inequality have a name?

Let $x_{1},\ldots,x_{n}$ be nonnegative numbers such that $m \leq x_{i} \leq M$. Let $$ S=\sum_{i=1}^{n}{x_{i}} $$ and $$ Q=\sum_{i=1}^{n}{x_{i}^{2}}. $$ Then $$ Q \leq S(M+m)-nMm. $$ This has ...
4
votes
3answers
443 views

Inequality of arithmetic, geometric and harmonic means

Let $a_1,\dots,a_n$ be positive numbers, does the following inequality holds? $$\frac{a_1+a_2+\cdots+a_n}{n}-\sqrt[n]{a_1a_2\cdots a_n}\geq\sqrt[n]{a_1a_2\cdots a_n}-\frac{n}{\frac{1}{a_1}+\frac{1}{...
2
votes
1answer
969 views

Inequality for the tail of normal distribution function

Let $ Ф(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{x} e^{-t^2/2} \, dt $ be the cumulative distribution function of the standard normal distribution. Numerical calculations suggest the following ...
13
votes
0answers
397 views

Determinant inequality involving Hermitian, positive definite matrices

Question: Let $A,B,C\in M_{n}(C)$ be Hermitian and positive definite matrices such that:$$A+B+C=I_{n}$$ Show that: $$\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n\det(A^2+B^2+C^2)$$ This question ...
1
vote
0answers
122 views

An inequality for moments of a random variable

I'm interested in a class C of $R^1$-valued random variables $\xi$ which satisfy an inequality of the type $$ (1) \qquad E|\xi|^p \leq F(E|\xi|^2), $$ where $p>2$, $F$ is a certain non-...
0
votes
1answer
81 views

angle inequality in n-dimensional vector space [closed]

Does anyone has answer for the following doc product problem? Let A,B,C be three vectors of magnitude of 1. Let A*B = Cos(x) ( * means dot product) B*C = Cos(y) ...
2
votes
1answer
530 views

An inequality involving traces and matrix inversions

The following question kept me wondering for some time: Given the symmetric matrices $A,B,C\in\mathbb{R}^{n×n}$ where $A$ and $C$ are positive definite (hence invertible), and $B$ is positive ...
5
votes
0answers
189 views

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 ...
1
vote
1answer
104 views

Positive Definiteness of a certain function

Recall that a function $f: \mathbb{R}^d \longrightarrow \mathbb{C}$ is positive definite, iff for all numbers $N$ and $x_1, \dots, x_N \in \mathbb{R}^d$, the matrix $(a_{ij})$ with entries $$a_{ij} = ...
2
votes
0answers
112 views

Variant form of the gronwall inequality

I know the following statement for gronwall inequality: Given $f$ non negative and absolutely continuous on $[0;T]$ and $\phi \in L^1(0;T)$if we have, $f' \leq \phi f$ and $f(0)=0$ then $f=0$ Now is ...
17
votes
1answer
403 views

A possible extension of a determinant inequality

It is well known that if $A, B$ are positive semidefinite matrices, then $$\det (A+B)\ge \det A+\det B.$$ I am considering a possible extension of this result. Let $\mathbb{M}_m(\mathbb{M}_n)$ ...
0
votes
1answer
291 views

A number theoretic inequality

Is this inequality true? : $$\displaystyle \prod_{i\le \left\lfloor{\frac{n-1}{2}}\right\rfloor}\left\lfloor{\frac{\left\lfloor{\frac{n-1}{2}}\right\rfloor}{i}}\right\rfloor\le \operatorname{lcm}(1,2,\...
23
votes
0answers
510 views

Are there any nontrivial near-isometries of the $n$-dimensional cube?

Consider the $n$-dimensional Hamming cube, $C = \{-1,1\}^n$. Given an $n \times n$ orthogonal matrix $O$, I'll measure "how close $O$ is to being an isometry of $C$" by the following scoring function:...
1
vote
0answers
143 views

Uniform bound for an alternating series of functions

I have mainly two questions, the first one being motivated by the second one. 1) Is there a way to prove that $F(x) = \sum_{k=1}^\infty \frac{(-1)^{k+1} x^{2k}}{(2k)!}$ is bounded on $\mathbb{R}_+$ ...
5
votes
0answers
771 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)...
3
votes
1answer
369 views

A similar Cauchy-Schwarz inequality with linear-algebra

Let $A$ be matrix in $M_{n}$ (i.e., $n\times n$ complex matrices), and $\|A\|\le 1$, we call it a contraction. Assume that $A$ and $B$ are contractions such that $I-AA^*$ and $I-BB^*$ are positive-...
1
vote
0answers
95 views

Nonstationary Markov chain maximal inequality

Let $X_i$ be a (finite-state, irreducible, aperiodic) Markov chain, not necessarily stationary. (That is, it doesn't start from the invariant distribution; I'm happy to have it be time-homogeneous if ...
7
votes
1answer
233 views

Sumsets and a bound

Let $q$ be a positive integer. Is it true there exists a constant $C_q$ such that the following inequality holds for any finite set $A$ of reals: $$\displaystyle |A+qA|\ge (q+1)|A|-C_q\qquad (1)$$ I ...
3
votes
0answers
125 views

Will relative entropy increase with majorization?

Suppose that three probability distributions have the relation $P\succ Q\succ R$, then do the following relations between their relative entropy valid? (assume that $p_1\geq p_2\geq\cdots \geq p_n$ ...
4
votes
1answer
242 views

Concentration inequalities in $\ell_{\infty}$ for sums of iid random (“nice”) functions?

I'm looking for "tail-bound-like" inequalities that look like this (I state a specific setting but more general settings are interesting): Let $D$ be a distribution on a set of "nice" functions $g$...
6
votes
2answers
873 views

Extension of the Azuma-Hoeffding inequality (when the differences are bounded with large probability)

Let $(X_i)$ be a super-martingale and suppose their differences are bounded ''with high probability'', that is $$\mathbb{P}(\exists\,i=1,\dots,n\text{ s.t. }|X_i-X_{i-1}|>c_i) \,\leq\, \epsilon$$ ...
2
votes
0answers
105 views

Positive, Uni-modal, Log-concave Combinatorics

We define a sequence, $\{a_n\}_{n=0}^\infty$, to be a uni-modal sequence if for some $m$, $$a_0<a_1<\cdots<a_m,\ \ \ \ a_m>a_{m+1}>a_{m+2}>\cdots.$$ We define a sequence, $\{a_n\}_{...
3
votes
0answers
105 views

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 \...
4
votes
0answers
130 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)...
2
votes
1answer
341 views

A functional inequality

$g:[0,1]\to[0,1]$ continuously differentiable and increasing such that for all integers $t>0$ and for all $r\in(0,1)$, $g(r^{t+1})>g(r)\cdot g(r^t)$. Does this imply that for all $r\in(0,1)...
6
votes
3answers
364 views

Is $1/\max(i,j)$ a bounded matrix on Hilbert spaces?

I would like to know if the infinite matrix $[\frac{1}{\max(i,j)}]_{i,j\geq 1}$ represents a bounded operator on $\ell^2(\mathbb{N}^\star)$. It would be sufficient to know if the Lehmer matrix $[\frac{...
39
votes
3answers
3k views

Absolute value inequality for complex numbers

I asked this question on stackexchange, but despite much effort on my part have been unsuccesful in finding a solution. Does the inequality $$2(|a|+|b|+|c|) \leq |a+b+c|+|a+b-c|+|a+c-b|+|b+c-a|$$ ...
0
votes
0answers
63 views

On the product of relatively prime number $< N$ [duplicate]

Let $FI(N)$ denote the product of all $\phi(N)$ [relatively prime numbers $<N$] . And define $SFI(N)$ as the product of remaining $N-\phi(N)$ numbers $\le N$ (Which are not relatively prime to $N$) ...
0
votes
0answers
111 views

Estimating when does a certain binomial sum exceed an upper bound

Given a fixed integer $n > 0$ and $0 \le m \le n$ let us define the numbers $$f_{n,m} = \sum_{i=\lfloor m/2 \rfloor}^m {n-2i \choose n - m -i}{i+1 \choose m - i +1}.$$ For example $f_{n,0} = 1,f_{...
1
vote
0answers
129 views

about an inequality which looks like a Hardy inequality

I would like to know if the following inequality can be true : consider a double sequence $(u_{i,j})_{(i,j)\in (\mathbb{N}^\star)^2}$ of real numbers and a real $p\geq 1$, do we have $$ \sum_{j\...
3
votes
1answer
242 views

An inequality involving Bessel functions of imaginary order

The following inequality: $$\frac{\pi k}{\sinh{(\pi k)}}\;|J_{ik}(\tau)|^2\le 1,\;\;\;k,\tau\ge 0,$$ for Bessel function $J_{ik}(\tau)$, I found in http://link.springer.com/article/10.1134%2F1.558677 (...
2
votes
2answers
157 views

Bounds for the fat tail after trimming the mean?

I am interested in the quantity $$f(X,t) = \int_t^\infty\negthinspace x\ p(x)\ dx,$$ where $p$ is a probability distribution for a positive variable $X$. 1) Does this quantity $f(X,t)$ have a name? ...
4
votes
0answers
166 views

Isoperimetric inequality, isodiametric inequality, hyperplane conjecture… what are the inequalities of this kind known or conjectured?

I duplicate here a question I asked on math.stackexchange. Question: Which inequalities similar to the famous isoperimetric inequality is known? conjectured? I recently learned about some ...
4
votes
0answers
163 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,\...
12
votes
2answers
295 views

Subset of vectors whose sum has a large norm

In Rudin - Real & Complex Analysis we have the following Lemma 6.3. If $z_1, \ldots, z_n \in \mathbb{C}$ then there is a subset $S \subseteq \{1,\ldots,n\}$ for which $$\left|\sum_{k \in S} z_k\...
3
votes
1answer
185 views

Is this bounded from below?

Let $u_1, u_2, u_3 \in \mathbb{Z}$ such that $u_1^2 + u_2^2 = u_3^2$. Is $(u_3 + \frac{u_1 + u_2}{\sqrt{2}})^2$ bounded from below? The irrationality of $\sqrt{2}$ certainly precludes zero, but can ...
8
votes
1answer
343 views

An inequality for positive definite matrices

Let $K$ and $K^\prime$ positive definite $n \times n$ matrices, such that for all vectors $f \ge 0$ with nonnegative coordinates we have $$\sum_{i,j} K_{ij} f_i f_j \le \sum_{ij} K^\prime_{ij} f_i ...