Questions on proving and manipulating inequalities.

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86 views

transforming nonlinear matrix inequality to LMI

I faced some nonlinearity in my problems. I need to check a matrix inequality condition in order to check the feasibility of designed controller through a continuous design problem. My problem is that ...
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63 views

Eigenvalues of sum of two particular matrices

Let $A$ be a matrix with real eigenvalues, its maximum eigenvalue is $0$ and it has sum for rows equals to zero. Let $B$ be a matrix $\mathrm{diag}([1\,0\, ...\, 0])$ and let $I$ be the identity ...
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50 views

Proving an inequality involving integral

Let $g: [a,b]\mapsto [0,1]$, with $\int_a^b|g'(t)|^2\,\mathrm{d}t\leq 1$. Suppose $b-a<\delta$, and define $$ \bar{g}=\frac{\int_{a}^{b}g\left(t\right)\,\mathrm{d}t}{b-a} $$ Show for ...
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153 views

Sharpness of the upper bound $(1-x)^n \leq 1 + \frac{nx}{2}$

Here is a known inequality: $$(1-x)^n\leq 1+\frac{nx}{2}\qquad \text{for} \, \frac 1n\leq x\leq 1 $$ I am wondering if there is a better upper bound than this? Thank you.
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29 views

Youngs inequality for any power of X and Y

So I am working on a condition where I want to find the maximum "p" and "q" values for the Youngs inequality shown below . I am looking to find the $p$ and $q$ which upper bounds $x^my^n$ by a ...
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37 views

Showing an inequality relating two Poisson tail-probabilities

In my research, I've discovered that a property that I am interested in is equivalent to an inequality involving two tail-probabilities of the Poisson distribution. I belive this inequality to be ...
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74 views

Inequality with Fibonacci numbers

The sequence $F_n$ of natural numbers defined by equation $F_{n+2}=F_{n+1}+F_{n}$, with $F_0=0, F_1=1$ is called the Fibonacci sequence. The n-th term in the sequence is called the n-th Fibonacci ...
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29 views

Proving $~\sum_{cyclic}\left(\frac{1}{y^{2}+z^{2}}+\frac{1}{1-yz}\right)\geq 9$

$a$,$b$,$c$ are non-negative real numbers such that $~x^{2}+y^{2}+z^{2}=1$ show that $~\displaystyle\sum_{cyclic}\left(\dfrac{1}{y^{2}+z^{2}}+\dfrac{1}{1-yz}\right)\geq 9$
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162 views

Choosing the vector that minimizes this sum related to the rearrangement inequality

The rearrangement inequality states that, for two sets of real numbers $x_1\leq\dots{}\leq x_n$ and $y_1\leq\dots{}\leq y_n$, the sum $\sum_{i=1}^n x_{\sigma(i)}y_i$ is minimized for the particular ...
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50 views

Prove or disprove an inequality involving statistics

Do we have any result in statistics like this: $$|\overline x - \mu_e| \leq \sigma$$ Here $\overline x$ denotes the usual mean of some given discrete observations, $\mu_e$ their median and ...
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58 views

Increasing fraction of sum of binomial coeffients

Let $n$ be a positive integer. Show that the quantity $$ \displaystyle \frac{ \displaystyle \sum_{i=1}^n { n+k \choose i-1 } }{ \displaystyle \sum_{i=1}^n { n+k+1 \choose i } } $$ is ...
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71 views

Sequence of probabilities with monotone function

Let $\{X_k\}_{k=1}^{\infty}$ be a sequence of i.i.d. random variables with finite support $S = \{ 1, 2, ..., N\}$. Let $P$ be the corresponding probability measure. For all $k \geq 1$, define $A_k := ...
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49 views

Generalized inequality with parameters $\alpha, \beta$

Let $d$ be a positive integer, and let $\alpha, \beta$ be positive real numbers such that $\alpha+\beta=1$. Consider the inequality in $k$ variables $x_1, x_2, …, x_k$, $$ \alpha \cdot \sum_{i=1}^k ...
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35 views

Bound on the difference of two convergent infinite products

Let $(\alpha_n)$ and $(\beta_n)$ be two sequences of non-zero complex numbers such that the products $\prod_n \alpha_n$ and $\prod_n \beta_n$ are convergent. How to prove the following inequality? $$ ...
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121 views

How to use Chebyshev's inequality or the law of large numbers to a probability question

Let x be a random bit string that takes values $\{1,0\}^n$. Let r be the value of the most significant (MSB) bit of x (and r is a r.v. 1 or 0 that are equally likely). Let g be our guess for the MSB ...
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70 views

An inequalities on order statistics

Let $X=(X_1,...,X_k)$ be a square integrable random vector in $\mathbb R^k$ and $X_{(j)}$ the $j$-th ordered value of $X$, i.e., $X_{(1)} \leq X_{(2)} \leq ... \leq X_{(k)}$. Prove (or disprove) that ...
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48 views

find conditions on input data such that a linear system has (no) feasible points

As a result of the apllication of Farkas' lemma I obtained the following problem: Let $ m,n,q \in \mathbb{N} $, $ b \in \mathbb{N}^m, l \in \mathbb{N}^m $ with $ l_i \mid q$ for all $ i=1,\ldots,m $. ...
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72 views

Find Minimum value of this expression?

Exercise 1: Let $a, b, c\ge 0$ satisfying $ab+bc+ca>0$. Find the minimum value of this expression: $P=\frac{1}{\sqrt{a(b+c)+2c^2}}+\frac{1}{\sqrt{b(a+4c)}}+2\sqrt{a+2b+4}+4\sqrt{c+1}$ Exercise 2: ...
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59 views

$L^{\infty}$ is p-concave

I want to show that the $L^{\infty}$ on a Banach lattice $X$ is $p$-concave with $M_{(p)}(L^{\infty})=1$. Where $L^\infty=L^\infty(X,\mathcal{M},\mu)$. Recall that a Banach lattice $X$ is said ...
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64 views

How prove this inequality $\sum \left|\frac{z_{1}z_{2}}{(z_{3}-z_{1})(z_{3}-z_{2})}\right|\ge\frac{9}{4}$

let $z_{1},z_{2},z_{3}\in \mathbb{C}$,and such $$z_{1}+z_{2}+z_{3}=0$$ show that ...
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72 views

Prove that $0\le\frac1{b-a}\int_a^b|f(x)|dx-\left|\frac1{b-a}\int_a^bf(x)dx\right|\le\frac{b-a}3\sup_{a\le x\le b}|f'(x)|$

Let $f'$ be integrable. Prove that $$0\le\frac1{b-a}\int_a^b|f(x)|dx-\left|\frac1{b-a}\int_a^bf(x)dx\right|\le\frac{b-a}3\sup_{a\le x\le b}|f'(x)|$$ Source: ...
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114 views

Inequalities involving liminf and limsup

I got homework about liminf and limsup. --Edit-- I guess that it would be better writing the question and my solution here... Question: Let $ a_n,b_n $ be sequences. Given that $ a=\underline{lim} ...
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36 views

How show that $\det(a^{b_{i}}_{i}+1)+\det(a^{b_{i}}_{i}-1)>0$

Let $1\le a_{1}<a_{2}<\cdots<a_{n},1\le b_{1}<b_{2}<\cdots<b_{n}$, show that $$\begin{vmatrix} a^{b_{1}}_{1}+1&a^{b_{2}}_{1}+1&\cdots&a^{b_{n}}_{1}+1\\ ...
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40 views

(Dis)prove that this system of inequalities has only $(0, 0, 0)$ as a solution

Consider the following system of inequalities: $$ \left\{ \begin{array}{c} \beta^2 + 4\gamma\theta \geq 0 \\ \theta^2 + 4\gamma\beta \geq 0 \\ \gamma^2 + 4\theta\beta \geq 0 \end{array} \right. $$ ...
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33 views

Modeling 4 people going to same place over 3 different places for at least 5 days

I'm trying to model a linear programming task with the condition 4 people going to the same place among 3 different places for at least 5 days. I have the variables for the time spend each person in ...
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58 views

How to know how many bounds an inequality has?

How can you know, before solving it, how many bounds an inequality should have? For example $$ \dfrac{x^2 + 2}{1-x^2} < 3$$ A priori to me it looks like it would have 2 bounds because it's a ...
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100 views

Is there such an example?

Is there an example of a sequence of point sets $\left\{ S_{n}\right\} _{n=1}^{\infty}$in which $S_{n}$ is a set of $n$ points inside the unit triangle, such that the minimum altitude of the triangles ...
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32 views

Expectation comparison of order statistics

$n$ variables $X_1,..., X_n$ are independently and identically drawn from pdf $f(x)$ and cdf $F(x)$ on the interval $[\underline{x}, \bar{x}]$. Denote $X_{(i)}$ as the $i$th smallest of all ...
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58 views

SOLVED — Equal or Equivalent Inequalities?

Suppose I have two inequalities: $2ab \leq a^{2} + b^{2}$ $0 \leq a^{2} + b^{2} - 2ab$ Now, obviously these inequalities are just rearrangements of one another. The question I have is: do I say ...
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61 views

Correct constant for Khintchine-Kahane inequality?

I'm reading this paper http://jmlr.org/papers/volume13/kloft12a/kloft12a.pdf In Lemma 1, it says the constant for the Khintchine-Kahane inequality is q*. But if so, I don't think the authors can get ...
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42 views

Ineqality regarding LCM of $1, 2, \ldots, n$

While going through F. Beukers proof of irrationality of $\zeta(3)$ I found the inequality $d_{n} < 3^{n}$ for all sufficiently large values of $n$ where $d_{n}$ denotes the LCM of all the numbers ...
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56 views

Prove that the length of this curve decreases as one of its parameters increases

The following is the problem statement of one of my assignment questions. Consider the $\partial_t (t,s) = K(t, s)N(t, s)$ for all $t \geq 0$, and for all $s \in [0, 1]$, where $T(t, s) = ...
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50 views

Intuition behind Gaussian isoperimetric inequality

I was wondering whether or not there's an intuitive way of understanding the Gaussian isoperimetric inequality. I have been studying the Classical isoperimetric inequality and I finally understand it. ...
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151 views

When $\int_a^b\left(f(x)\right)^kdx=\left[\int_a^bf(x)dx\right]^k$

Under what conditions on $f(x)$ the following equation holds? $$\int_a^b\left(f(x)\right)^kdx=\left[\int_a^bf(x)dx\right]^k$$ with $k\in\mathbb{N}$ and $k\gt1$. I know the following inequality holds: ...
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44 views

Proving Muirhead-like inequalities

Let $T_{m,n,p}(x,y,z)=\sum_{Sym} x^m y^n z^p$. For $x,y,z>0$, prove $2T_{6,3,0}(x,y,z)+T_{3,3,3}(x,y,z)+3T_{4,4,1}(x,y,z)\geq 6T_{5,2,2}(x,y,z)$. I tried to prove that by using AM-GM ...
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97 views

Generalization of Minkowski's inequality

Let $$\mathcal{M}_\phi(x)=\phi^{-1}\left(\sum_i q_i \phi(x_i)\right)$$ where $x=x_1,...x_n$ is a sequence of positive, real numbers, $\sum q_i=1$ and $\phi(\cdot)$ is a twice continuously ...
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168 views

Inequality that may use Simon's Favorite Factoring Trick

Let $a,b,c,d,e$ be positive integers. When is $abcd>abc+abd+acd+bcd+ab+ac+ad+bc+bd+cd+a+b+c+d$? Additionally, when is $abcde>abcd+abce+...a+b+c+d+e$?
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57 views

Linear Complementarity Problem - multiple solutions, which one will it find?

If I have a inequality constrained system: w = Mz + q <= 0, z<=0, z^T w = 0 that for some given properties M and ...
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44 views

What are the best and most elementary bounds for $n!$?

What this question is looking for is bounds on $n!$ that are elementary in nature (I seem to have a fetish for these type of proofs). In general, as the results become more complicated, they also ...
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28 views

How find this $m$ Value range

let $a\ge\dfrac{2^{m-1}-1}{m-1}$and such $$\left(\dfrac{\dfrac{3}{4}(\dfrac{3}{4}+a)(\dfrac{3}{4}+2a)\cdots(\dfrac{3}{4}+(m-1)a)}{(1+a)(1+2a)\cdots ...
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79 views

Certain matrix inequalities

I want to solve the following inequalities: \begin{equation} \left| Tr\left( \frac{(X\otimes Y).A.(X\otimes Y)^*.B}{Tr((X\otimes Y).A.(X\otimes Y)^*)}\right)\right|>2, \quad\text{given} \quad ...
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56 views

convolution inequality on R

Let $\nu$ be a complex Radon measure on $\mathbb{R}$ such that $$ \int_{\mathbb{R}} \check{\overline{f}}*f\ d\nu\geq 0 $$ for any complex continuous function $f$ with compact support, where ...
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Inequality of $h(s)=\int_{0}^{\infty}t^{N-1}e^{-t}\prod_{i=1}^M(1+\lambda_{i}ts)^{-1}dt.$

Suppose \begin{align} h(s)=\int_{0}^{\infty}t^{N-1}e^{-t}\prod_{i=1}^M(1+\lambda_{i}ts)^{-1}dt. \end{align} where $M$ and $N$ are positive integers and $N<M$, $\lambda_i$ are distinct positive ...
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60 views

Simplistic method for proving inequlaities

I have noticed that many inequalities posed in olympiads or otherwise were solvable using Lagrange Multipliers. However, the method might get tedious in which case, I had noticed that in inequalities ...
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97 views

Lower bound for $\pi(x)$

Is there a way to show that $$\frac{x}{\ln x} < \pi(x),$$ for sufficiently large $x$, using only elementary calculus? Apparently it is true for $x \geq 17$ (see this article). However, I am looking ...
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118 views

Normal distribution inequality

Let $n(x) = \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}}$, and $N(x) = \int_{-\infty}^x n(t)dt$. Prove the following inequality. $$(x^2+1)N + xn-(xN+n)^2>N^2$$ where the dependency of $n$ and $N$ on ...
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50 views

inequality with expectation of a function

Question $X$ is an RV. Let $f\colon \Bbb R \to \Bbb R$ be a strongly ascending function. $\forall X: m<f(X)$ a. Show that if $f(X)$ has a finite expectation then $P(X\ge t)\le\frac ...
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61 views

How prove this$\left|\frac{\sin{(\sqrt{\lambda}\cdot\tau)}}{\sqrt{\lambda}}\right|\le e^{|\mathrm{Im}\sqrt{\lambda}|\cdot\tau}$

let $\lambda $ is a any complex numbers,and $\tau\in[0,1]$ show that $$\left|\dfrac{\sin{(\sqrt{\lambda}\cdot\tau)}}{\sqrt{\lambda}}\right|\le e^{|\mathrm{Im}\sqrt{\lambda}|\cdot\tau}$$ my idea: ...
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41 views

Proving Weierstrass sine representation mass bound: $\inf_n \sup_{|w| \geq m} |s_n(w)| \leq 4^{1-m}$

If one defines $$ s_n(w):=\frac{\sin\pi w}{\pi w}\prod_{j=1}^n\left( 1-\frac{w^2}{j^2}\right)^{-1}=\prod_{j=n+1}^\infty\left( 1-\frac{w^2}{j^2}\right),$$ how may one show that, for every $m \geq 0,$ ...
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156 views

How prove the following inequality

Let $x,y,z\ge 0$,$x+y+z=3$,prove: $$3{{x}^{2}}(1+2y){{(1+z)}^{3}}+3{{y}^{2}}(1+2z){{(1+x)}^{3}}+3{{z}^{2}}(1+2x){{(1+y)}^{3}}\le {{\left( 3+xy+yz+zx \right)}^{3}}$$ my idea:let ...