Questions on proving, manipulating and applying inequalities.

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80 views

Probability Inequality Exercise

Let $n \in \mathbb{N}$ be non-negative. Show that if $E[|X|^{n}]$ is finite, then: $\lim\limits_{x\rightarrow\infty}x^{n}P(|X|\geq x) = 0$ Attempt at Solution By Markov's inequality, we have: $x^{...
1
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3answers
110 views

Use proof by induction to prove $ \frac{1}{n!}<\frac{1}{2^n-1} $ for all $n\geq 4$

Use proof by induction to prove that that $ \frac{1}{n!}<\frac{1}{2^n-1} $ for all $n\geq 4$, .\Base case: $$\frac{1}{4}=\frac{1}{24}\leq \frac{1}{2^4-1}$$ Inductive hypothesis: Assume there ...
2
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3answers
287 views

A generalization of this type of mean to power > 2

Let $a,b \in (0, \infty)$ and $0 < \theta < 1$. Then $$ (a + b)^2 \leq \frac{1}{\theta} a^2 + \frac{1}{1 - \theta} b^2. $$ Can you generalize to a power $m \geq 2$? (The question is not ...
3
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3answers
94 views

Show that $n x (1-x)^n < 1$ for any $x \in [0,1]$

Why $n x (1-x)^n < 1$ for any $x \in [0,1]$?
3
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3answers
90 views

Evaluate $\frac{x^3-x^2-5x-3}{x^3-x^2-15x} \ge 0$

I've been trying to simplify the problem but I can't. I tried long division factoring and perfect cube but I can't still solve it. My calculator shows $x=3$ and $x=-1$ but wait how? this is for the ...
0
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1answer
34 views

System of linear inequalities

Consider the following system of inequalities. $$x_1 − x_2 ≤ 3\\ x_2 − x_3 ≤ −2,\\ x_3 − x_4 ≤ 10,\\ x_4 − x_2 ≤ α,\\ x_4 − x_3 ≤ −4,$$ where $\alpha$ is a real number. A value for $\alpha$ for which ...
1
vote
1answer
103 views

Use chebyshev inequality to find the probability $P[|X-E[X]| \ge k\sigma]$

 For an arbitrary random variable $X$, use the Chebyshev inequality to show that the probability that $X$ is more than $k$ standard deviations from its expected value $E[X]$ satisfies $$P[|X-E[...
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3answers
44 views

How to solve the inequality from $n-1$ to $n$?

I would like to solve the inequality using induction from $n-1$ to $n$: $\sum_{i=1}^n$ $\frac{1}{i}\leq 7+\sqrt n$ I tried: $\sum_{i=1}^{n-1}$ $\frac{1}{i}\leq 7+\sqrt{n-1}+\frac{1}{n}$ However, ...
0
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0answers
42 views

For any two positive numbers, $p$ and $q$, which is larger $p^q$ or $q^p$?

For $p>0$ and $q>0$, which is larger $p^q$ or $q^p$? Give a complete analysis of the function $f(p,q)=p^q-q^p$, for $p>0$ and $q>0$. You may need to make bounds on your values of $p$ and $...
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2answers
99 views

Putnam and Beyond AM-GM help

From Putnam and Beyond: The Solution is: The only part I do NOT understand is how: $a_k + b_k = 1$ for every $k$? The problem just specifies nonnegative numbers?
1
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1answer
65 views

Outer measure of product of sets

If $A\subseteq R^n$ and $B\subseteq R^m$, such that $A \times B\subseteq R^{n+m}$ Prove that $μ^{*}_{n+m}(A\times B)\leq μ^*_n(A)μ^*_m(B)$, where $μ^*_q$ is the outer measure of $ \mathbb{R}^q $. ...
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2answers
49 views

Showing that the Triangle Inequality holds for the $L_\infty$ norm as a metric.

We have a metric on $\mathbb{R}^2$ defined as: $d(x,y) = \max(|x_1-y_1|,|x_2-y_2|)$ where $x = (x_1,x_2)$ and $y = (y_1,y_2)$. To satisfy the triangle inequality, we must show that $\max(|x_1-y_1|,|...
0
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1answer
37 views

A functional inequality

Suppose that $\varphi (t), \psi (t), w(t)$ are continuous functions on $[a,b]$ such that $w(t)>0$. If the inequality $$\varphi (t)\leq \psi (t) + \int_a^t w(s)\varphi (s)ds$$ holds on $[a,b]$, ...
1
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2answers
119 views

Inequality in Hanoi Open Mathematics Competition [closed]

$\frac{1}{bc} + \frac{1}{ab} + \frac{1}{ca} \geq 1$. Prove $\frac{a}{bc} + \frac{c}{ab} + \frac{b}{ca} \geq 1$, where $a,b,c$ are positive real numbers
3
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1answer
57 views

Prove: $\frac{b+c}{a^2+bc}+\frac{c+a}{b^2+ac}+\frac{a+b}{c^2+ab}\leq \frac{1}{a}+\frac{1}{b}+\frac{1}{c}$

Let $a;b;c>0$. Prove that : $\frac{b+c}{a^2+bc}+\frac{c+a}{b^2+ac}+\frac{a+b}{c^2+ab}\leq \frac{1}{a}+\frac{1}{b}+\frac{1}{c}$ I think: $\frac{b+c}{a^2+bc}+\frac{c+a}{b^2+ac}+\frac{a+b}{c^2+...
3
votes
3answers
198 views

Prove that $a^2 + b^2 \geq 8$ if $ x^4 + ax^3 + 2x^2 + bx + 1 = 0 $ has at least one real root.

If it is known that the equation $$ x^4 + ax^3 + 2x^2 + bx + 1 = 0 $$ has a (real) root, prove the inequality $$ a^2 + b^2 \geq 8. $$ I am stuck on this problem, though, it is a very easy problem for ...
5
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1answer
216 views

Trigonometric inequalities: when to reverse sign

I have the inequality $$\cos^{-1}{x^2 \over 2x-1} \ge {\pi\over 2} $$ Now, by multiplying both sides by cos, I get. $$ {x^2 \over 2x-1} \ge 0$$ However, I SHOULD be getting $$ -1\le {x^2\over2x-1} \...
1
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1answer
71 views

Let $a>0$ and $b>0$. Prove that $\sqrt{ab} \le (a+b)/2$. [duplicate]

Let $a>0$ and $b>0$. Prove that $\sqrt{ab} \le (a+b)/2$. Here is what I have tried: Let $a \le b$. Multiplying both sides of this inequality by $a$ results in $a^2 \le ab$. It follows that $a \...
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2answers
39 views

Necessary and Sufficient Conditions for $p_1p_2 \leq q_1q_2$

Given that $p_1, p_2, q_1, q_2$ are all non-negative and that $p_1 + p_2 = q_1 + q_2$, what are the necessary and sufficient conditions that $p_1p_2 \leq q_1q_2$?
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6answers
206 views

Summation inductional proof: $\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\ldots+\frac{1}{n^2}<2$ [duplicate]

Having the following inequality $$\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\ldots+\frac{1}{n^2}<2$$ To prove it for all natural numbers is it enough to show that: $\frac{1}{(n+1)^2}-\frac{1}{n^2}...
0
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1answer
35 views

Positivity of a Sine transform of a positive function

Consider a function $f(t)$ with $f(t>0)>0$ and $f(-t)=-f(t)$. Can I make any statement about the positivity of the Sine transform $$\hat{f}(\omega) = \int_{0}^{\infty} \sin(\omega t) f(t) \...
2
votes
0answers
134 views

Bound for variance of maximum of normal random variables

Suppose that $(X_1,\ldots,X_n)=\mathbf{X}\sim N(\mathbf{0},\Sigma)$ is an $n$-dimensional normal random vector. I want to show the bound $$ \text{Var}\left(\max_{i\leq n} X_i\right)\leq \max_{i\leq n} ...
2
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3answers
106 views

Prove that $\sqrt{a^2+(1-b)^2}+\sqrt{b^2+(1-c)^2}+\sqrt{c^2+(1-a)^2}\geq\frac{3\sqrt{2}}{2}$

Prove that the inequality $$\sqrt{a^2+(1-b)^2}+\sqrt{b^2+(1-c)^2}+\sqrt{c^2+(1-a)^2}\geq\frac{3\sqrt{2}}{2}$$ holds for arbitrary real numbers $a, b, c.$ Someone says, "It's very easy problem. It can ...
2
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1answer
67 views

Two inequalities in a triangle

I'm trying to prove that in a triangle with side lengths $a,b,c$, median lengths $m_a, m_b, m_c$ and circumdiameter $D$ the following inequality holds: $$ \frac{a^2+b^2}{m_c}+\frac{b^2+c^2}{m_a}+\frac{...
2
votes
3answers
68 views

Complex numbers modulus inequality

Let $a, b \in \mathbb{C}$. Prove that $|az + b\bar{z}| \leq 1$, for any $z \in \mathbb{C}$, $|z| = 1$, if and only if $|a| + |b| \leq 1$. I want to know if my demonstration is correct. Here is how I ...
2
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1answer
49 views

Asymptotic solution to inequality $x < k \ln(1+x)$

What is an upper-bound on $x$, given that $x < k \ln(1+x)$? I believe that the solution is something of the form $\mathcal{O}(k \ln k)$ but I am unable to prove this. This is my first encounter ...
1
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2answers
45 views

inequalites of an acute triangle angles $ 180^{180}*a^b*b^c*c^a \le (a^2+b^2+c^2)^{180} $

If $a,b,c$ are an acute angle of triangle the prove that $ 180^{180}*a^b*b^c*c^a \le (a^2+b^2+c^2)^{180} $ No idea
0
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1answer
46 views

Solving an inequality graphically

$$\frac{2x - 5}6 < \frac{x^2 - 1}4 - \frac{x(x+1)}3$$ Should I solve it by turning it firstly into an equation of the form $ax^2+bx+c>0$ and then draw the graph or draw the graph of $\...
0
votes
1answer
16 views

showing an inequality regarding expectation of a random variable

Suppose $h: \mathbb{R} \to [0, \alpha] $ is bounded and $0 \leq a < \alpha $. Then $$ P( w : h(X(w)) \geq a ) \geq \frac{ \mathbb{E}(h(X(w)) - a )}{\alpha - a } $$ I was trying to use the fact ...
2
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3answers
215 views

How to get this upper bound of this sum of squares?

Given n non-negative values. Their sum is k. $x_1+x_2+⋯+x_n=k$ Given the constraints $x_i \leq \sqrt{k}$ (thus, $n \geq \sqrt{k}$) Is it possible to prove that $x_1^2 + x_2^2 + ... + x_n^2 \leq ...
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5answers
430 views

Proving $\frac{n^n}{e^{n-1}}<n!<\frac{(n+1)^{n+1}}{e^{n}}$ by induction for all $n> 2$.

I am trying to prove $$\frac{n^n}{e^{n-1}}<n!<\frac{(n+1)^{n+1}}{e^{n}} \text{ for all }n > 2.$$ Here is the original source (Problem 1B, on page 12 of PDF) Can this be proved by ...
0
votes
1answer
61 views

Schwarz Inequality?

Is the following inequality correct $$\bigl|\sum_i a_i h_i\bigl|^2 \leq \sum_i |a_i|^2 |h_i|^2 $$ I am assuming it is using the triangle inequality, if yes can someone explain to me intuitvely why? ...
0
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3answers
101 views

Prove that $|x+y+z| \le |x|+|y|+|z|$

Prove the following: $$|x+y+z| \le |x|+|y|+|z|$$ It is so trivial that I do not have idea how to show it. Thus, how do I show it?
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1answer
107 views

How to obtain $||x|-|y||\le|x-y|$ from $|x|-|y|\le |x-y|$? [duplicate]

Having the following inequality: $$|x|-|y|\le |x-y|$$ does it imply that $||x|-|y||\le|x-y|$ if it does (i think it does) how to prove it?
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4answers
82 views

An inequality of functions

If $f(0)f(2) \ge f(1)^2$ and $g(0)g(2) \ge g(1)^2$, does it imply that $$ (f(0)+g(0))(f(2)+g(2)) \ge (f(1)+g(1))^2 $$ I have figured out that this is equivalent to proving that: $$f(2)g(0)+g(2)f(...
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0answers
43 views

Find this maximum value the sum $x_1+\cdots+x_n$ may achieve

I had read a hard problem on AOPS Given an integer $n\geq 2$, determine the maximum value the sum $x_1+\cdots+x_n$ may achieve, as the $x_i$ run through the positive integers, subject to $x_1\leq x_2\...
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0answers
37 views

Relation between norms [duplicate]

if A is an $m\times n$ matrix prove that: $$\frac{1}{\sqrt{m}} \|A\|_1 \leq \|A\|_2 \leq \sqrt{n} \|A\|_1 $$
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3answers
139 views

Using the AM-GM to find the minimum of $x^2+(4-2x)^2$

I would like to get a min value of the parabola expression $x^2+(4-2x)^2$ with AM-GM inequality. ($x$ is a real number) $$x^2+(4-2x)^2\geq 2\sqrt{x^2(4-2x)^2 }$$ with equality when $x^2=(4-2x)^2$. ...
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0answers
25 views

Visualising relations between inequalities and solution criteria.

Is there any intuitive, visual explanation of the following lemma: Lemma: Let $\{ \alpha_{ij} : i = 1, \ldots, m, j = 1,\ldots, n \}$ be an $m \times n$ matrix, $\alpha_i = (\alpha_{i1}, \ldots, \...
1
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2answers
108 views

Proof by induction; inequality $1\cdot3+2\cdot4+3\cdot5+\dots+n(n+2) \ge \frac{n^3+5n}3$

Ok so I'm kind of struggling with this: The question is: "Use mathematical induction to prove that 1*3 + 2*4 + 3*5 + ··· + n(n + 2) ≥ (1/3)(n^3 + 5n) for n≥1" Okay, so P(1) is true as 1(1+2)=3 and (...
0
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1answer
27 views

Need help to find where I was wrong while solving for inequality

Here is inequality and we should solve for n: $$c_1n^2 - 2c_1n + c_1 + c_0n \le c_1n^2$$ The answer should be $n \ge \frac{c_1}{2c_1 - c_0}$ Here is my solution (btw, my way of solving inequality is ...
2
votes
4answers
82 views

Why does $(x+3)/(x-4) \geq 0$ not include 4 in the interval result?

Wolfram alpha, the book I am going through, an other sources all give the resulting interval for: $$ \frac{x+3}{x-4} \geq 0 $$ as: $$ (-\infty, -3] \cup (4, \infty). $$ I am struggling to ...
0
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5answers
87 views

If $|a+b|≤1,$ then $ |a|≤|b|+1.$

How can I prove If $|a+b|≤1,$ then $|a|≤|b|+1$ in real analysis ? I try to use Triangle inequality
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2answers
70 views

How to solve the following equation: $a(x^2 (x-1)b+c) = d x(1-x e) f^{x-1}$

Please, I need to solve the following equation (w.r.t. $x$): $a(x^2 (x-1)b+c) = d x(1-x e) f^{x-1}$, where $a$, $b$, $c$, $d$, $e$, $f$ are positive constants and $x \ge 1$ . ( I am trying to solve ...
8
votes
6answers
178 views

Prove that $(n+1)^{n-1}<n^n$

How would one prove that $$(n+1)^{n-1}<n^n \ \forall n>1$$ I have tried several methods such as induction.
1
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1answer
41 views

Integral bounded question

Hey there I am having some trouble understanding a practice problems and problems of this type in general, the question asks to show, and or give an argument for the inequality below $$ \int_1^n lnx ...
0
votes
2answers
53 views

if $2a+3b \geq 12m+1$, then either $a \geq 3m+1$ or $b \geq 2m+1$

Not sure how to go about proving this. So far I've declared the contrapositive but can't seem to get further... Let $a,\ b$ and $m$ be integers. Prove that if $2a+3b \geq 12m+1$, then $a \geq 3m+1$ ...
1
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0answers
21 views

Is it true that $2\max_{|z|=1} | z^n+ (\sum_{ i=2}^{n-1} a_iz^i ) + a_1 z+a_0| \geq \max_{|z|= 1} |a_1z+a_0|$?

I come across to prove or disprove an inequality which is $$2\max_{|z|=1} | z^n+ (\sum_{ i=2}^{n-1} a_iz^i ) + a_1 z+a_0| \geq \max_{|z|=1} |a_1z+a_0|,$$ where $z\in \mathbb{C}$ and $a_i$ are real. ...
4
votes
3answers
128 views

How can I prove that $2/9<x^4+y^4<8$?

$x$ and $y$ are real numbers. Given that $1<x^2-xy+y^2<2$, how can I show that $\frac 29<x^4+y^4<8$ ? Then can I use that to prove that for any natural number $n>3$ $$x^{2^n}+y^{2^n}&...
3
votes
0answers
46 views

A bound on the squares of primes

If $p_n$ is the $n$th prime, what is the best known upper bound on $m$ such that $p_n \cdot p_{m+n} < p_{n+1}^2$?