Questions on proving, manipulating and applying inequalities.

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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 ...
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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 ...
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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 ...
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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: ...
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193 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 ...
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302 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 ...
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87 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 ...
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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 ...
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22 views

Transformation that preserves an increasing ratio between vectors

Consider two vectors $x=(x_1,x_2,x_3)$, $y= (y_1,y_2,y_3)$ such that all $x_i,y_i>0$ and \begin{align} \frac{y_1}{x_1}\le \frac{y_2}{x_2}\le \frac{y_3}{x_3} \end{align} Now consider an upper ...
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45 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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39 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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27 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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34 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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16 views

$\|h\|_{L^{p}} \leq C \|f\|_{L^{p}} \implies \|g\ast h \|_{L^{p}} \leq C_1 \|g\ast f\|_{L^{p}}$?

Suppose that $f, h \in L^{p}(\mathbb R) (1\leq p \leq \infty)$ so that $\|h\|_{L^{p}} \leq C \|f\|_{L^{p}}$ for some constant $C$. Take $g\in \mathcal{S}(\mathbb R^{d})$ (Schwartz Space). We note ...
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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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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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48 views

How to show AM-GM inequality using rearrangement inequality?

Wikipedia article on Rearrangement inequality (link to the current revision) says (without giving any citation for this claim): Many famous inequalities can be proved by the rearrangement ...
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9 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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19 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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53 views

How do I show that $\frac 4{abcd} \ge \frac ab + \frac bc + \frac cd + \frac da$ for $a + b + c + d = 4$ using only AM-GM?

Let $a, b, c, d$ be non-negative numbers such that $a + b + c + d = 4$. Show that $$\frac 4{abcd} \ge \frac ab + \frac bc + \frac cd + \frac da.$$ Edit: The question already has an answer here (and ...
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60 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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23 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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29 views

Model linearly: Determine amount of units for production

A company produces 2 products in a week. Let $x_i$ denote the number of units of product $i$ to produce. Each product requires liters of Chemical X to make. Info is given below: \begin{array}{|c|c|} ...
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37 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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9 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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11 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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21 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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51 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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19 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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27 views

Quality of $E(f(X))\approx f(EX)+\frac 1 2 f''(EX)\sigma_X^2$ approximation

For convex $f$, we have Jensen's lower bound $Ef(X)\ge f(EX)$. What conditions do we need to put on $f,X$ so that the second order expansion in the title would be an upper/lower bound/good ...
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26 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 ...
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64 views

Closed form representation of an exponential series

Let $a\in\mathbb{R}$, $t\in\mathbb{R}\ge 0$ and consider the following series $$ f(t)=\sum_{n=1}^{\infty}\frac{\exp({-a^2 t})-\exp({-n^2 t})}{n^2-a^2} . $$ The series is well defined in the poles at ...
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44 views

Cyclic inequality

How can we prove that: $$a^{60} c^{10} +b^{60}a^{10}+c^{60}b^{10}+a^{50} c^{20} +b^{50}a^{20}+c^{50}b^{20}\geq 2(a^{51}b^{9}c^{10}+b^{51}c^9 a^{10}+c^{51}a^9 b^{10}), \ \forall\ a,b,c\geq 0.$$ I ...
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84 views

A double inequality for $\frac{\pi}{2}$

Approximating $\frac{\pi}{2}$ from above Since $$\left(\frac{\pi}{2}\right)^9\approx 58.220897$$ the root $$58^\frac{1}{9}\approx 1.5701$$ is not far from $$\frac{\pi}{2}\approx 1.570796$$ This ...
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49 views

Showing a predictable gambling strategy gives a supermartingale

I am stuck on the following problem. Your winnings per unit stake on game $n$ are $\epsilon_n$, where the $\epsilon_n$ are IID RVs with $$ P(\epsilon_n=1)=p, \ P(\epsilon_n=-1)=q, \text{ where ...
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39 views

solve an inequality with Log terms

I get the following inequality to solve (after some manipulations/transformations). Special I want to write $x$ or $y$ in terms of other parameters. I even tried to convert this into form which we can ...
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56 views

prove: $f(x_2)-f(x_1)>(x_2-x_1)f'(\frac{x_1+x_2}{2})$

Given $f:\mathbb{R}\rightarrow\mathbb{R}$, so that $f'''(x)>0$ for all $x\in \mathbb{R}$. Need to show that for all $x_2>x_1$: $f(x_2)-f(x_1)>(x_2-x_1)f'(\frac{x_1+x_2}{2})$ Any ideas? ...
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36 views

Strict inequalities in an LP problem

So I have to formulate an LP problem out of this scenario converting 3 variables into just 2 variables in order to use the graphical method. I guess I do that by using the demand constraint: I ...
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58 views

A power inequality with convexity / majorization flavor

Let $a,b,c,d\in\mathbb R_+$ be nonnegative reals. Define the function $f:\mathbb N_+\to\mathbb R$ as $$ f(k):=\left(\frac{(a+b+c+d)^k+(a-b+c-d)^k-(a+b-c-d)^k-(a-b-c+d)^k}{4}\right)^{1/k} $$ Is it ...
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25 views

Invariance of polynomial inequality over addition

Problem:Let $r$ be a real number and $A$ be the set of polynomials of $\mathbb{R}$ which satisfy 1) $f(0) \geq 0$ 2)if $f(0)=0$ then $f'(0)=0$ and $f"(0) \geq 0$ 3) $f(0)f''(0)-(f'(0))^2 \geq ...
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35 views

It is possible a Skewes number between twin primes? Can you discard such extreme question?

Let $p_n$ the nth prime number. A prime number $p=p_{k}$ is called twin prime if $2+p_k$ is also a prime number. Since $1+p_k$ is even then we have obviously that $2+p=p_{k+1}$. My purpose with this ...
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38 views

Triangle inequality in norms

Let $N:V\to\mathbb{R}$ be a sequence of norms and $d(x,y)=\sum_{j=1}^{\infty}2^{-j}\frac{N_{j}(y-x)}{1+N_j(x)+N_j(y)}$ Show that $d(x,z)\leq{d(x,y)}+d(y,z)$ I tried the following ...
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68 views

Approximately not equal

What terms do you consider appropriate for the relations denoted by symbols like these: $$\Large 1.≈\qquad 2.≉\qquad 3.⪅\qquad 4.⪉$$ The first one should be easy: “almost equal to” and ...
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28 views

Criteria for inequality

I am working with an inequality and I need to prove something of the shape $$c\cdot a+d\cdot b \leq a\cdot b$$ The numbers $a$ and $b$ have a specific form, but for the $c$ and $d$ I only know that ...
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74 views

Making a Matrix singular

During my research I came across the following problem. Intuitively this should be an easy one. However, the simplest version of it looks like this: Let $C \geq \frac{1}{2}$ be some fixed ...
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34 views

Use the arithmetic-geometric inequality for this list to deduce the arithmetic-geometric inequality for $n$.

Suppose that $n$ is not a power of two. Let $2^k$ be a power of $2$ that exceeds $n$ and consider the list $$a_1,\dots,a_n,\underbrace{A,A,\dots,A}_\text{$2^k-n$ times}$$ of length $2^k$. Use the ...
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20 views

Comparing irrational powers

I want to compare two numbers of the form $a^b$ and $b^a$ for $a,b, >1$. The simpler way that I know is to start from: $$ a^b\le b^a \quad \iff \quad b\log a\le a \log b \quad \iff \quad ...
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36 views

Prove $\frac{3}{64}(ab+bc+ca)^3\geq (de)^3+(ef)^3+(fd)^3$ where $a, b, c$ are three sides of and $d, e, f$ three angle bisectors of a triangle.

A triangle has sides $a, b,c$ and angle bisectors $d, e, f$ where each pair of $a$ and $d$, $b$ and $e$, $c$ and $f$ intersect. Prove that $\frac{3}{64}(ab+bc+ca)^3\geq(de)^3+(ef)^3+(fd)^3$. I was ...
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16 views

Upper bound of $\left|\sum_{n,k} c_{n,k}\, \alpha_k\right|^2$.

We have a double finite series, such that: $$\sum_{n,k} |c_{n,k}|^2=1$$ What can we conclude about upper bound of $\left|\sum_{n,k} c_{n,k}\, \alpha_k\right|^2$ for $|\alpha_k|<M$? I have tried to ...
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14 views

expressing a certain relation by means of linear inequalities

I have a question and don't know whether there is a (good) solution. Given two distinct integer vectors $ p \neq q \in \mathbb{Z}_{+}^n $ with $ p_i \le q_i $ for each component. Let $ K \in ...