The inequalities tag has no wiki summary.

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### Estimate of a Sobolev norm of p-form

$\underline{\mathrm{NOTATIONS}}$
Let $(M,g)$ be a compact connected Riemannian malifold of $d$ dimensional.
$A^p(M)$ denotes the set of $p$-forms on $M$.
$g_{\wedge^p}$ denotes the fiber metric on ...

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43 views

### Bounding the absolute value of a polynomial involving a Diophantine equation

Let $\mathbf{z}\in\mathbb{C}^n$ with entries $z_1,z_2,\ldots,z_n$. I would like to bound the following quantity
\begin{equation*}
...

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**1**answer

90 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 ...

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**1**answer

88 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 ...

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**1**answer

209 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 ...

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336 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 ...

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119 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 ...

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177 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 ...

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82 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 ...

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63 views

### Bounding multiplications of PSD random matrices

Consider the following setup,
$(X, \hat{X}, Y, \hat{Y})$ are four $n \times n$ real, symmetric, full-rank, positive-definite matrices with entries between zero and one and operator norm $O(n)$.
The ...

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69 views

### Inequality related to regularized incomplete beta function

I am trying to analyze the following inequality related to regularized incomplete beta function $I$:
$$p^{c}/2\geq I_{\frac{p}{1+p}}(\alpha,\alpha)= ...

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56 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)
...

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**1**answer

358 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 ...

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115 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 ...

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**1**answer

88 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} = ...

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### 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 ...

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**1**answer

255 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 $\mathbb{M}_m(\mathbb{M}_n)$ ...

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264 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 ...

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394 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 ...

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### 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 $\mathbb{R}_+$ ...

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336 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:
...

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**1**answer

217 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 ...

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52 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 ...

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votes

**1**answer

163 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 ...

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### 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 $p_1\geq p_2\geq\cdots \geq p_n$ ...

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### 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 ...

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201 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
$$\mathbb{P}(\exists\,i=1,\dots,n\text{ s.t. }|X_i-X_{i-1}|>c_i) \,\leq\, \epsilon$$
...

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### 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, ...

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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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### 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} ...

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323 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 ...

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203 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 ...

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### 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
$$2(|a|+|b|+|c|) \leq |a+b+c|+|a+b-c|+|a+c-b|+|b+c-a|$$
...

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### 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 $N$)
...

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### 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} = ...

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### 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
$$ ...

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### 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 ...

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119 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? ...

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### 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 ...

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137 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:
...

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### 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} ...

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### 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 ...

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259 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 ...

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305 views

### Inequality for a gamma function

Let $s=\sigma+it$ and $\Gamma(s)$ be the Euler gamma function.
Does the inequality holds?
$$
\left|\frac{\Gamma(s)}{\Gamma(2-s)}\right|\leq |s|^{2(\sigma-1)},\, 1<\sigma<2,\, t>t_0>0.
$$
...

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**1**answer

158 views

### Inequality for Laguerre polynomials

Let $L_n$ be the $n$-th Laguerre polynomial defined by
$\quad
L_n
(x)=\frac{e^x}{n!}\frac{d^n}{dx^n}(x^n e^{-x}).\quad
$
I want to prove that
$$
\forall n\in \mathbb N,\forall x\ge 0,\quad \sum_{0\le ...

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### Local-to-global inequalities for measures: Brunn-Minkowski, Ahlswede-Daykin, what else?

This question is motivated by an obvious formal analogy between two well-known inequalities:
Log-concavity and Brunn-Minkowski inequality
Let $\mu(dx) := m(x) dx$ be an absolutely continuous ...

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### Inequality regarding sum of gaussian on lattices

When S is a subset of an inner product space, let d(S) denote ${\sum\limits_{s \in S} e^{- \langle s,s \rangle}}$
Suppose L is a discrete additive subgroup of $\mathbb{R^n}$, M is a subgroup of L, ...

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### Bounds or approximations for the conditional probability of an event involving correlated random variables

Let $\tilde{\gamma_1}, \tilde{\gamma_2}, \ldots, \tilde{\gamma_N}$ be exponential random variables (RVs) that are correlated with each other. Let $\gamma_n$ be another exponential RV that is ...

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541 views

### A sumset inequality

A friend asked me the following problem:
Is it true that for every $X\subset A\subset \mathbb{Z}$, where $A$ is finite and $X$ is non-empty, that $$\frac{|A+X|}{|X|}\geq \frac{|A+A|}{|A|}?$$
...

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**1**answer

199 views

### About generalized Minkowski inequality

For which functions $f:\mathbb{R}^+ \to \mathbb{R}^+$ does the inequality
$f^{-1}\left(\sum\limits_{k=1}^n f(x_k+y_k)\right) \leq f^{-1}\left(\sum\limits_{k=1}^n f(x_k)\right) + ...