Questions on proving and manipulating inequalities.

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0
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2answers
13 views

The distribution of the product of Gaussian variable and Rademacher variable.

I have two independent variables: $X$ follows from standard Gaussian distribution $N(0,\sigma^2)$; $Y$ follows from Rademacher distribution, i.e., $Y$ can be either $-1$ or $1$ with the same ...
1
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1answer
17 views

Bounds for double exponential integrals

I understand that the double-exponential integral $$ F(a,b,C) := \int_{C}^\infty \exp(-a \exp(b x)) \, dx \quad \text{(with $a,b>0$ and $C \geq 0$)} $$ can in general not be solved in closed-form. ...
0
votes
0answers
31 views

What are the numbers in an inequality called?

Summand is to addition what multiplicand is to multiplication, but what is the terminology for the quantities of an inequality, such as 1<4? My best guess is simply "quantity" for both parts of ...
2
votes
0answers
35 views

“Triangle” inequality for integrals

I have got two questions: 1) Let $f:\mathbb{R}^2 \rightarrow \mathbb{R}$ be any continuous function. Let $\Gamma$ be a piecewise smooth curve on $\mathbb{R}^2$. The following inequality holds: ...
0
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2answers
39 views

Largest integer $x$ that satisfies $\dfrac{4x+19}{x+5}<\dfrac{4x-17}{x-3}$

Find the largest integral $x$ that satisfies $\dfrac{4x+19}{x+5}<\dfrac{4x-17}{x-3}$ I tried $ \dfrac{4x+19}{x+5} < \dfrac{4x-17}{x-3}\\~\\ (4x+19)(x-3)<(4x-17)(x+5)\\~\\ x<-7 ...
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votes
0answers
17 views

Establishing consistency

I need to establish the (weak) consistency of an estimator of the mean, $T=a+b\bar{X}$. I tried to apply Chebyshev's inequality, but I couldn't do much because the parameter that subtract in the ...
2
votes
0answers
33 views

Do more generalizations of Schur's inequality exist?

I meet this following problem If $$n\ge 3,\sum_{i=1}^{n}\left(\prod_{j\neq i}(a_{i}-a_{j})\right)\ge 0$$ where $a_{i}$ are real numbers. when $n=3$, it is Schur's inequality so which $n$ ...
0
votes
1answer
16 views

Replacing a fractional quantity by one in inequalities

I was looking through proofs for the product law for limits and I stumbled upon a very clear one and managed to follow all the steps involving algebra and limits, but, near the end, in the proof, a ...
1
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2answers
55 views

solve $\sqrt{x+7}<x$ for $x\in \mathbb{R}$

solve $\sqrt{x+7}<x$ I tried $\sqrt{x+7}<x\\ x+7<x^2\\ x^2-x-7>0\\ x\in \left(-\infty, \dfrac{1-\sqrt{29}}{2}\right) \cup \left( \dfrac{1+\sqrt{29}}{2},+\infty\right) $ I m not ...
3
votes
1answer
51 views

A Real Matrix, its Kernel and Image

This is an old exam problem: For an $m \times n$ real matrix $A$, define $\ker A = \{x \in \mathbb{R}^n \mid Ax=0 \}$ and $\operatorname{Im} A = \{Ax \mid x \in \mathbb{R}^n \}$. Show that for all $b ...
3
votes
3answers
36 views

solve $|x-6|>|x^2-5x+9|$

solve $|x-6|>|x^2-5x+9|,\ \ x\in \mathbb{R}$ I have done $4$ cases. $1.)\ x-6>x^2-5x+9\ \ ,\implies x\in \emptyset \\ 2.)\ x-6<x^2-5x+9\ \ ,\implies x\in \mathbb{R} \\ 3.)\ ...
3
votes
2answers
24 views

Check proof of some simple inequality

Can you check please my proof of this inequality? It's all right?
-1
votes
1answer
58 views

Proof inequality using Lagrange Multipliers

Is it possible: $a,b,c$ are non-negative real numbers for which holds that $a+b+c=3.$ Prove the following inequality: $$ 4\ge a^2b+b^2c+c^2a+abc $$ Is it possible using Lagrange Multipliers. I ...
8
votes
3answers
93 views

Prove that $\frac{x}{y}+\frac{y}{z}+\frac{z}{x} \geq 3$ for $x,y,z>0$

By considering that $$\frac{x}{y}+\frac{y}{x} \geq 2$$I can show that $$\frac{x}{y}+\frac{y}{x}+\frac{x}{z}+\frac{z}{x}+\frac{y}{z}+\frac{z}{y} \geq 6$$ But how would one go from here to prove the ...
0
votes
2answers
28 views

squaring both side for an absolute inequalites on only one side

this is about squaring both side for an absolute inequalities on only one side problem. For example: $|6-2x|< x+4$ when solved both by squaring both sides and by defining it$ -(x+4)<6-2x< ...
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2answers
80 views

Proving inequalities using Calculus

In general how do you prove inequalities using calculus, I believe it is using maxima or minima right? For example $$a^2b+b^2c+c^2a \le 3, \qquad a,b,c \ge 0,\quad a+b+c=3.$$ How would you use ...
2
votes
1answer
74 views

Graham's Number versus another large number

I recently read this article about the most damage you can do in a single turn in Magic the Gathering. According to the current version of the deck, that damage is about a) $2 \rightarrow 17 ...
0
votes
3answers
38 views

complete the proof for this statement

$$\forall x \in \mathbb{R}, x \neq 0 \implies \frac{1}{x^2\:+3}\:<\:\frac{4}{5}\: $$ I thought of doing the contrapositive but not sure what to do next. $$ \frac{1}{x^{2\:}+3}\:\ge ...
2
votes
3answers
49 views

Find the general values of $x$ satisfying the trigonometric equation

Find the general values of $x$ satisfying $$ \frac{\tan^2 x \sin^2 x}{1-\sin^2 x \cos2x}+\frac{\cot^2 x \cos^2 x}{1-\cos^2 x \cos2x}+\frac{2\sin^2 x}{\tan^2 x+\cot^2 x}=\frac{3}{2} $$ It ...
3
votes
5answers
287 views

Intuition behind Chebyshev's inequality

Is there any intuition behind Chebyshev's inequality or is that only pure mathematics? What strikes me is that any random variable (whatever distribution it has) applies to that. $$ ...
0
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2answers
61 views

Prove that $\frac{8}{5}\le 2a+b\le 8$ [on hold]

Let $a,b,c,d,e$ be real numbers such that $$\begin{cases} a+b+c+d+e=8\\ a^2+b^2+c^2+d^2+e^2=16 \end{cases}$$ Prove that: $$\dfrac{8}{5}\le 2a+b\le 8$$
3
votes
1answer
35 views

Solve $x^2-|5x-3|-x<2,\ \ x\in \mathbb{R} $

Solve $x^2-|5x-3|-x<2,\ \ x\in \mathbb{R} $ I tried $x^2-|5x-3|-x<2$ , case $1$ , $x^2-(5x-3)-x<2,\ x\geq 0 \\ x^2-6x+1<0 \\ 3-2\sqrt2 < 3+2\sqrt2 \\ 0.17<x<5.8\\ $ ...
16
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2answers
140 views
0
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2answers
32 views

Calculus: simpler way of showing that derivative is negative?

I want to show that $\frac{1-(1-\beta)^N}{\beta}$ is strictly decreasing in $\beta$ for $\beta \in (0,1)$ and $N \geq 2$. My approach so far is as follows: I take the derivative with respect to ...
2
votes
2answers
293 views

Proving the an expression is larger than a simplified quadratic

Let p and q be positive real numbers. Prove that $$ (p + 2)(q+2)(p+q) \ge 16pq $$ Any explanation/answer would be extremely helpful. Thanks : )
-1
votes
1answer
28 views

Two parameters inequality problem

Find $p \gt 0$ for which $f(x) =x\ln p - p\ln x$ is $\ge0$ for $x \gt 0$ They also say $p$ is real number.
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votes
1answer
26 views

Simple case of Young's inequality

I have a question concerning Young's inequality stated as follows: $||a∗b||_{ℓ_q}≤||a||_{\ell_1}||b||_{ℓ_q},~~~~ 1≤q≤∞$. Here you can find something on $\ell_q\big(\mathbb{Z}\big)$: Young's ...
2
votes
4answers
64 views

Prove the inequality $e^x \geq x^e$ for $x > 0$ [duplicate]

Prove that $e^x \ge x^e$ for $x \gt 0$ I applied the natural logarithm to simplify the function and I get $$\frac{x}{\ln x}\ge e$$ How to solve these types of problems?
4
votes
2answers
97 views

Prove this inequality $\frac{1}{1+a}+\frac{2}{1+a+b}<\sqrt{\frac{1}{a}+\frac{1}{b}}$

Let $a,b>0$ show that $$\dfrac{1}{1+a}+\dfrac{2}{1+a+b}<\sqrt{\dfrac{1}{a}+\dfrac{1}{b}}$$ It suffices to show that $$\dfrac{(3a+b+3)^2}{((1+a)(1+a+b))^2}<\dfrac{a+b}{ab}$$ or ...
1
vote
1answer
42 views

Inequality using integrals and absolute values

Let $u,v$ be continous functions in $[a,b]$ a compact interval and let $c > 0$. Suppose that $\forall x\in [a,b]$, the following inequality is true: $$|u(x)-v(x)|\leq c\int^x_a|u(t)-v(t)|dt$$ ...
6
votes
0answers
26 views

Lower bound on absolute value of determinant of sum of matrices

I needed to find a lower bound on $|\det(A+B)|$ where $|.|$ is the absolute value operator. Because I was unable to get such a bound so I was trying to guess a bound and prove it. But ...
1
vote
1answer
62 views

Prove that:$\sum_{i=1}^n\frac{1}{x_i}-\sum_{i<j}\frac{1}{x_i+x_j}+\sum_{i<j<k}\frac{1}{x_i+x_j+x_k}-\cdots+(-1)^{n-1}\frac{1}{x_1+\ldots+x_n}>0.$?

Is there someone show me how do I prove this , I guess this inequality hold only if $x=0$ . Let $x_1,x_2,\ldots,x_n>0$. Prove that ...
3
votes
1answer
34 views

How to prove the following inequality $|\prod_{i=1}^{i=n}a_i-\prod_{i=1}^{i=n}b_i| < n\delta$?

The constraints are $0 \le a_1,a_2....a_n,b_1,b_2....b_n \le 1$. $|a_i-b_i|< \delta$ for all $1 \le i \le n $ How do I go about proving the following $$|\prod_{i=1}^n a_i-\prod_{i=1}^n b_i| ...
6
votes
3answers
421 views

Some trouble with the induction

Prove, that for any positive integer $n \geqslant 2$ we have the inequality $$ \frac{ 4^n }{ n+1 } < \frac{ (2n)! }{ (n!)^2 }.$$ For $n=2$ the inequality is true. Directly just take and ...
0
votes
0answers
20 views

Order of 'Strength' of inequalities

There have been times when we solve an inequality and we get the reverse sign of inequality. The reason is quite simple- we did not choose a strong inequality. So my question is- Is there an order of ...
0
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3answers
71 views

Prove sum of $\sin$ of angles is greater than $\sin$ of sum of angles

It seems that $\displaystyle \sum_{x_i \in X} \sin\left(x_i\right) \geq \sin\left(\sum_{x_i \in X} x_i\right)$ where $X$ is a set of angles where $\displaystyle \sum_{x_i \in X} x_i \leq \pi$ radians ...
5
votes
2answers
38 views

Doubt with Absolute Value Inequality

Problem: Find all values of $x$ for which $\dfrac{|x-2|}{x-2}>0$ My incorrect attempt: Using the definition the Modulus, $|x-2|=x-2$ for all $x\ge2$ and $|x-2|=-x+2$ for all $x\le2.$ ...
0
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1answer
38 views

Finding a metric such that $\Phi$ becomes a contraction

Let $$ f:\:[0,1]\rightarrow\mathbb{R};\\ \Phi: \mathcal{F}([0,1], \mathbb{R}) \rightarrow \mathcal{F}([0,1], \mathbb{R}),\\ f\mapsto\Phi(f):=Φf := \left(x\mapsto\left\{ ...
1
vote
1answer
37 views

Help with a summation+inequality problem.

I need help in solving for all possible x values for the below inequality: (Note: $x \in N)$ $$\sum^x_{k=1}\frac{k^2+k+1}{k(k+1)(k+1)!} \leq \frac{599}{600}$$ I think the series is telescopic; I'm ...
1
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0answers
19 views

search on a split data structure

I have the following problem: Part 1: Lets say I have n items in a data structure and I want to search for them. I know that a subset of my data $r \cdot ...
0
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0answers
22 views

An inequality of probability [on hold]

Suppose $X$ is a random variable. $E(X^2)=1$, $E|X|\ge a>0$. Let $\lambda\in[0,1]$, prove that $$P(|x|\ge\lambda a)\ge(1-\lambda)^2a^2.$$ I have tried to use Chebyshev's inequality, but I didn't ...
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votes
0answers
25 views

Quadratic recurrence inequality

I have the recurrence relation: $r_{k+1} \leq r_k^2+ (1/2)r_k \quad (k =1,2,\ldots)$, where each $r_k$ is non-negative and $r_1<1$. I have the following questions in this regard: A simple plot ...
-1
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1answer
28 views

an inequality for multiplication of cubic numbers

I need to know is there a positive constant $c$ such that $$ |a_1 \cdots a_n|^3 \leq c\big(|a_1|^3+\cdots+|a_n|^3\big), $$ where $a_{i}\neq 0$ ? I tried geometric and arithmetic inequalities but I ...
1
vote
2answers
43 views

Concluding three statements regarding $3$ real numbers.

$\{a,b,c\}\in \mathbb{R},\ a<b<c,\ a+b+c=6 ,\ ab+bc+ac=9$ Conclusion $I.)\ 1<b<3$ Conclusion $II.)\ 2<a<3$ Conclusion $III.)\ 0<c<1$ Options By ...
6
votes
2answers
138 views

Find the minimum value of $A=\frac{2-a^3}{a}+\frac{2-b^3}{b}+\frac{2-c^3}{c}$

Let $a, b$ and $c$ three positive real numbers such that $a+b+c=3$. Find the minimum value of $$A=\frac{2-a^3}{a}+\frac{2-b^3}{b}+\frac{2-c^3}{c}.$$ Here is my attempt. By symmetry we can assume that ...
5
votes
1answer
73 views

Inequality with reciprocals of $n$-variable sums

Let $a_1,a_2,\ldots,a_n$ be positive real numbers. Is it always true that $$\sum_{i=1}^n\frac{1}{a_i}-\sum_{1\leq i<j\leq n}\frac{1}{a_i+a_j}+\sum_{1\leq i<j<k\leq ...
2
votes
3answers
41 views

Reasoning about numbers close to two other numbers $a,b$ (inequalities)

Let $a < b$ and $0 < \varepsilon < (b - a)$ and let $x, y \in \mathbb R$ be such that $$ | x - a | < \frac{(b - a) - \varepsilon}{2}, \qquad | y - b | < \frac{(b - a) - ...
0
votes
1answer
32 views

An inequality involving logarithmic function and floor function [closed]

How do I show that $${\log_2(\log_2(x))}\geq\lfloor{\ln(\ln(x))}\rfloor.$$
0
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3answers
56 views

Proving that $|a-b|≤|a|+|b|$ [closed]

Can someone prove this to me: $$|a-b|≤|a|+|b|$$ I am in 8th grade and I have this for my homework. Thanks for helping.
3
votes
2answers
110 views

Show that $\frac{(x^2 + y^2 )}{4} \leq e^{x+y-2}$

Show that \begin{equation} \frac{x^2 + y^2}{4} \leq e^{x+y-2} \end{equation} is true for $x,y \geq 0$. As far, I have prove that \begin{equation} x^2 + y^2 \leq e^{x}e^{y}\leq e^{x+y} ...