Questions on proving and manipulating inequalities.

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0
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0answers
9 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
45 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 ...
3
votes
0answers
44 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
vote
2answers
29 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 ...
0
votes
0answers
42 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
votes
1answer
18 views

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

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.
11
votes
4answers
161 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 ...
-1
votes
2answers
35 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
vote
1answer
37 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 ...
1
vote
0answers
27 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
vote
1answer
21 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).
-2
votes
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
26 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
vote
3answers
54 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
votes
1answer
29 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 ...
1
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0answers
21 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
votes
1answer
65 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
vote
3answers
79 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
vote
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
vote
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 ...
-2
votes
4answers
89 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
vote
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 ...
-4
votes
2answers
25 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
votes
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
votes
6answers
101 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
votes
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 ...
-1
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
votes
1answer
24 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
vote
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 ...
-1
votes
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
votes
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
29 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 ...
-3
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
151 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 ...
1
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0answers
14 views

solving non-linear systems inequalities

I am trying to solve a non-linear systems of 14 inequalities with 12 variables, involving exponential and polynomial functions. I have been searching over the web for leads,but without any success.I ...
3
votes
0answers
48 views

Find $\Big\{ (a,b)\ \Big|\ \big|a\big|+\big|b\big|\ge 2/\sqrt{3}\ \text{ and }\forall x \in\mathbb{R}\ \big|a\sin x + b\sin 2x\big|\le 1\Big\}$ [on hold]

Find all (real) numbers $a $ and $b$ such that $|a| + |b| \ge 2/\sqrt{3} $ and for any $x$ the inequality $|a\sin x + b \sin 2x | \le 1$ holds. In other words, find the set $Q$ defined as ...
6
votes
1answer
104 views

Seeking a More Elegant Proof to an Expectation Inequality

Let $X$ and $Y$ be i.i.d. random variables, and $\mathbb E[|X|]<\infty$, prove that $$\mathbb E[|X+Y|]\geq\mathbb E[|X-Y|].$$ This question is a re-posting of An expectation inequality. I can ...
1
vote
2answers
32 views

Inequality $\left|z+w\right|\geq\left||z|-|w|\right|$ when $|w|\leq A|z|$.

Let $z,w\in\mathbb C$ and $|w|\leq A|z|$ for $A>0$. I want estimate from below $|z+w|$. I proceeded as follows. Since $$\left|z+w\right|\geq\left||z|-|w|\right|$$ and $-|w|\geq -A|z|$, I write ...
-3
votes
1answer
37 views

The integer part of $x+1$ is the integer part of $x$ plus $1$ [closed]

How do you solve the proof: If $x$ is a real number, then: $[x+1] = [x] + 1$. For my proof, I tried to describe the interior of the argument inside the parentheses, but I was unsuccessful. Please ...
2
votes
0answers
19 views

$\sum_{i=1}^n\sum_{j=1}^n|c_i+c_j|a_ia_j\geq\sum_{i=1}^n\sum_{j=1}^n|c_i-c_j|a_ia_j$

Let $c_i\in\mathbb R$, $a_i\geq0$ with $\sum_{i=1}^n a_i=1$, prove $$\sum_{i=1}^n\sum_{j=1}^n|c_i+c_j|a_ia_j\geq\sum_{i=1}^n\sum_{j=1}^n|c_i-c_j|a_ia_j$$ This inequality comes from there, when $X$ is ...
1
vote
2answers
38 views

Solve for real value of $x$: $|x^2 -2x -3| > |x^2 +7x -13|$

Here I have a question: Solve for real value of $x$: $$|x^2 -2x -3| > |x^2 +7x -13|$$ I got the answer as $x = (-\infty, \frac{1}{4}(-5-3\sqrt{17}))$ and ...
2
votes
5answers
108 views

Determine whether $f(x)$ is increasing or decreasing

Let $f(x) = -x + (x^3/3!) + \sin(x)$ How do I determine if $f(x)$ is increasing or decreasing? I have already found the derivative of this function which is: $f'(x) = -1 + (x^2/2) + \cos(x)$ And I ...
0
votes
1answer
34 views

About inequality $\sum_{k=1}^n |a_k|^2 \lessgtr \sum_{k\neq s} |a_k| |a_s|$

Let $a_k$ a sequence of complex number. We have $$\left(\sum_{k=1}^n |a_k|\right)^2 \geq \sum_{k=1}^n |a_k|^2$$ It is a basic fact because $$\left(\sum_{k=1}^n |a_k|\right)^2 = \sum_{k=1}^n |a_k|^2 + ...
2
votes
1answer
34 views

Find an example that the following equality doesn't apply

I need a sequence $(f_n)_{n\in\mathbb{N}}, f_n:X\to\mathbb{R}^\star$ which is measurable, $f_n \ngeq 0$ and doens't accept the following equality: $$\int_X\sum_{n=1}^\infty f_n \, d\mu = ...
5
votes
2answers
73 views

Proving that $\frac{1}{a^3(b+c)}+\frac{1}{b^3(c+a)}+\frac{1}{c^3(a+b)}\ge\frac32$ using derivatives

Let $a,b,c\in\mathbb{R}^+$ and $abc=1$. Prove that $$\frac{1}{a^3(b+c)}+\frac{1}{b^3(c+a)}+\frac{1}{c^3(a+b)}\ge\frac32$$ This isn't hard problem. I have already solved it in following way: Let ...
1
vote
2answers
41 views

prove using Lagrange multipliers that for $x,y>0,\space n\in \mathbb N,\space (\frac{x+y}2)^n \leq \frac{x^n+y^n}2 $

I have been asked to prove using Lagrange multipliers that for \begin{equation*} \space (\frac{x+y}2)^n \leq \frac{x^n+y^n}2,~x,y>0,~n\in \mathbb {N} \end{equation*} I am familiar with the ...
1
vote
1answer
49 views

Struggling to prove inequality

I've been given to following inequality to prove: (The hint given was not to evaluate the integral) \begin{equation*} \frac{1}{4} \leq \int_{\frac{\pi}{6}}^{\frac{\pi}{3}}\frac{sin(x)}{x}dx\leq ...