The inequalities tag has no wiki summary.

**16**

votes

**2**answers

410 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 ...

**3**

votes

**2**answers

115 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 ...

**0**

votes

**1**answer

149 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 ...

**1**

vote

**1**answer

662 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 ...

**3**

votes

**1**answer

221 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|=
...

**3**

votes

**0**answers

152 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 ...

**-1**

votes

**2**answers

355 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
...

**0**

votes

**1**answer

183 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, ...

**6**

votes

**2**answers

269 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

323 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
...

**1**

vote

**0**answers

202 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
...

**4**

votes

**1**answer

287 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:
...

**2**

votes

**1**answer

591 views

### Are there such numbers?

Maybe better to ask for help on this question here: Are there eight numbers $0< x_{1},\ldots,x_{4},y_{1},\ldots,y_{4}<1$,
such that
$$
...

**0**

votes

**1**answer

260 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 ...

**11**

votes

**2**answers

581 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**

vote

**2**answers

278 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

363 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**

votes

**1**answer

648 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**

votes

**0**answers

415 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

798 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

233 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 ...

**5**

votes

**1**answer

474 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
$\text{trace}(X^{-1})\geq\text{trace}([\text{diag}(X)]^{-1})$,
...

**5**

votes

**1**answer

507 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**

votes

**1**answer

90 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

115 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:
...

**5**

votes

**2**answers

140 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**

vote

**1**answer

169 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

209 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 ...

**3**

votes

**3**answers

312 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?
...

**1**

vote

**2**answers

277 views

### how to solve a singular integral equation involving the kernel $1/x$

Dear all,
Suppose we know that $f(x)$ is nonnegative and Hölder continuous at zero with exponents $1/2$. We also know that
$$
f(x) \le g(x) + \int_0^x \frac{f(y)}{y} d y,\quad\forall x>0,
$$
...

**15**

votes

**4**answers

803 views

### Proving univariate polynomials (defined by sums, binomial coeffs, etc.) are nonnegative: is it 'routine'?

My colleagues and I are working on a project related to an old paper of C. Borell and we have boiled it down to the following problem:
Show, for all integers $1 \leq i \leq k$, that the univariate ...

**3**

votes

**2**answers

371 views

### Is there a software to prove or deduce symbolic inequalities?

I have a bunch of inequalities, and I'm trying to see if another inequality can be deduced from the first bunch. For example, assuming that $a \leq b$ and $c \leq d$, we can deduce that $a + c \leq b ...

**0**

votes

**1**answer

706 views

### Difficult Inequality [closed]

Suppose $x,y\geq 0$ and $b,c,d\geq 1$ are integers. Prove or find a counterexample to the following inequality
$$
\frac{1}{1+\frac{b+d}{\sqrt{x^{2}+c^{2}}}} + ...

**0**

votes

**2**answers

176 views

### inequality of a function

Hello everyone, could someone gives the conditions for $\lambda$ that the following inequality is correct for any $0\leq x\leq\alpha$
$$\lambda x-1+e^{-\lambda x}\leq\lambda^2x\sqrt{\alpha x}$$
...

**0**

votes

**1**answer

155 views

### Strengthening an inequality

Let $k$ be an integer. The following inequality is standard.
$$
(a+b)^{k+1} - b^{k+1} \leq (k+1)a(a+b)^k
$$
for $a,b > 0$.
However, does the following inequality still hold
$$
(a+b)^{k+1} - ...

**7**

votes

**1**answer

438 views

### Shapiro inequality for $n=23$

Is the Shapiro inequality for $n=23$ an open problem? The reason why I am asking is I have two contradictory pieces of information from two different articles.
The first article titled "The validity ...

**5**

votes

**1**answer

222 views

### Convexity of a specific semialgebraic set

I have an engineering problem which maybe resolved with semi-definite programming optimization.
I have a set which I would like to know if is convex:
Being $m \in \mathbb{R}^+$ a positive real ...

**1**

vote

**1**answer

146 views

### inequality for a symmetric nonnegative matrix

Given $A$ symmetric and semidefinite positive, for each $x$
$$ x'Ax \geq \frac{1}{\Vert A\Vert} \Vert Ax \Vert^2 $$
This inequality appears at page 24 of "Introduction to Optimization" from Boris T. ...

**1**

vote

**1**answer

235 views

### Mapping multivariate polynomial inequalities system to subspace

What I will ask, more than a solution, is better mathematical definition of my problem and directions to find the solution.
I have a set of linear equations, e.g.:
\begin{align}
d_1 &= L_1 - ...

**3**

votes

**2**answers

346 views

### anyone help me with this inequality

I'm have some difficulties in bounding the following inequality:
I want to find a c as small as possible s.t.
$$\sum_{i=1}^nx_i^4\sum_{i=1}^nx_i^2 -\sum_{i=1}^nx_i^6 \leq ...

**1**

vote

**2**answers

275 views

### An inequality with $\ell_p$ norm

I encounter the following claim in my research for which I couldn't get a solution for a long time. I asked a more general version of the question at math.stackexchange which did not attract much ...

**0**

votes

**1**answer

296 views

### A Polynomial Inequality Proof

Given $x_1,x_2, \ldots, x_n \ge 0, \alpha \ge 1$, show that
$\sum_{i}\alpha x_i(\sum_{j \le i}{x_j})^{\alpha-1} \ge (\sum_{i}x_i)^\alpha$
We're pretty sure the ineuqality holds for the given ...

**7**

votes

**2**answers

952 views

### (sharp)Garding's inequality and inequality with lower bounds

The origin of Garding's inequality was an effort to solve Dirichlet's problem for linear elliptic operators of high even order.Let $$P(x,D)= \sum a_{\alpha}(x)D^{\alpha}$$ with principal part ...

**3**

votes

**1**answer

356 views

### Can these logarithmic inequalities show existence of a prime between (x-1)^2 and x^2

SOME PROPERTIES OF THE SERIES OF COMPOSED NUMBERS, p2 gives the bounds
$$l(x)=\frac{x}{\log(x)-28/29}<\pi(x) < u(x)=\frac{x}{\log(x)-1.12} \qquad (1)$$
for $x \geq 3299$.
TWO GENERALIZATIONS ...

**1**

vote

**1**answer

344 views

### sobolev embedding theorem in the smooth metric measure space

we know the sobolev embedding theorem of Saloff-Coste
$\Big(\int_B|F|^{2q}d\mu\Big)^{\frac1q}\le e^{C(1+\sqrt KR)}V^{-2/n}R^2\int_B\Big(|\nabla F|^2+R^{-2}F^2\Big)d\mu
$
wtih $Ric\ge-(n-1)K$, for ...

**10**

votes

**2**answers

1k views

### Proving a messy inequality

EDIT:
After much work I was able to reduce the inequality to a single variable function which I need to show is non-positive. That function is (for $0\leq p\leq\frac{1}{2}$)
$$\frac{p^2(\log(p))^2 - ...

**2**

votes

**0**answers

299 views

### Does this inequality of negative relative entropy and quantum relative entropy hold?

Hello, everyone!
Question
I have a question about the relationship between general relative entropy and general quantum relative entropy: Given a unit vector $|i\rangle$ and two Hermitian matrices ...

**4**

votes

**1**answer

369 views

### Is it possible to extend this inequality about Euclidean distance &Frobenius norm to more general Bregman divergence such as relative entropy & von Neumann divergence?

Motivation- A Special Case
Supposing $A,B\in\mathbb{S}^{m\times m}$ are symmetric positive semi-definite (SPD) matrices and $\mathbf{x}\in\mathbb{R}^m$ is a unit vector where $\|\mathbf{x}\|=1$, we ...

**0**

votes

**1**answer

223 views

### Is there relationship between $f\left({\sum_i(\mathbf{v}_i^\top\mathbf{x})^2\lambda_i},\sum_j{(\mathbf{u}_j^\top\mathbf{x})^2\theta_j}\right)$ and $\sum_i\sum_j{(\mathbf{v}_i^\top\mathbf{u}_j)^2f(\lambda_i,\theta_j)}$ if $f$ is jointly convex?

Hello, everyone!
As we know that by Jensen's inequality, for jointly convex function $f$ and $\sum_ix_i^2=1$, we have
...