Questions on proving and manipulating inequalities.

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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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55 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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46 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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150 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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40 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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94 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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157 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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53 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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43 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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27 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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77 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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55 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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58 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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96 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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115 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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60 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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150 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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61 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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175 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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242 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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44 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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101 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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85 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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56 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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91 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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45 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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79 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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128 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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201 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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136 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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59 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 ...