Questions on proving and manipulating inequalities.

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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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54 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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17 views

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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56 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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94 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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112 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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49 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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59 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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40 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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149 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 ...
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59 views

Algebra inequalities

I'm trying to prove the following inequalities: $$ (1+a^2)^s \leq (1+(a-b)^2)^s + (1+b^2)^s$$ $$ 1 +a^2 \lesssim (1+(a-b)^2)(1+b^2).$$ They're in a set of notes I'm reading, just stated in passing. In ...
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40 views

Estimation of a scalar product

I encountered the following, which shouldn't be that hard, but I can't get my head around it. The problem is the following estimate (part of a bigger equation, but here's just the difficult part): ...
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173 views

A puzzling inequality involving exp, erf, and log

For $0<y<x<\infty$, I believe the following inequality is true (I've tested it numerically with random values for $x$ and $y$) but have been unable to analytically confirm: \begin{align} ...
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45 views

$ | \;\overline{z_1}^{3-\alpha} z_1^\alpha - \overline{z_2}^{3-\alpha} z_2 ^\alpha | \leq C ( |z_1 - z_2 |^2 + |z_2| ^2 )|z_1 - z_2|$?

I questioned about this inequality before, but how about weaker one: For $\alpha = 0,1,2,3$, does this inequality always hold for any complex number $z_1, z_2$? $$ | \;\overline{z_1}^{3-\alpha} ...
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49 views

conversion from psi function to prime counting function

Can we convert $\psi(x)$ to $\pi(x)$ without using integrals. Also if $\psi(x)>\psi(y)$ when we can say that $\pi(x)>\pi(y)$ . It seems that $\theta(x)>\theta(y)$ so $\pi(x)>\pi(y)$ but ...
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21 views

Some questions about a “relaxed” invariant probability problem $|\mu(P-I)|\leq \epsilon$

Let's consider the set $\mathcal{M}=\{\mu:|\mu(P-I)|_1\leq \epsilon\}$ where $\mu$ is a probability vector, $P$ is the transition matrix of a discrete homogeneous Markov chain, $I$ is the identity ...
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241 views

proving inequality for combinatorial sum

If somone can prove the following for every $d\leq r$ (for $d=0,1$ its easy, see below, the case d=r may be also simple, I didn't find something helpful) $$\frac{(d!)^2}{2^{n-2d}}\sum_{k=0}^{n}{n ...
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42 views

Inequality of Partial Taylor Series

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} \frac{x^k}{k!} ...
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41 views

Find the small value of the following functions

Choose $1<x_1<x_2<\cdots<x_M<2$ , such that $$\left|\sum\limits_{i=1}^{M}x_{i}^{2013}\dfrac{1}{\prod\limits_{1\leq p\leq2013\,,\,,p\neq i}(x_{i}-x_{p})}\right|\leq2$$ where $M=100$
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99 views

Inequalities involving regularized incomplete Gamma functions

I am new to the world of the Gamma functions and am wondering if there exist positive functions $f_1(x)>0$ and $g_1(x)>0$, and non-negative functions $f_2(x)\geq0$ and $g_2(x)\geq0$ such that ...
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136 views

How prove this $\dfrac{x^y}{y^x}\ge (1+\ln{3})x-(1+\ln{3})y+1$?

let $x>0,y>0,z>0$, and $x+y+z=1$,prove that $$\dfrac{x^y}{y^x}+\dfrac{y^z}{z^y}+\dfrac{z^x}{x^z}\ge 3$$ my idea: let $f(x,y)=\dfrac{x^y}{y^x}$ then we consider $$f(x,y)\ge g(x,y)=px+qy+r$$ ...
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84 views

Graphical representation for triangular inequality

Is it possible to represent all the problems graphically that involve triangular inequality to solve them. For example, to prove that for any series $a_n \to a$ for every $m\gt n$ hence there is only ...
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55 views

inequality for series

Let $j \in Z_+$. Set $$ a_j^{(1)}=a_j:=\sum_{i=0}^j\frac{(-1)^{j-i}}{i!6^i(2(j-i)+1)!} $$ and $a_j^{(l+1)}=\sum_{i=0}^ja_ia_{j-i}^{(l)}$. Let $X(i)=|a^{(2i)}_j|j!$. Verify that $X(i)\leq X(1)$ for ...
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89 views

How was the isoperimetric inequality formulated?

I'm tyring to understand how the isoperimetric inequality came into existence. It seems like finding the region which yields maximum area when enclosed by a curve of fixed length is an old problem. ...
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51 views

The cdf of a beta variable, evaluated at the mean

Consider a Beta random variable $X$ with shape parameters $k/2$ and $(d-k)/2$, where the parameters $k, d$ are integers that satisfy $0 < k < d$. What is the best possible upper bound for the ...
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46 views

How do I bound this sum with Chernoff?

I have a random variable $U$ equal to the sum of $j$ identical, independent other random variables $U_1$ through $U_j$, all of which have mean 0. $j$ is going to be a very big number. For some ...
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178 views

how to solve the following expectation? closed-form expression or approximation

Suppose there is a binomial random variable $X\sim B(n-1,p)$,how to solve the following expectation $$E[(1- b^{X})^{m}]$$ where $b\in (0,1]$ and $m\in \mathbb{N} $ are all constants.I have tried my ...
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114 views

Proof of an inequality problem

Wise men or women over the world!! I badly ask you to help me. Let $N$ and $B$ be two positive integers such that $1\le B\le \frac{N}{2}$ and $N=ug$ (for convenience, assume that $N$ is even.) For ...
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74 views

Inequality with Expectation and vectors

Let $x=(x_1,\ldots, x_n)\in R^n$. Let $r_i, i=1, \ldots, n$ be Rademacher i.i.d. random variables (i.e. $P(r_i=1)=P(r_i=-1)=1/2$). It is a well-known inequality that: $$ E\left(\left|\sum_{i=1}^n ...
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112 views

Inequalities of integrals of periodic functions

I have a function that has a shape similar to $\sin(x)^2$ (could be periodic extensions of $(x/(\pi/2))^2$ defined between $-\pi/2$ to $\pi/2$ for example). Let's call it $g(x)$. I want to show that ...
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74 views

Reference for Khinchine inequality

I am looking for the proof of Khinchine inequality (see http://en.wikipedia.org/wiki/Khintchine_inequality for example), using martingales and the Azuma inequality. Can you please help me to find a ...
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44 views

Show inequality for modified bilinear form

Let $\Omega_h$ denote to the domain that is bounded by a polygon, and $V_h$ to the space of all $c\in C^0(\Omega)$ such that $v_{|T}$ is linear on any (curved) triangle T and $v=0$ in the vertices of ...
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92 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 ...
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78 views

Inequality, with quotient substitution

I do not know how to prove this inequality: Suppose that $x_i>0$ and $x_1\cdot ...\cdot x_n=1$, show that $$\frac{1}{1+x_1+x_1x_2}+...+\frac{1}{1+x_n+x_nx_1}>1$$ The hint is to use quotient ...
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127 views

An inequality involving exponential of compact self-adjoint operator (SOLVED)

I've run into a tricky functional analysis problem. Here it is: Suppose $$A: H \to H$$ is a compact self-adjoint operator on a Hilbert space H. Assume that the spectrum of $A$ is located in the open ...
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200 views

Cauchy-Schwarz inequality - find minimum.

I'm stuck at proof of finding minimum of the expression $\ \sum_{k=1}^{n}a_{k}^{2}+\left(\sum_{k=1}^n a_k\right)^2\\ \sum_{k=1}^{n}p_{k}a_{k}=1\\ $ So my first thought is to square ...
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135 views

Harnack Inequality…

Consider the eigenfunction $\varphi_R>0$ $$L\varphi_R=\lambda_R\varphi_R, \ \ \ in \ \ B_R,$$ and $$\varphi_R=0, \ \ \ in \ \ \partial B_R,$$ where $L$ is a elliptic operator and $\lambda_R$ is the ...
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180 views

Does this hold?

Strayed on the following question. Assume that $x_{1}$,$\ldots$, $x_{d}\ge0$ with $x_{1}+\ldots+x_{d}=1$ and $y_{1},\ldots,y_{d}\in\mathbb{R}$. Does $$ \min_{1\le i\ne j\le ...
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90 views

Proving a simple inequality

Can someone show that the inequality bellow holds? $$ f(n) \leq f(n+1) \ $$ Where $$ \frac{\sum\limits_{k=1}^n \Lambda(k) {k}/{n}\lceil{n}/{k}\rceil{}\{ n/k \}}{\sum\limits_{k=1}^n \Lambda(k)}=f(n)$$ ...
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58 views

Inequality help

Can someone help me prove the inequality, $$ \frac{\sum\limits_{k=1}^n \Lambda(k) \frac{k}{n}\lceil\frac{n}{k}\rceil\{ \frac{n}{k} \}}{\sum\limits_{k=1}^n \Lambda(k)}<\ \frac{\sum\limits_{k=1}^n ...
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120 views

An integral inequality related to maximal function

Suppose that $f\in C^1([0,\infty),\mathbb{R})$,and $F(x)=\max_{x\leq y\leq 2x}|f(y)|$,then show that $$ \int_{0}^{\infty}F(x)dx\leq \int_{0}^{\infty}|f(x)|dx+\int_{0}^{\infty}x|f'(x)|dx $$ EDIT ...
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143 views

second order stochastic dominance

Let the nonnegative random variables $X$ and $Y$ have distribution functions $F$ and $G$ and density functions $f$ and $g$, respectively. Suppose $X$ is second-order stochastically dominant over $Y$, ...
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108 views

Binomial coefficient intervals (inequality)

For given $N$, $x$ and $k$ such that $0\leq x<N$ and $2\leq k\leq \left\lfloor \frac{N+1-2x}{2}\right\rfloor $, does it exist $p,$ $2\leq p\leq \left\lfloor \frac{N+1}{2}\right\rfloor $ such that ...
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118 views

Bounding the product of a quadrilateral's side lengths in terms of the lengths of its diagonals

Suppose we've a convex quadrilateral both of whose diagonals have length 2. Is it true that the product of the lengths of the quadrilateral's sides must be less than or equal to 5? If we require, in ...
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39 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: ...
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104 views

How to prove an inequality for a special structure of strictly triangular matrix

The problem I cause is attached below. I am trying to prove the inequality. By using small $M$, I found that the terms on the left side of the inequality are part of the terms of the expansion on ...
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116 views

Help proving that $\frac{2}{n+\frac{1}{2}} \leq \int_{\frac{1}{n+1}}^{\frac{1}{n}}\sqrt{1+(\sin(\frac{\pi}{t}) -\frac{1}{t}\cos(\frac{\pi}{t}))^2}dt$

I am trying to prove that for $n\geq 1$: $$\frac{2}{n+\frac{1}{2}} \leq \int_{\frac{1}{n+1}}^{\frac{1}{n}}\sqrt{1+\sin^2(\frac{\pi}{t}) - \frac{2}{t}\sin(\frac{\pi}{t})\cos(\frac{\pi}{t}) + ...
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97 views

Inequality estimation

Let $B$ be an open unit ball in $\mathbb{R}^d$ centered at the origin and $u$ be a twice continuously differentiable function on $\bar{B}$ with $u|_{\partial B} = 0$. Know $$\Delta u = f.$$ How can I ...
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44 views

Minimal modulus for the finite field NTT

I need your support. Suppose I am performing an NTT in a finite field $GF(p)$. I assume it contains the needed primitive root of unity. I am using it to compute the convolution of two vectors of ...
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97 views

Find the nearest point that lies within a polygon specified by 6 linear inequalities

I have six linear inequalities that together specify a polygon. For a given point $P$, how can I find the nearest point $P'$ that satisfies all six inequalities (if $P$ itself does not)? [edit: ...