The inequalities tag has no usage guidance.

**5**

votes

**1**answer

144 views

### Rate of Convergence of Borwein Algorithm for computing Pi

In a book "Pi and the AGM" in 1987, authors, Jonathan Borwein and Peter Borwein, introduced a magical algorithm to compute $\pi$. However there is a problem that I couldn't understand and couldn't ...

**0**

votes

**0**answers

70 views

### Conditional version of martingale difference concentration inequality

Let $M_n$ be a $\mathscr{F}_n=\sigma(\eta_m,\theta_m, m\leq n)$ measurable martingale difference sequence. Then is it possible to find a exponential tail bound for the following
$$P(|M_{n+1}| > u|\...

**2**

votes

**0**answers

130 views

### system of complex number equations

Let $a_1,a_2,a_3,a_4\in \mathbb{C}$ be distinct such that
$$a_1^3+a_2^3+a_3^3+a_4^3=0$$
$$(1+|a_1|^2)a_1^2+(1+|a_2|^2)a_2^2+(1+|a_3|^2)a_3^2+(1+|a_4|^2)a_4^2=0$$
$$(1+2|a_1|^2+2|a_1|^4)a_1+(1+2|a_2|^...

**2**

votes

**1**answer

82 views

### Majorization of cyclic products

Let $k,m,n\in\mathbb N$ such that $n>k$. For a partition $\alpha=(\alpha_1,\dots,\alpha_k)\vdash m$ with $\alpha_1\ge\dots\ge \alpha_k>0$ and nonnegative $ x_1,\dots,x_n$ define $x^\alpha :=\...

**2**

votes

**0**answers

77 views

### Octahedron and System of trigonometric equations

Could somebody help me to prove the following?
$$\sum_{k=1}^6 \cos(2 \theta_k) (\cos(2\phi_k)-1))=0$$
$$\sum_{k=1}^6 \sin(2 \theta_k) (\cos(2\phi_k)-1))=0$$
$$\sum_{k=1}^6 \cos (\phi_k)=0$$
$$\sum_{k=...

**1**

vote

**0**answers

73 views

### Projecting on a a special polyhedron

Let $X$ be an $n$-by-$p$ matrix and consider the closed convex polyhedron
$$\mathcal P_X := \{y \in \mathbb R^n | \|X^Ty\|_\infty \le 1\}.$$
Notice that $\mathcal P_X$ is symmetric about the origin.
...

**13**

votes

**2**answers

459 views

### regular polygon question

Let $a_1,a_2,\ldots,a_n$ be distinct points on the complex plane $\mathbb{C}$ and $L$ be a circle in $\mathbb{C}$ such that
$$f(z):=\sum_{i=1}^n|z-a_i|^{2n-2}$$
is constant on $L.$ Could somebody ...

**0**

votes

**0**answers

183 views

### Can we find an upper bound?

Let $f\in C^1(\mathbb R)$ with $f(0)=0$ and $|f'(x)|\le m$, where $m\in (1,2]$.
Let $x(0)\in\mathbb R$ be arbitrary, and define $x(n),y(n)$ recursively by
$$
x(n+1)=f(x(n)) , \quad\quad
y(n)=\frac{x(n+...

**2**

votes

**2**answers

353 views

### A logarithmic cotangent inequality

I must be a terrible googling searcher but I cannot find a reference to the following inequality:
$$ \forall_{\phi\in(0;\frac \pi 4)}\ \ln(\cot(\phi)))\, <\, \cot(2\!\cdot\!\phi) $$
I have just ...

**2**

votes

**1**answer

155 views

### Proofs of inequalities used by Erdos-Renyi in their Random Graphs Paper 1

Please refer to this, it is Erdos-Renyi 1959 paper 1 on Random Graphs. I am currently working on this, but I am stuck on the fifth page, where they use two estimates. More specifically, here's the ...

**17**

votes

**3**answers

723 views

### Deceptively simple inequality involving expectations of products of functions of just one variable

For a proof to go through in a paper I am writing, I need to prove the following deceptively simple inequality:
$$(*)\qquad E(X^a) E(X^{a+1}\log X) > E(X^{a+1})E(X^a\log X) $$
where $X>e$ has ...

**2**

votes

**0**answers

58 views

### Minimal condition $\sum \rho_{ij} \Psi_{ij} s_i s_j < \sum s_i s_j $, $\mid \rho_{ij} \mid \leq 1$, $s_i \in \mathbb R$ and $\Psi_{ij} \in \{0,1\}$

Consider a sequence of real number $\{s_i\}_{i\leq n}$. Now consider the real numbers $F$, $G$ and $\alpha$ defined below
$$F= \sqrt{ \left( \sum ~\rho_{ij} ~\Psi_{ij}~ s_i ~s_j \right)^+}, $$
$$G = ...

**4**

votes

**1**answer

123 views

### concentration inequality for $d$-dimensional martingale

Are any concentration inequality available for $d$-dimensional martingale. It is easy to find such inequality using the inequalities for single dimension, but that will contain the dimension $d$ in ...

**4**

votes

**2**answers

275 views

### Extending inequality for $\ell^p$ from integer $p$ to real $p$

Suppose $(a_n)_{n=1}^\infty$ and $(b_n)_{n=1}^\infty$ are two decreasing sequences of positive numbers such that
$a_1<b_1$ and
$$\sum_{k=1}^\infty a^p_k\leq \sum_{k=1}^\infty b^p_k<\infty$$ ...

**14**

votes

**3**answers

619 views

### An inequality improvement on AMM 11145

I have asked the same question in math.stackexchange, I am reposting it here, looking for answers:
How to show that for $a_1,a_2,\cdots,a_n >0$ real numbers and for $n \ge 3$:
$$\sum_{k=1}^{n}\...

**6**

votes

**1**answer

353 views

### Inequality for the maximum of Gaussian variables

Let $X=(X_1,\dots,X_n)$ and $Y=(Y_1,\dots,Y_n)$ be centered Gaussian vectors
with variance matrix $\Gamma_X$ and $\Gamma_Y$. We assume that the matrix
$\Gamma_Y-\Gamma_X$ is positive definite. Is it ...

**2**

votes

**1**answer

86 views

### Complexity of Deciding Feasibility of a system of linear inequalities over restricted variables

I am working out an interesting problem and would like some help with this particular sub problem:
Suppose we have a matrix $ M =\left\lbrace a_{ij}\right\rbrace $ of size $n\times m$ where $ a_{ij}\...

**0**

votes

**0**answers

73 views

### Brunn-Minkowski Inequality : References request or A Particular Example of a 2 dimensional set

I had this question on Mathematics StackExchange unanswered for a month or so. Hence, it came here which it seems is a more natural place.
I have an inequality - $(R_{C+D})^{2} \geqslant (R_{C})^{2} +...

**1**

vote

**2**answers

260 views

### Looking for (information about) long diamonds

I was given an open problem as a birthday present recently. While I can probably handle spoilers at this point, what I really want are literature and other references. Also acceptable would be ...

**8**

votes

**0**answers

92 views

### Which ordering of factors is needed to obtain this kind of determinantal inequalities?

Let $A$ and $B$ be $n\times n$ Hermitian positive definite matrices. The curious determinantal inequality given here, which can be stated as $$\det (A^{4}+ ABBA+BAAB+B^{4})\ge\det(A^{4}+ AABB+BBAA+B^{...

**2**

votes

**3**answers

414 views

### Hard inequality and number theory [closed]

Let $m,n \in \mathbb{N}$ and $m \geq n \geq 2$ and $x_1,x_2,...x_n \in \mathbb{N}_{\geq 1}$ such as $x_1+x_2+...+x_n=m$.
Find $\min P$ with $P= \sum_{i=1}^{n} x_i^2.$

**0**

votes

**1**answer

129 views

### For a set of positive integers, is this inequality always true?

The input consists of a set of positive integers $\{b_1,...,b_2\}$ such that $$\sum_{i=1}^nb_i=CK,$$ with $C$ and $K$ two positive integers.
The question is the following, is there $k\in\{1,...,n\}$ ...

**8**

votes

**1**answer

304 views

### Exchange determinant and integral of a matrix-valued function

Assume $A(x)=(a_{ij}(x))_{k\times k}$ is a Hermitian matrix function on some manifold $M$, is there any inequality relates the integral of its determinant $\int_M det(A)$ and the determinant of its ...

**2**

votes

**1**answer

110 views

### Proof of Binet-Cauchy identity through the polarization transformation [closed]

This questions is motivated by exercise 3.7 in Steele's "The Cauchy-Schwarz Master Class." This is not a homework (I am trying to learn some math by myself) and I have already posted the question on ...

**2**

votes

**0**answers

98 views

### Intuitive (?) inequality extremal inequality

Consider $N$ pairs of random variables $(X_i, Y_i)$. $X_i$ are iid, with $EX_i=0$ and $EX_i^2=1$. The same conditions hold for $Y_i$. Moreover all $X_i$ are independent of all $Y_j$. It seems very ...

**0**

votes

**0**answers

146 views

### Asymptotics to Taylor expansions?

I posted a question on MSE about approximating Taylor series but Despite a bounty I did not receive any answers or comments.
Maybe you guys can help.
http://math.stackexchange.com/questions/1440931/...

**2**

votes

**1**answer

74 views

### interpret of Picone inequality for non-regular functions

Assume $\Omega \subset \mathbb{R}^N$, $ N>4 $ is open set.
There is a well-known picone identity that says
Let $u,v \in C^2(\Omega)$ satisfy $v>0$ and $-\Delta v \geq 0$ in $\Omega$. The ...

**2**

votes

**1**answer

138 views

### maximal inequalities for dependent random variables

I want to know literature about maximal inequalities for dependent random variables i.e. upper bound for $P(\max_{n\ge k\ge 1}\sum_{i=1}^{k}X_i > \delta)$ where $X_i$ are dependent random ...

**-3**

votes

**1**answer

625 views

### How i show this beautiful inequality :$\frac{x^n}{x^m+y^m}+\frac{y^n}{y^m+z^m}+\frac{z^n}{z^m+x^m}\geq \frac{3} {2}(\frac{1}{\sqrt{3}})^{n-m}$? [closed]

This question accross to this question from SE which there some answers but they r n't
enough to me hop to see MO what can they say about it .
let $m,n$ be integers, show that if $ n>m\geq 0 $ :
...

**3**

votes

**0**answers

154 views

### Dynamics of an inequality

The dynamics $D\ni(r_i,r_{i+1})\mapsto(r_{i+1},r_{i+2})\in D$ on the set $D:=\{(x,y)\in\mathbb{R}^2\colon x>0,y>x^2/2\}$ is given by the recurrence
$$r_{i+2}=\frac{r_{i+1}^2}2+\frac1{r_{i+1}^3}
...

**21**

votes

**3**answers

1k views

### Curious inequality

Set
$$
g(x)=\sum_{k=0}^{\infty}\frac{1}{x^{2k+1}+1} \quad \text{for} \quad x>1.
$$
Is it true that
$$
\frac{x^{2}+1}{x(x^{2}-1)}+\frac{g'(x)}{g(x)}>0 \quad \text{for}\quad x>1?
$$
The ...

**2**

votes

**0**answers

136 views

### $\frac{1}{2}<\sigma<1$, is $f(n) = \Bigl| \,1+ \frac{1}{2^{\sigma + i n}} + \cdots + \frac{1}{n^{\sigma + i n}} \Bigr|$ from $O(\log n)$?

We have $\frac{1}{2} < \sigma < 1$ and
$$
f(n) = \Bigl|\, 1+ \frac{1}{2^{\sigma + i n}} + \cdots + \frac{1}{n^{\sigma + i n}} \Bigr|
$$
. My goal is proving this statement that $|f(n)|$ is $O(\...

**4**

votes

**1**answer

510 views

### inequality with exponents

We are given a graph $G$, each vertex $v$ has an assigned value $\gamma_v\in [0,1]$, and it happens that for every $v$ we have $\gamma_v+\sum_{u\in \delta(v)} \gamma_u = 1$. Assume that $\sum_v \...

**10**

votes

**6**answers

693 views

### Identities and inequalities in analysis and probability

Usually, at the heart of a good limit theorem in probability theory is at least one good inequality – because, in applications, a topological neighborhood is usually defined by inequalities. Of course,...

**3**

votes

**1**answer

186 views

### Gagliardo Nirenberg inequality for the laplacian

It is a classical result due to Gagliardo and Nirenberg that there exists a constant C such that it holds
$$
||\nabla \psi|| _{L ^\infty (\mathbb{R}^2)} ^2
\le ||D ^2 \psi|| _{L ^\infty (\mathbb{R}^2)}...

**0**

votes

**1**answer

101 views

### Forbidden Tripartite Graphs

I was looking at extremal graph theory. I have understood the proofs of upper bounds for the Zarankiewicz problem which basically states: What can you say about the edges of a graph with $n$ vertices ...

**0**

votes

**0**answers

91 views

### Integral inequality with Hardy's inequality

Let $f\in C_{0}^{\infty}((-1,1))$. Prove that for any $t\in (-1,1)$ we have
$$(f(t))^4\le \left(\int_{-1}^{1}\dfrac{[2(1-|x|)f'(x)-f(x)][2(1-|x|)f'(x)+f(x)]}{4(1-|x|^2)}dx\right)\cdot\left(\int_{-1}^{...

**0**

votes

**0**answers

104 views

### Norm inequalities between difference operators

Assume $(v_i)$ to be a sequence in $\ell_\infty(\mathbb{R})$ for $i=1,\dotsc,N$. Define the difference operator as $\Delta v_i:=v_{i+1}-v_i$ and $\Delta^n v_i:=\Delta(\Delta^{n-1} v_i)$. Then, how can ...

**2**

votes

**0**answers

106 views

### Vector inequation problem [closed]

$${A_i} = \left( {\begin{array}{*{20}{c}}{{A_{i1}}}\\{{A_{i2}}}\\ \vdots \\{{A_{in}}}\end{array}} \right),{B_i} = \left( {\begin{array}{*{20}{c}}{{B_{i1}}}\\{{B_{i2}}}\\ \vdots \\{{B_{in}}}\end{array}}...

**4**

votes

**0**answers

104 views

### Dimension reduction for low-order moments of Rademacher-weighted sums of vectors

Let $x_1,\dots,x_n$ be vectors in a Euclidean space $H$. Let $\varepsilon_1,\dots,\varepsilon_n$ be independent Rademacher random variables (r.v.'s), so that $P(\varepsilon_i=\pm1)=1/2$ for all $i$.
...

**0**

votes

**1**answer

96 views

### Constrained optimal control problem

I am trying to solve an optimization problem which is probably reminiscent of optimal control theory but all of this is not exactly my field of expertize and I am a little bit lost in translation. If ...

**1**

vote

**1**answer

75 views

### A bound on the $\mathrm{L}^p$ norm in terms of the $\mathrm{L}^2$ norm in periodic Sobolev spaces

(The question was originally posted on Math StackExchange.)
Preliminaries: Given ${s \geq 0}$, let ${\mathrm{H}_P^s}$ denote the ${\mathrm{L}^2}$-based fractional-order Sobolev space of ${P}$-...

**5**

votes

**0**answers

134 views

### Inequality about moments of a random variable and of its conditional expectation

This is a follow-up to a question I asked earlier: Moments of a random variable and of its conditional expectation
My claim turned out to be false. Here is a new claim.
Let $X$ be a bounded random ...

**2**

votes

**1**answer

137 views

### Finding matrices $A$ such that the entries of $A^n$ have specified signs

What techniques are there for ensuring nonnegativity of various entries of matrix powers?
Specific Question: Consider a matrix $A\in SL_2(\mathbb R)$. Let $(A^n)_{i,j}$ denote the $(i,j)$ entry of ...

**7**

votes

**1**answer

213 views

### Moments of a random variable and of its conditional expectation

Let $X$ be a bounded random variable with $\mathbb{E}X=0$. Since $X$ is bounded, all its moments exist. Let $\mathcal{G}$ be any $\sigma$-field and let $Y:=\mathbb{E}[X|\mathcal{G}].$ I am interested ...

**0**

votes

**0**answers

45 views

### Column Inner Products vs. Row Inner Products

Given two matrices $A,B\in\mathbb{R}^{n\times r}$ where $A$ has orthogonal columns and $A^TB$ is symmetric, are there any non-trivial interesting relationships / inequalities between the following ...

**1**

vote

**2**answers

177 views

### Lower bound for $ \sum_{i=1}^n x_i f(x_i)$ when $\sum_{i=1}^{n}x_i = K$

Considering,
the set of all n dim. vectors $\{x_i\}_{i=1,...,n} $ such that $x_i \geq 0 $ and $\sum_{i=1}^{n}x_i = K$
Any continuous and strictly increasing function $f^+(x)$ : $ \mathbb R^+ \to \...

**4**

votes

**0**answers

139 views

### Do more generalizations of Schur's inequality exist?

I meet this following problem
If $$n\ge 3,\sum_{i=1}^{n}\left(\prod_{j\neq i}(a_{i}-a_{j})\right)\ge 0$$
where $a_{i}$ are real numbers.
when $n=3$, it is Schur's inequality
so which $n$ such ...

**10**

votes

**0**answers

154 views

### When is alternating sum $\sum_{i}f(a_i)-\sum_{i<j}f(a_i+a_j)+\ldots+(-1)^{n-1}f(a_1+\ldots+a_n)$ always positive?

Let $a_1,a_2,\ldots,a_n\geq 1$, and let $f:\mathbb{R}^+\rightarrow\mathbb{R}^+$. Consider the sum
$$S(f)=\sum_{i}f(a_i)-\sum_{i<j}f(a_i+a_j)+\sum_{i<j<k}f(a_i+a_j+a_k)-\ldots+(-1)^{n-1}f(a_1+...

**1**

vote

**1**answer

71 views

### An inequality with critical Sobolev exponent

Let $\Omega\subset \mathbb{R}^n, n\geq 3$ be a nice bounded domain and $2^*=2n/(n-2)$ the critical Sobolev exponent. One may expect that $\forall \epsilon>0$, $\exists C_\epsilon<\infty$ such ...