The inequalities tag has no wiki summary.

**4**

votes

**2**answers

195 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 ...

**12**

votes

**6**answers

2k views

### subadditive implies concave

Let $f:R_+\to R_+$ be smooth on $(0,\infty)$, increasing, $f(0)=0$ and
$\lim_{x\to\infty}=\infty$. Assume also that $f$ is subadditive:
$f(x+y)\le f(x)+f(y)$ for all $x,y\ge 0$. Must $f$ be concave? ...

**5**

votes

**0**answers

104 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

**1**answer

84 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} = ...

**5**

votes

**1**answer

229 views

### Inequality for a gamma function

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

**2**

votes

**0**answers

70 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 ...

**1**

vote

**0**answers

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

**13**

votes

**1**answer

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

**14**

votes

**6**answers

9k views

### Determinant of sum of positive definite matrices

Say $A$ and $B$ are symmetric, positive definite matrices.
I've proved that $\det(A+B) \ge \det(A) + \det(B)$ in the case that $A$ and $B$ are two dimensional.
Is this true in general for ...

**0**

votes

**1**answer

255 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 ...

**20**

votes

**0**answers

304 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 ...

**4**

votes

**0**answers

287 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:
...

**2**

votes

**0**answers

84 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, ...

**2**

votes

**1**answer

193 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 ...

**0**

votes

**0**answers

49 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

**1**answer

154 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 ...

**2**

votes

**1**answer

94 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 ...

**3**

votes

**0**answers

89 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

**2**answers

163 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
$$\mathbb{P}(\exists\,i=1,\dots,n\text{ s.t. }|X_i-X_{i-1}|>c_i) \,\leq\, \epsilon$$
...

**3**

votes

**0**answers

89 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

**0**answers

99 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} ...

**10**

votes

**2**answers

325 views

### Inequalities for averaging over partially ordered sets

Let's start from a classical inequality:
If $0\le a_1\le\cdots\le a_k$ and $0\le b_1\le\cdots\le b_k$ then
$(a_1+\cdots+a_k)(b_1+\cdots+b_k)\le k(a_1b_1+\cdots+a_k b_k)$.
It can be written also in ...

**2**

votes

**1**answer

313 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 ...

**29**

votes

**3**answers

1k 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

**0**answers

61 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

**0**answers

85 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**

vote

**0**answers

94 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
$$ ...

**3**

votes

**1**answer

152 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

**2**answers

117 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? ...

**2**

votes

**0**answers

77 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

**0**answers

135 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:
...

**7**

votes

**1**answer

483 views

### Proof of the “Neo-classical Inequality”

I came across the following inequality, dubbed the "Neoclassical Inequality" which holds uniformly in $p\geq 1, n$:
$\frac{1}{p^2}\sum_{j=0}^n\frac{a^{j/p}b^{(n-j)/p}}{(j/p)!((n-j)/p)!}\leq ...

**10**

votes

**1**answer

281 views

### Hölder's inequality for matrices

I was wondering if the HÃ¶lder's inequality was true for matrix induced norms, i.e. if
$$\|AB\|_1 \leq \|A\|_p\|B\|_q, \quad\forall p,q \in [1,\infty] \text{ s.t. } \tfrac{1}{p}+\tfrac{1}{q} = 1.$$
But ...

**0**

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} ...

**5**

votes

**4**answers

648 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 ...

**11**

votes

**2**answers

242 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} ...

**3**

votes

**1**answer

162 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

**1**answer

252 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 ...

**6**

votes

**1**answer

151 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 ...

**0**

votes

**0**answers

55 views

### 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 ...

**7**

votes

**2**answers

170 views

### 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 ...

**7**

votes

**0**answers

173 views

### 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, ...

**19**

votes

**3**answers

517 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 $$\frac{|A+X|}{|X|}\geq \frac{|A+A|}{|A|}?$$
...

**2**

votes

**4**answers

422 views

### Sum over integer compositions

Sorry if the question is trivial - are there closed form expressions or good approximations for the sum of a symmetric function taken over all integer compositions (into given number of parts) of a ...

**3**

votes

**1**answer

176 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) + ...

**7**

votes

**2**answers

510 views

### Is this ergodic inequality true?

Is anything similar to the following inequality true,
$\displaystyle P\{\max_{n \leq k \leq m} |A_k f - A_n f| > \epsilon\} \leq C \frac{||A_m f - A_n f||_1}{\epsilon}$
where $A_n f = ...

**0**

votes

**1**answer

193 views

### Is this Holder-type inequality true $\;(\int fg )\,(\int f^2 + g^2)\leq \int f^2g+fg^2\;$? [closed]

Suppose $f$ and $g$ are non-negative functions such that $\int f(x)dx = \int g(x)dx=1$. Is it true that
$$\Big(\int fg \Big)\,\Big(\int f^2 + g^2\Big)\leq \int f^2g+fg^2\quad?$$

**3**

votes

**1**answer

269 views

### An elementary inequality: reference request

Consider the problem of minimizing $\sum_{i=1}^{n}{x_{i}}$ under the constraints $\sum_{i=1}^{n}{x_{i}^{2}}=1$ and $x_{i} \geq 0$. Obviously the solution is given by the vector $(1,0,\ldots,0)$.
Now ...

**7**

votes

**3**answers

297 views

### Rosenthal like inequality for weak $\mathbb L^p$-norms

Let $p$ be a real number greater than $1$. It is well known (see Hall and Heyde's Martingale limit theory and its applications, Theorem 2.10) that there exists a constant $C_p$ such that if ...

**4**

votes

**1**answer

291 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}
...