Questions on proving, manipulating and applying inequalities.

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2
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37 views

For which minimal $k$ true is that ${4}^{k}\cdot n\leq \displaystyle\sum^{n}_{i=1}{a}_{i}^{k}\leq {5}^{k}\cdot n$, ${a}_{i}\in {1,2,3,4,5,6}$?

I've got the following inequality, which bounds Minkowski distance. ${4}^{k}\cdot n\leq \displaystyle\sum^{n}_{i=1}{a}_{i}^{k}\leq {5}^{k}\cdot n$ and values of ${a}_{i}\in {1,2,3,4,5,6}$ We know all ...
2
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0answers
65 views

Lower bound on diophantine system of inequalities with all but one non-linear constraint

I have a system of $n+1$ diophantine inequalities, in the following form: $$f_{1}(x_1, x_2, \dots, x_n) \geq 0$$ $$f_{2}(x_1, x_2, \dots, x_n) \geq 0$$ $$\vdots$$ $$f_{m}(x_1, x_2, \dots, x_n) ...
2
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0answers
343 views

Find a tight upper bound of the following expectation.

I am stuck in finding a tight upper bound (as tight as possible) of the following expectation $$E\left [ (1-a\cdot b^{X})^{m} \right ]$$ where $X\sim B(n-1,p)$ is a binomial random variable.In ...
2
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219 views

How to solve systems of polynomial inequalities?

I am currently working on a project that deals with systems of inequalities and so far I have found algorithms for the basic case of a system of inequalities as well as the non-strict linear ...
2
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0answers
127 views

Inequality with binomial coefficient

Let $n$ be a natural number, $m\in [-n, n]$. Let $p=0,\ldots, \frac{n+m}{2}$. Show, that for all $p$, $$ {n \choose \left[{\frac{n+m}{2}}\right]}\geq \frac{2^{n+1/2}}{\sqrt{n-p/2}}. $$ Thank you for ...
2
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55 views

Symmetric matrices $M$ such that $x,y\geq 0$ implies $(x^t M y)^2\geq (x^t M x)(y^t M y)$

Is there a name for $n$-by-$n$ symmetric matrices $M$ such that for all $n$-dimensional non-negative-valued vectors $x,y$ we have $$(x^t M y)^2 \geq (x^t M x)(y^t M y)?$$ In particular I am ...
2
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0answers
46 views

Randomized Solution to a System of Inequalities

Given a set of $\mathbf v_i \in \{0,1\}^k$ for $i=1,\dots,n$ and a vector $\mathbf x \in [0,1]^k$, we want to decide if the following inequality holds or not: $$ \mathbf x \le \sum_{i=1}^n \alpha_i ...
2
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117 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 ...
2
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39 views

The sign of a sum of integrals over a probability measure

Let $\mu$ be a probability measure, $f$ a function taking values in $[0,1]$. I am trying to determine the sign of the expression $$3\left( \int f^2 d\mu \right)\left( \int f d\mu \right) - 2 ...
2
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138 views

Inequality involving norms.

Suppose $p,q,r \in[1, \infty)$ and ${1\over r} = {1\over p} + {1\over q}$ . How can I use Minkowski's Inequality for prove below? $$||fg||_r \le ||f||_p||g||_q$$
2
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0answers
102 views

Inner product and inequalities

Suppose $p:[0,1]\to \mathbb C$ is a curve where $p(t)=u(t)+iv(t)$ and $u,v$ are smooth functions of $t$. Why then is $$\left(\int_0^1 \langle \dot{p},\dot{p}\rangle^{1\over 2} dt\right)^2\le \int_0^1 ...
2
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0answers
121 views

Generalizing an approach to proving AMGM

This problem is Exercise 5.5.30 of "The Art and Craft of Problem Solving" by Paul Zeitz. The problem asks to use the identity $$ a^3+b^3+c^3-3abc = (a+b+c)(a^2+b^2+c^2-ab-ac-bc) $$ to prove the AMGM ...
2
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0answers
148 views

Can anyone point me to an elegantly simple proof of Muirhead's Inequality?

I am currently finishing a project for a module I have in Mathematical Investigations. I have been looking at inequalities and ways to produce true inequalities in homogeneous symmetric form. I have ...
2
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0answers
161 views

another inequality involving complex numbers.

Let $\{z_i\}$, $i=1,2,\ldots,n$ be a set of complex numbers. Then I know that there is a set $J$ such that $$\left|\sum_{j\in J} z_j\right|\ge \frac{1}{\pi} \sum_{k=1}^n |z_k|. $$ However, how do I ...
2
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192 views

Help with an integral inequality involving an incomplete beta function

I would like to determine if the following inequality is true: ...
2
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0answers
194 views

A trigonometric inequality involving sine

Let $0<a<\pi/2,0<b<\pi/2$, $0<\lambda<1, \mu=1-\lambda$. Does anyone see a good proof of the inequality: $$\sin(\lambda a)\sin(\lambda b)+\sin(\lambda a)\sin(\mu ...
2
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0answers
303 views

Two vague steps in the proof of Harnack inequality

I am reading the book Elliptic and Parabolic Equations and the proof is excerpted from page 133-136. In Theorem 5.1.3: it claims that ...
2
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88 views

Single solutions to an inequality

Suppose we had an inequality $ax < by < cx$ where $a,b,c,x,y \in \mathbb Z$. If we fix $a,b,c$ and let $x,y$ vary how can we find the values of $x$ for which only a single $y$ satisfies the ...
2
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295 views

Sharp (Reverse) Harmonic-Arithmetic Mean Bounds

Let $\mathbf{x} =$ {$x_{i}$} be a set of $n$ positive reals. In every good book on inequalities, one finds the classical result \begin{eqnarray} AM(\mathbf{x}) \geq GM(\mathbf{x}) \geq ...
1
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0answers
27 views

$f \in C^2(\mathbb R)$ , $(f(x))^2 \le 1$ ; $(f'(x))^2+(f''(x))^2 \le 1 $ ; then is $(f(x))^2+(f'(x))^2 \le 1 $?

Let $f \in C^2(\mathbb R)$ be such that $$(f(x))^2 \le 1 ; (f'(x))^2+(f''(x))^2 \le 1 , \forall x \in \mathbb R$$ Then is it true that $(f(x))^2+(f'(x))^2 \le 1 , \forall x \in \mathbb R$ ? I ...
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20 views

Constant of Holder-type Inequality for Polynomial Function

Is anybody aware of an inequality in the following form $$ \Vert f \Vert_{L_p(\Omega)} \leq C(p) \Vert f \Vert_{L_q(\Omega)} $$ where $f$ is a polynomial function of degree $p$ on $\Omega \subset ...
1
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0answers
31 views

Prove trigonometric inequality with sin

Let $ n\in \mathbb{N}^{*},x\in \mathbb{R} $. Prove that $ sin^{2}(x)\cdot sin^{2}(2x)\cdot ...\cdot sin^{2}(2^{n}x)\leq \left ( \frac{3}{4} \right )^{n},\forall x\in \mathbb{R} $. The only result I ...
1
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0answers
41 views

Minimum variance, fixed mean , discrete random variable

Consider the ordered set $\mathcal{S}$ $=$ $\{0,a_i,a_2,\ldots,a_n\}$, where $a_i$ are all stricly positive real numbers and $a_i< a_{i+1}$ forall indices i. What is the random variable $X$ which ...
1
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0answers
17 views

Trying to find the asymptotic behaviour of an inequality involving integers

Let $m,q,v$ be integers with $m\geq 2$, and $v|q-1$. A certain result that I have which is not important for this question, holds when $$q^{\frac{m}{2}-2}(q-mv)\geq v^{m-1}. \quad (1)$$ I would like ...
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0answers
54 views

Prove that $(xy+xz+yz)^2 \leq (2x^2 + y^2)(2z^2 + y^2)$

What is the easiest way way to prove that $$(xy+xz+yz)^2 <= (2x^2 + y^2)(2z^2 + y^2)$$ holds for real $x$, $y$, $z$? I've solved it using some "advanced" mathematics, but I would like to know ...
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33 views

Does this inequality $e^{H_n}\ln[H_n(H_n+0.5)]-e^{H_n-1}(H_n+1)\le 2n$ Hold?

Valid for all values of $n\ge 1$ Does this inequality hold? $$e^{H_n}\ln[H_n(H_n+0.5)]-e^{H_n-1}(H_n+1)\le 2n$$ Where $H_n=\sum_{i=1}^{n}\frac{1}{i}$ Let $A=H_n$ I simplified to ...
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0answers
15 views

Terminology for asserting truth of equality/inequality based on symbolic equalities/inequalities

This may seem silly, but I am curious about algorithms used to computationally assert the truthiness (true, false, or unknown) of symbolic statements subject to a set of inequality constraints, for ...
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0answers
54 views

Is it true for $a,b,c$ positive reals that $\frac{a}{\sqrt{a^2+2b^2}}+\frac{b}{\sqrt{b^2+2c^2}}+\frac{c}{\sqrt{c^2+2a^2}} \ge \sqrt3$

Is it true for $a,b,c$ positive reals that $$\frac{a}{\sqrt{a^2+2b^2}}+\frac{b}{\sqrt{b^2+2c^2}}+\frac{c}{\sqrt{c^2+2a^2}} \ge \sqrt3$$ My thoughts: The LHS is equal to ...
1
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0answers
32 views

Ineqaulity $\int_{\partial B(x,r)}\mathbb 1_{(0,1)}(y)\ dS(y)\le \int_{\partial B(0,1)}1\ dS(y)$

$\int_{\partial B(x,r)}\mathbb 1_{B(0,1)}(y)\ dS(y)\le \int_{\partial B(0,1)}1\ dS(y)$ for any $x\in\mathbb R^3, r>0$ How can I shove the above inequality ? Is the RHS area of the ball with ...
1
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0answers
68 views

$a,b,c >0$, and $ab+bc+ca=3$, prove $(a^ab^bc^c)^{\frac{3}{a+b+c}} \geqslant \sqrt[3]{\frac{a^3+b^3+c^3}{3}}$

$a,b,c >0$, and $ab+bc+ca=3$, prove $$(a^ab^bc^c)^{\frac{3}{a+b+c}} \geqslant \sqrt[3]{\frac{a^3+b^3+c^3}{3}}$$ I think the equality is only achieve when $a=b=c=1$. The condition $ab+bc+ca=3$ is ...
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0answers
57 views

Prove $a^{ab}b+b^{bc}c+c^{ca}a \geqslant \sqrt[6]{5}$

$a,b,c >0$, and $a+b+c=3$, prove $$ a^{ab}b+b^{bc}c+c^{ca}a \geqslant \sqrt[6]{5}$$ I try to substitute $c=3-a-b$ to reduce the number of variables, but cannot further proceed to solve the ...
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0answers
38 views

Upper-bounding $\sum_{i=1}^n \sum_{j = i}^{i+a_i} \frac{1}{\sqrt{j}}$?

Suppose $a_1, ..., a_n \in \mathbb{N}$ are arbitrary integers. Is it possible to bound $$ A =\sum_{i=1}^n \sum_{j = i}^{i+a_i} \frac{1}{\sqrt{j}} $$ with either of the following: $$ B = ...
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0answers
48 views

$1+x^4\leq 2(y-z)^2$ and switching of $x,y,z$

Find all triples of real numbers $x,y,z$ such that $1+x^4\leq 2(y-z)^2$, $1+y^4\leq 2(z-x)^2$, and $1+z^4\leq 2(x-y)^2$. Beside $(1,0,-1)$ and permutations, I can't find any others. We cannot have ...
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0answers
40 views

Prove that if $f(\lambda x) = \left(\frac{1}{\lambda}\right)^N f(x)$ than $|f(x)| \leq c \left|\frac{1}{x}\right|^N$

The task is: Knowing that $\forall \lambda >0, x \neq 0$ $f(\lambda x) = \left(\frac{1}{\lambda}\right)^N f(x)$ prove that: $|f(x)| \leq c \left|\frac{1}{x}\right|^N$ I would really appreciate ...
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0answers
74 views

System of Equations which can be solved by inequalities: $(x^3+y^3)(y^3+z^3)(z^3+x^3)=8$, $\frac{x^2}{x+y}+\frac{y^2}{y+z}+\frac{z^2}{z+x}=\frac32$.

S367. Solve in positive real numbers the system of equations: \begin{gather*} (x^3+y^3)(y^3+z^3)(z^3+x^3)=8,\\ \frac{x^2}{x+y}+\frac{y^2}{y+z}+\frac{z^2}{z+x}=\frac32. \end{gather*} Proposed by ...
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0answers
28 views

Use Chebyshev’s inequality to choose $n$ such that $P(\bar{X} > 4) > 0.9$

Use Chebyshev’s inequality to choose n such that $$ P(\bar{X_n} > 4) > 0.9 $$ where $$ E[\bar{X_n}] = 5 \ \ \ \ \ Var[\bar{X_n}] = \frac{4}{n} $$ The problem I am having when using Chebyshev's ...
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0answers
35 views

Prove the inequality about $Re(z)$

Consider three different vectors $x$,$y$ and $z$ in $\mathbb{C}^{n}$. So $x = (x_{1} \dots x_{n})$ and this is the same for $y$ and $z$. Now we have $\langle x,y\rangle = ...
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0answers
14 views

Azuma's inequality conditional version

Let $(\Omega, \mathcal{F},P)$ be a probability space. Consider a martingale $M_n$ with filtration $\mathcal{F}_n$. Let $B \in \mathcal{F}$. On $B$, $a_n \leq |M_n - M_{n-1}| \leq b_n$ a.s. Can we ...
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0answers
41 views

Computing limit using definition of e

I want to show that for fixed $n\in \mathbb{N_0}$, there exists some $C>0$ such that $c>0$ implies $(1-\frac{1}{2^n})^{cn}\le \frac{1}{4c}$. EDIT: Realize now from comments that the limit ...
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0answers
11 views

Removing the dimension factor in Fannes inequality

Given two distributions $x=(x_1,\ldots, x_n),y=(y_1,\ldots y_n)$ on $[n]$, it is known by Fannes inequality that $H(x)-H(y)\leq O(\|x-y\|_1\log n)$, where $H(\cdot)$ and $\|\cdot\|_1$ represent ...
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0answers
22 views

Q: On Minkowski's inequality

It is well known that by applying the function $f(x)=x^p$, $0<p<1$, the inequality $$2^{1-p}(a+b)^p \ge a^p+b^p$$ holds My question is I want an equivalent form for the reverse inequality but ...
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0answers
90 views

Inequality for symmetric $n \times n$-matrix with non-negative elements.

Let us consider a symmetrix $n \times n$ - matrix $A$ with non-negative elements $a_{ij} \geq 0$. Furthermore, we look at a non-negative vector $x \in \mathbb{R}^n$ with $x_i \geq 0$. Then we want to ...
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0answers
24 views

Poisson process deviations

How can one prove the following inequalities for a standard Poisson process $\mathbf{N}(t)$ ? $\mathbb{P}\bigg[\bigg|\frac{\mathbf{N}(\lambda)}{\lambda}-1 \bigg| > \varepsilon\bigg] \leq ...
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0answers
42 views

$4$ or more type $2$ implies $3$ or less type $1$

I'm having difficulties with the logic with the last part of the reformulation part of the problem below. Let $x_i$ be the the number of ships of type $i$ to purchase. For $4a:$ (the ...
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0answers
12 views

On a criterion for almost perfect numbers using the abundancy index

Let $\sigma(x)$ denote the sum of the divisors of $x$, and let $I(x) = \sigma(x)/x$ be the abundancy index of $x$. A number $y$ is said to be almost perfect if $\sigma(y) = 2y - 1$. In a preprint ...
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0answers
12 views

Does there exist an optimal solution $(x^*,y^*)$ to $\max x^TAy$ such that $x^*=y^*$?

Given two positive integers $n \le m$ and non-negative real constants $a_{ijkl} \ (1\le i,k\le n,1\le j,l\le m)$. Let $M$ be the set of $X\in\mathbb{R}^{n\times m}$ satisfying: $X\ge 0$, The sum ...
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0answers
22 views

Cauchy Schwarz inequality on scalar terms

The following equations are derived from Quantum information theory, which requires the use of Cauchy Schwarz inequality for a proof. I am quite puzzled by the second term in the summation, which ...
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0answers
54 views

Application of Gronwall Inequality

Let $T>0$ and $f\in C(\mathbb R, L^{2}(\mathbb R))$ with the following property: Put $g(t):= \sup\limits_{0\leq \tau\leq t} \|f(\tau)\|_{X},$ where $X \subset L^{2}$ and $X$ is a Banach Space. ...
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0answers
20 views

How to show this inequality with matrix norms?

Let $A$ be invertible and $M$ of the same size as $A$. Show that if $$\frac{\|M\|}{\|A^{-1}\|}\le c \le \frac{1}{2}$$ then $A+M$ is invertible and $$\frac{\|(A+M)^{-1} - A^{-1} \|}{\|A^{-1}\|}\le ...
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0answers
27 views

Proof of an inequality using Jensen's inequality

I want to prove the following inequality. $$ \mathbb{E}\left[\ln(1+\exp(\ln(x)))\right] \geq \ln(1+\exp(\mathbb{E}[\ln(x)])),$$ where $\mathbb{E}[\cdot]$ is the expectation operator and $x$ is a ...