Questions on proving and manipulating inequalities.

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2answers
33 views

Solve Inequality for $ |x| $

Given $$\big|\frac{(x-2)}{(x+3)}\big| < 4,$$ solve for $x.$ \ My solution $$|x - 2| < 4|x + 3|$$ Since, $ |x - 2| \ge |x| - |2| $ and $ |x + 3| \le |x| + |3| $ according to triangle ...
1
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2answers
21 views

Inequality involving inner product and an othonormal set of vectors

$ \newcommand{\ip}[2]{\left\langle #1,#2 \right\rangle} $ Here is the statement of the problem: Suppose that $V$ is a real inner product space with an inner product $\langle\cdot,\cdot\rangle$, and ...
1
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2answers
30 views

Requesting constructive feedback on my proof of a problem from Apostol Vol.1.

If x is an arbitrary real number, prove that there is exactly one integer n which satisfies the inequalities $n \le x < n+1$. Let S be the set of all $t \in \mathbb{Z}$ such that $t \le x$ for an ...
0
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2answers
15 views

Solving inequality(limit)

Can someone explain how we get from $(x - 3) < \varepsilon/8$ and $x < 4$ to: $(x-3)(x+3) < (\varepsilon/8)(4+3) = (7\varepsilon)/8$
10
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1answer
73 views

$\forall x,y>0, x^x+y^y \geq x^y + y^x$

Prove that $\forall x,y>0, x^x+y^y \geq x^y + y^x$ A friend of mine told me none of the teachers in my school have succeeded in proving this seemingly simple inequality (it was asked at an ...
2
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3answers
31 views

Method for proving polynomial inequalities

Let $x\in\mathbb{R}$. Prove that $\text{(a) }x^{10}-x^7+x^4-x^2+1>0\\ \text{(b) }x^4-x^2-3x+5>0$ Possibly it can be proved in a few different ways, but I have first tried to prove it ...
2
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0answers
23 views

upper bound of a differential equation solution

Let $A(t)$ be a bounded singular values matrix that is function of time, and $f(t)$ and $L^\infty$ function of time. And consider the ODE $$ \dot x = A(t) x + f(t) $$ How we can describe qualitatively ...
3
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1answer
34 views

Inequality and Trigonometric Substitution [duplicate]

Prove that for all positive real $a,b,c$, we have $$(a^2+2)(b^2+2)(c^2+2) \geq 9(ab+bc+ca).$$ Because of the term $a^2+2$, this motiveates me to substitute $a=\sqrt{2}\tan A, b=\sqrt{2}\tan B, ...
11
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2answers
93 views

Why is $\int\int f(x)f(y) |x-y|dxdy$ negative?

The Setup Let $f:\mathbb{R} \to\mathbb{R}$ be a smooth function with support in the interval $[-R,R]$ and satisfying $\int f = 0$. By manipulating some integrals, I found the surprising inequality ...
1
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1answer
41 views

Is this a correct way to use triangle inequality

If I have: $$|g_1(x) - g_2(x) - (g_1(a) - g_2(a))| \leq f(x^*)$$ Can I proceed to say: $$|g_1(x) - g_2(x) - (g_1(a) - g_2(a))| \leq |g_1(x) - g_2(x)| - |(g_1(a) - g_2(a))|$$ $$ \implies |g_1(x) - ...
0
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0answers
24 views

If positive definite matrices $A>B>0$ and $C>0$, then is $AC>BC$ true? [on hold]

Suppose I have positive definite matrices $A$, $B$ and $C$. If $A>B$, can we conclude $AC>BC$?
0
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1answer
31 views

Exponential type of $\sin z$

An entire function $f$ is of exponential type if $\,\lvert\, f(z)\rvert\le C\mathrm{e}^{\tau\lvert z\rvert},\,$ for all sufficiently large values of $\lvert z\rvert$. The exponential type of $f$ is ...
4
votes
1answer
26 views

Multiplying two inequalities

Suppose we have two inequalities $$a\leq x\leq b\tag{1}$$ $$c\leq y\leq d\tag{2},$$ where $a,b,c,d>0$. Then can I conclude that $$ac\leq xy\leq bd\quad ?$$ My attempt: Since $a,b,c,d>0$ and ...
1
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0answers
31 views

Local estimates for $|(x+\epsilon)^{-1} - x^{-1}|$

I am interested in a local pertubation bound for the reciprocal function. How can you estimate the difference $|(x+\epsilon)^{-1} - x^{-1}|$ where $x > 0$ and $\epsilon > 0$ is small? Even ...
5
votes
2answers
54 views

Can every (convex) polygon be described by a single inequality (involving absolute values)?

For example, $$ |x| + |2x + y| + |x + 2y| + |y| + |x+y| < 4 $$ describes an octagon. I'm wondering whether an equation of this form always exist for any convex polygon, and if so, whether there ...
5
votes
2answers
93 views

$\frac{x}{\sqrt{yz}+\sqrt{3}}+\frac{y}{\sqrt{xz}+\sqrt{3}}+\frac{z}{\sqrt{yx}+\sqrt{3}}\leq \frac{1}{4\sqrt{3}xyz}$

Let $x;y;z>0$ such that: $xy+yz+zx=1$. Prove that: $\frac{x}{\sqrt{yz}+\sqrt{3}}+\frac{y}{\sqrt{xz}+\sqrt{3}}+\frac{z}{\sqrt{yx}+\sqrt{3}}\leq \frac{1}{4\sqrt{3}xyz}$ I think: ...
1
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2answers
31 views

How to apply the AM-GM inequality?

What is the minimum value of $8x^3+36x+54/x+27/x^3 $ for positive real numbers x? Express your answer in simplest radical form. I attempted to make an equation between the product of the terms and ...
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0answers
46 views

How to compute P(|X - E_Y[h(y)]| < c)?

Consider the discrete random variable $Y$, the continuous random variable $X$, and a constant $c$. The goal is to find $$P(|X - E_Y[h(y)]| < c),$$ when we are only given $P(y)$, function $h(y)$, ...
0
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1answer
24 views

For which $x$ the inequality $ax+be^{x/2}>c$, where $a,b,c,x>0$ and $c>2b$ holds [on hold]

For which $x$ the inequality $ax+be^{x/2}>c$, where $a,b,c,x>0$ and $c>2b$ holds. Can someone help me for this. Thank you.
12
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4answers
205 views

Prove that $\sinh(\cosh(x)) \geq \cosh(\sinh(x))$

Prove that $$\sinh(\cosh(x)) \geq \cosh(\sinh(x))$$ I tried to tackle this problem by integrating both lhs and rhs, in order to get two functions who show clearly that inequality holds. I've ...
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votes
2answers
37 views

Calculate the greatest inequality solution. [on hold]

Can someone help me with this task, please? :) By the way, it's not my homework or etc. I want to learn some new things. :) Calculate the greatest inequality $10^x \leq 16 \cdot 5^x$ solution.
1
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1answer
39 views

Varied use of the AM-GM inequality

This question appeared in the IMO some year. I have done it in 2 different ways that seem absolutely correct. Please tell me which one is right and why. I fell both are very interesting. The question ...
2
votes
2answers
51 views

Bound on the integral of a function with multiple zeros

This is a follow-up to this If $f(0)=f(1)=f(2)=0$, $\forall x, \exists c, f(x)=\frac{1}{6}x(x-1)(x-2)f'''(c)$ Let $f:[0,2]\to \mathbb R$ be a $C^3$ function such that ...
1
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1answer
22 views

tight estimate for a log-linear inequality

Given $q>0$ and $p$, how do we get a tight estimate for the smallest $x$ such that $x\log(x)+px \geq q$? (such an $x$ always exists).
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0answers
18 views

I have 4 in-equations with 4 variables in each of the in-equations. how to find the minimum value of each variable?? [on hold]

Please tell me the answer with solution. I don't know how to start it.completely blank.
0
votes
3answers
30 views

Are there any integral solutions to this inequality?

Are there any integral solutions to this inequality? $$\frac{n\sqrt{3} + 1}{n\sqrt{3}} + {\left(\frac{2n}{n + 1}\right)}^{1/2} < 1 + \sqrt{3}$$ WolframAlpha appears to give an inconsistent ...
1
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3answers
55 views

Proof of sum in an inequality

I was having hard time solving this one, any help will be greatly appreciated. prove that: $$ {39\over e^2}\le\sum_{n=1}^\infty {4n^2-1\over e^n}-{3\over e}\le{54\over e^2} $$
3
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1answer
31 views

If $q^k n^2$ is an odd perfect number with Euler factor $q^k$, can $q = 73$ hold?

Call a number $N$ perfect if $\sigma(N)=2N$ where $\sigma$ is the classical sum-of-divisors function. If $N = q^k n^2$ is an odd perfect number, can $q = 73$ hold? Here is my attempt: Since ...
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0answers
22 views

Dvoretzky–Kiefer–Wolfowitz inequality holds for discrete distributions?

I am wondering whether Dvoretzky–Kiefer–Wolfowitz inequality holds for discrete distributions? Any comments or references would be greatly appreciated.
2
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1answer
70 views

Proof: $|||AB|||^ \frac{1}{2} \leq |||A ^ \frac{1}{2}B ^ \frac{1}{2}||| $?

I am looking for a proof of the following: \begin{equation*} |||AB|||^ \frac{1}{2} \leq |||A ^ \frac{1}{2}B ^ \frac{1}{2}||| \end{equation*} Where A, B are positive, hermitian matrices, and $|||⋅|||$ ...
1
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3answers
80 views

Prove inequality $ab+bc+ca\ge 3,\ abc=1$

How can I prove \begin{equation*} ab+bc+ca\ge 3,~a,b,c \in\mathbb{R},~ a,b,c>0\ \end{equation*} and the product $abc=1$? I obtained only $(a+b+c)^2-(a^2+b^2+c^2)\ge6\ and \ ...
1
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0answers
22 views

Poincarè inequality in probability

I'm looking for a proof of the poincarré inequality in a probabilitic setting. That is to say, let $\mu$ be a probability on $\Bbb R^n$, what are the hypothesis in order to have, for any f smooth ...
1
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0answers
23 views

First moment inequality and time-average limits

Suppose $\{A(t)\}_{t \geq 0}$ and $\{B(t)\}_{t \geq 0}$ are two non-negative stochastic processes such that $$ \frac{1}{T} \int_{s=0}^T A(s) \, {\rm d} s \stackrel{\text{a.s.}}{\rightarrow} a \in ...
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votes
4answers
91 views

Prove the inequality $a^2 + b^2 +c^2 \ge ab +bc +ac$ [on hold]

How do I prove the inequality \begin{equation*} a^2 + b^2 +c^2 \ge ab +bc +ac \end{equation*} where $ a,b,c\in\mathbb{R} $ and $a,b,c>0$? I obtained only $(a+b+c)^2\ge 3(ab+bc+ac)$ and some ...
1
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0answers
18 views

Maximum density linear combination chi squares

I have a positive linear combination of chi square variables \begin{equation*} X=\sum_{i=1}^k \lambda_i \chi^2(r_i), \end{equation*} the degrees of freedom satisfy $r_i>1$. I need an upperbound ...
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votes
2answers
26 views

help with alternating series test [on hold]

How do I show that $$\large{\frac{\ln(n)}{n} \geq \frac{\ln(n+1)}{(n+1)}}$$ for $\large{n \geq 1}$?
0
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0answers
36 views

Inequality on complex polynomial

For every $a\geq 0$, let $p_a(z)=1-z+az^3$. What is the maximal value of $a$ such that $$ p_a(|z|)\leq |p_a(z)| $$ for all $z\in \mathbb C$? EDIT: I claim that $a=\frac{4}{27}$ is the maximal value. ...
3
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6answers
103 views

Visualize $z+\frac{1}{z} \ge 2$

As we know, always $$z+\frac{1}{z} \ge 2,~~~~~~~~~ z\in \mathbb{R}^+$$ However, is there any geometric way to visualize this equation for some one who is not that expert in math? I know this ...
0
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1answer
38 views

Prove using mathematical induction that $n^2 > n+1$ for all $n \ge 2$

I have proved for the initial case $P(2)$ that this is true, but I'm stuck at substituting in $n=k+1$, $(k+1)^2 > (k+1)+1$ = $k^2 + 2k + 1 > k+2$, where do I go from here or have I made a ...
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votes
2answers
30 views

How many books can fit in a box? [on hold]

Each book is 450kg An empty box is 200kg The total mass of the book(s) and box cannot exceed 6,500kg. How many books can fit in a box?
0
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1answer
25 views

Semilinear Poisson PDE - proving a (hopefully) simple inequality

This is from page 557 of PDE Evans, 2nd edition. My question is at the bottom of this post, but for now, here is some context for my question: LEMMA 2 (Boundary estimates). Let $u \in ...
1
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1answer
27 views

Two exercises about hypermetric spaces

Take $S$ to be the collection of all subsets of $\{1,\dots,n\}$. If $x, y$ are in $S$, define $d(x,y)$ as the number of elements of the symmetric difference $x\triangle y$. Exercise 2.1. Show ...
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1answer
29 views

Doubt with Intervals and Inequalities

This doubt has been bothering me for ages. I would be truly grateful for any help. Problem 1: $\dfrac{2}{|x-4|}>1$ Express the solutions using intervals Solution: $x\in(2,4)\cup(4,6)$ ...
0
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3answers
39 views

Show that $\frac{ n^{1/3} }{n-1} > \frac{ (n+1)^{1/3} }{n}$

I am trying to demonstrate that: $$\frac{ n^{1/3} }{n-1} > \frac{ (n+1)^{1/3} }{n}$$ for $n>0$ I am really struggling. I can get to the point $-2n^3+2n-1>0$ but I am really unsure of how to ...
0
votes
1answer
18 views

System of inequalities any real numbers

Here is a system of inequalities I've been trying to solve, and nothing so far; $y>x,$ $1>y,$ $1>x,$ $x>y/2, x>(2-y)/3, y>0, x>0, 1>x+y$
2
votes
3answers
30 views

Absolute Value Inequality Problem

Problem: $\dfrac{2}{|x-4|}>1$ Express the solutions using intervals My attempt using the Definition of Modulus: $$\dfrac{2-|x-4|}{|x-4|}>0$$ $$CASE A:x-4\ge 0\Rightarrow x\ge4\Rightarrow ...
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votes
2answers
40 views

How should I go about this proof? [on hold]

Let $a,b > 0$ be real numbers. Prove that $2ab \leq (a+b)\sqrt{ab}$. I'm new to proofs and would like some help understanding how to approach this proof. Thank You.
0
votes
1answer
70 views

Upperbound on the following logarithmic function with matrix

I am trying to find an upperbound the expression below with a function $f$ that is a function of the identity matrix $$\log(1+\mathbf{h}^* \mathbf{\Sigma} \mathbf{h}) \leq f( {\bf I},{\bf h })$$ ...
9
votes
2answers
153 views

Proving $\sqrt{2}(a+b+c) \geq \sqrt{1+a^2} + \sqrt{1+b^2} + \sqrt{1+c^2}$

I've been going through some of my notes when I found the following inequality for $a,b,c>0$ and $abc=1$: $$ \begin{equation*} \sqrt{2}(a+b+c) \geq \sqrt{1+a^2} + \sqrt{1+b^2} + \sqrt{1+c^2} ...
0
votes
1answer
17 views

How to prove the inequality on relative entropy?

Here is the definition of Relative Entropy Now I am only interested in the simplest condition that the index set is finite and discrete, as the naive probability distribution vectors. Now if the ...