Questions on proving, manipulating and applying inequalities.

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0
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1answer
13 views

How to prove if $5/2 < x < (5/4)(1+sqrt2)$, then $25/(x(2x-5)\ge 8$

if $\frac52 < x < \frac54(1+\sqrt2)$, then $\frac{25}{x(2x-5)} \ge 8$ First I unpacked the conclusion to: $$ 16w^2-40w-25 \le 0 $$ I attempted to solve by manipulating the interval (squaring, ...
0
votes
1answer
12 views

Do there exist $a,b,c,d,e,f$ such that $ax^2+by^2+cxy+dx+ey+f > 0 \quad\forall 0<x\le 1, 0< y\le 1$ and…

Do there exist $a,b,c,d,e,f$ satisfying: \begin{cases} ax^2+by^2+cxy+dx+ey+f > 0 \quad\forall 0<x\le 1, 0< y\le 1\\ a+b+c+d+e+f \le 0\\ a+d+f \le 0\\ b+e+f \le 0\\ f\le 0 \end{cases}? ...
0
votes
1answer
20 views

Optimizing the area of a rectangle with one side against a wall using the am-gm inequality

Given 300 meters of fence, how can I find the dimensions of a rectangle that is built against a wall the encloses the maximum area. I found this question in a calculus book and saw a simple solution ...
-2
votes
2answers
79 views

Prove or disprove that $(a_1+a_2+\ldots+a_n)\leq n\sqrt{a_1^2+\ldots+a_n^2}$, by showing that $RHS-LHS\geq 0$ if possible. [on hold]

Prove or disprove that $$\left|a_1\right|+\left|a_2\right|+\ldots+\left|a_n\right|\leq n\sqrt{a_1^2+\ldots+a_n^2}$$ Where $a_1,\ldots,a_n\in\mathbb{R}$ and $n\in\mathbb{N}$. EDIT: I was hoping there ...
1
vote
0answers
12 views

Proving Holder's inequality for Schatten norms

Sticking to the finite dimensional case, Holder's inequality for Schatten norms is given by $$\left\|AB\right\|_{S^1}\leq\left\|A\right\|_{S^p}\left\|B\right\|_{S^q}$$ for $A,B$ $n\times n$ ...
4
votes
6answers
165 views

Prove that $\frac{7}{12}<\ln 2<\frac{5}{6}$ using real analysis

I studying in Real Analysis 2, but I have no idea how to solve this problem. My guess is to use Mean Value Theorem or a similar theorem? Could any one help me? Thanks.
1
vote
1answer
44 views

An inequality on the rank of a block matrix

Let $\mathbb F$ be a field, and let $r_1, r_2, s_1, s_2$ be positive integers. Consider the matrix $$X:=\left[\begin{array}{cc} A & B \\ C & D \end{array} \right],$$ where $A \in \mathbb ...
4
votes
0answers
103 views

A monotonically increasing series for $\pi^6-961$ to prove $\pi^3>31$

This question is motivated by Why is $\pi$ so close to $3$?, Why is $\pi^2$ so close to $10$? and Proving $\pi^3 \gt 31$ There are series with all terms positive for $\pi-3$ and $10-\pi^2$ ...
3
votes
1answer
29 views

Which inequalities are there with stochastic integration?

Which inequalities can I use with stochastic integration? For example, with the standard lebesgue integral we have $$\left|\int_\Omega f(x) dx\right| \le M |\Omega|$$ (where $M$ is the maximum of ...
0
votes
1answer
23 views
1
vote
1answer
455 views

What is the upper-bound for this?

I am looking at a paper and am trying to understand how this bound was driven. The first part is clear, but not sure how you can extend it to the second part. So here is the first part: Assume ...
0
votes
2answers
1k views

Combining inequalities into one inequality

Let's say we are given $a$, $b$, $d$ with $1 \leq a, b, d \leq 1000$ and inequalities $x \geq a$, $y \geq b$, and $a+b < x + y \leq a+b+d$. I need to combine all this and the following into one ...
11
votes
5answers
279 views

This inequality $a+b^2+c^3+d^4\ge \frac{1}{a}+\frac{1}{b^2}+\frac{1}{c^3}+\frac{1}{d^4}$

let $0<a\le b\le c\le d$, and such $abcd=1$,show that $$a+b^2+c^3+d^4\ge \dfrac{1}{a}+\dfrac{1}{b^2}+\dfrac{1}{c^3}+\dfrac{1}{d^4}$$ it seems harder than This inequality $a+b^2+c^3\ge ...
2
votes
5answers
119 views

How do I prove $\frac 34\geq \frac{1}{n+1}+\frac {1}{n+2}+\frac{1}{n+3}+\cdots+\frac{1}{n+n}$

How do I prove the following inequality $$\frac 34\geq \frac{1}{n+1}+\frac {1}{n+2}+\frac{1}{n+3}+\cdots+\frac{1}{n+n}$$ without the help of induction? Thanks for any help!!
1
vote
1answer
29 views

Finding the maximum value of a divergent series [on hold]

I came across this divergent sum- $$\sum_{n=1}^\infty\frac{1}{n+1}$$ Now,a divergent sum does not a limit.So is it possible to get a maximum value for the sum or more specifically prove that ...
2
votes
0answers
44 views

I am trying to show an inequality involving the product of three inner product terms

Define the inner product $\langle\cdot,\cdot\rangle$ for continuous functions defined on $[0,1]$ as: $$\langle\,f\mid g\rangle=\int_{0}^{1}f(x)g(x)e^{\rho x}dx,$$ where $\rho$ is a real number. I ...
2
votes
5answers
379 views

Proving $\pi^3 \gt 31$

$$\large \pi^3 \gt 31$$ Using a calculator, $\pi^3/31 \approx 1.0002$, so I thought this may be challenging to do by hand. It is extremely easy with the use of any calculator, so I was wondering ...
-5
votes
2answers
189 views

Why is $\pi^2$ so close to $10$?

Noam Elkies explained why $\pi^2=9.8696...$ is so close to $10$ using an inequality on Euler's solution to Basel problem $$\frac{\pi^2}{6}=\sum_{k=0}^{\infty} \frac{1}{\left(k+1\right)^2}$$ to form ...
2
votes
3answers
48 views

Proving the inequality $ \frac {x+y}{x^2+y^2}\leq \frac 12 \left(\frac {1}{x}+\frac{1}{y}\right)$

Let x and y be definitely two positive numbers : Prove that : $$ \left( \frac {x+y}{x^2+y^2}\right) \leq \frac 12 \left(\frac {1}{x}+\frac{1}{y}\right)$$ I answered this one by squaring the two ...
0
votes
4answers
61 views

Proof that $|x|+|y|\leq\sqrt{2(x^2+y^2)}$

How do I prove that for $x,y\in\mathbb{R}$ we have $|x|+|y|\leq\sqrt{2(x^2+y^2)}$? I thought that $(|x|+|y|)^2=x^2+y^2+2|x||y|\leq2(x^2+y^2)$, but I'm not sure why that holds.
2
votes
0answers
32 views

Prove that $(n-1)!S_m\geq (n-m)!m!P_m.$

If $a_1, a_2,\cdots a_n$ be all positive rationals such that $S_n=a_1^m+a_2^m+\cdots +a_n^m$, $P_m=\sum a_1a_2\cdots a_m$ (the sum of products m taken m at a time). Prove that $$(n-1)!S_m\geq ...
0
votes
2answers
34 views

If $\left| x \right| \ge \left| y \right|$ then Can we say $\left| {x + y} \right| \ge \left| x \right| - \left| y \right|$?

Let $x,y\in \mathbb{R}$ and $\left| x \right| \ge \left| y \right|$. Can we say $\left| {x + y} \right| \ge \left| x \right| - \left| y \right|$?
1
vote
3answers
46 views

Cauchy like inequality $(5\alpha x+\alpha y+\beta x + 3\beta y)^2 \leq (5\alpha^2 + 2\alpha \beta +3\beta ^2)(5x^2+2xy+3y^2)$

Problem: Prove that for real $x, y, \alpha, \beta$, $(5\alpha x+\alpha y+\beta x + 3\beta y)^2 \leq (5\alpha^2 + 2\alpha \beta +3\beta ^2)(5x^2+2xy+3y^2)$. I am looking for an elegant (non-bashy) ...
3
votes
2answers
56 views

Finding the minimum of $x^2+y^2$ when $(x^2y-xy^2)(x^3-y^3)=x^3+y^3$

If $x,y \in \mathbb {R}$, find the minimum of $x^2+y^2$ when $(x^2y-xy^2)(x^3-y^3)=x^3+y^3$ and $xy>0$. This problem was inspired by a problem which asked if $x,y \in \mathbb {R}$ and $xy \neq ...
0
votes
1answer
597 views

Does this numerical series have any special name?

I don't have enough background to find the answer for this on my own, so I am posting it here with the hope to get some pointers. Assume a descending sequence of K numbers $\{n_1, ..., n_K\}$ where ...
1
vote
3answers
66 views

Prove $\left(1+\frac{x}{n}\right)^n < e^x$, where $x$ is any positive real number and $n$ is any positive integer.

I am having trouble with my homework problem, it says: Suppose that $n$ is a positive integer and that $x > 0$. Show that $$\left(1+\frac{x}{n}\right)^n < e^x.$$ I have proved the base ...
1
vote
7answers
67 views

Proving $\frac{n}{n+1} < \frac{n+1}{n+2}$ by induction?

I have the inequality $\frac{n}{n+1} < \frac{n+1}{n+2}$ I'm not sure how to go about proving it. I've started by testing with n = 1, which results in $\frac{1}{2} < \frac{2}{3}$ which is true ...
5
votes
1answer
177 views

Proof that $\frac{2}{3} < \log(2) < \frac{7}{10}$

Positive integrals $$\int_{0}^{1}\frac{2x(1-x)^2}{1+x^2}dx=\pi-3$$ and $$\int_0^1\frac{x^4(1-x)^4}{1+x^2}dx=\frac{22}{7}-\pi $$ (http://math.stackexchange.com/a/1618454/134791) prove that ...
1
vote
3answers
55 views

proving an inequality related to $AM\ge GM$

$$a^2+ab+b^2\ge 3(a+b-1)$$ $a,b$ are real numbers using $AM\ge GM$ I proved that $$a^2+b^2+ab\ge 3ab$$ $$(a^2+b^2+ab)/3\ge 3ab$$ how do I prove that $$3ab\ge 3(a+b-1)$$ if I'm able to prove the ...
1
vote
3answers
63 views

Prove $\ln x \ge \frac{x-1}{x}$

Prove that for every $x>0$: $$\ln x \ge \frac{x-1}{x}$$ What I did: $$f(x) = \ln x, \text{ } g(x) = \frac{x-1}{x} $$ $$f(1) = g(1) = 0 $$ So it's enough to prove that $$ f'(x) \ge g'(x)$$ ...
0
votes
2answers
77 views

Is there a sequence of $v_{n}$ such that $G=\prod_{n=1}^{\infty} \frac{2\tan^{-1}(v_{n})}{\pi}> 1$?

Let $G=\prod_{n=1}^{\infty} \frac{2\tan^{-1}(v_{n})}{\pi}$, where $v_{n}$ is an increasing, monotonic sequence of natural numbers: is it true that there is no sequence of $v_{n}\in\mathbb{N}$ such ...
3
votes
0answers
76 views

Prove that $a+b^2+c^3+d^4 \ge \frac{1}{a}+\frac{1}{b^2}+\frac{1}{c^3}+\frac{1}{d^4}$ [duplicate]

If $0 < a \le b \le c \le d$ and $abcd = 1$ prove that $$a+b^2+c^3+d^4 \ge \frac{1}{a}+\frac{1}{b^2}+\frac{1}{c^3}+\frac{1}{d^4}$$ I first thought of multiplying both sides with ...
0
votes
1answer
33 views

Name of the inequality $|x|+|y| \geq |x+y|$?

What is the name of the inequality $|x|+|y| \geq |x+y|$? I remember seeing this inequality and thinking it was the triangle inequality, but that only holds if $x,y,z$ are the side lengths of a ...
28
votes
1answer
640 views

Prove that $a\sqrt{a^2+bc}+b\sqrt{b^2+ac}+c\sqrt{c^2+ab}\geq\sqrt{2(a^2+b^2+c^2)(ab+ac+bc)}$

Let $a$, $b$ and $c$ be non-negative numbers. Prove that: $$a\sqrt{a^2+bc}+b\sqrt{b^2+ac}+c\sqrt{c^2+ab}\geq\sqrt{2(a^2+b^2+c^2)(ab+ac+bc)}.$$ I have a proof, but my proof is very ugly: it's ...
18
votes
1answer
423 views

Existence of two real numbers satisfying $f(x-f(y))>yf(x)+x$

Let $f:\mathbb{R} \longrightarrow \mathbb{R}$ be a function. Is it always the case that for some $x,y \in \mathbb R$, the inequality $f(x-f(y))>yf(x)+x$ holds? Thanks in advance.
2
votes
1answer
40 views

Normed Linear Space - maximum norm vs. $||f||_1$

For $f$ in $C[a,b]$ define $$|| f ||_1 =\int_a^b |f|.$$ a. Show that this is a norm on $C[a,b]$. b. Show that there is no number $c \geq0$ for which $$||f||_{max} \leq c ||f||_1 \ for \ all \ f \ ...
0
votes
0answers
27 views

Understanding ADMM: how is it applied to this particular problem?

While reading Boyd's paper on ADMM I encountered an issue. Consider the following problem: Problem. Minimize $f(u) + g(v)$ subject to $Au + Bv = c$, where $f$ and $g$ are closed, proper, convex and ...
1
vote
1answer
55 views

Prove ${20n \choose 10n}\ge {2n-1 \choose n-1}^{10}$

As the title says, I can't prove that, no matter what I try. What I've tried so far: induction: seemed the most obvious method, since we already had a lot of tasks with it, but using the esimates ...
1
vote
1answer
35 views

Prove Jensen Inequality holds for a function

Given function $$f:\mathbb{R}^n_{+} \rightarrow \mathbb{R}, \ f(x) = \sum_{k=1}^K c_k x_1^{a_{1k}} x_2^{a_{2k}} \cdots x_n^{a_{nk}}$$ Show that for any $x, y \in \text{dom} \ f, \theta \in [0,1]$, ...
0
votes
4answers
58 views

Prove the following Inequality [3]

Suppose $m>x$, where $x$ belongs to positive Real Numbers and letting $n\ge m$, show that $${x^n\over n!}\le{x^n\over m^n}\cdot{m^m\over (m-1)!}$$ Frankly, I consider myself as a beginner in ...
0
votes
2answers
29 views

How to solve a quadratic inequality that acts like a quadratic equality?

This will be largely a trivial question. But how do I solve an inequality like this: $3x^4 - 4x^2 + 1>0$ ? Of course, I can treat it like a quadratic inequality by saying $t=x^2$ So I can solve ...
1
vote
0answers
64 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 ...
-1
votes
2answers
69 views

How does triangle inequality imply this?

I am banging my head against the wall over this easy equation, but I can't see it right now. How the triangle inequality imply $|x_0 + y_0| = |x_0| + |y_0|$? EDIT: I got it, Basically $2|y_0| ...
0
votes
1answer
31 views

Showing that $\log{\log^d{3n}} = O(\log{\log^d{n}})$

I'm trying to show this: $$\log{\log^d{3n}} \leq q\cdot \log{\log^d{n}} \;\;\exists\, q,k > 0,\forall n>k, \text{where } d \text{ is a constant} > 0$$ This is what I have so far ...
1
vote
1answer
38 views

A density argument in the proof of Hardy's inequality for $H^1(\mathbb R^3)$

I understand $C^{\infty}_0 (\mathbb{R}^3)$ is dense in $H^1 (\mathbb{R}^3)$. But I don't understand the reason it is enough to prove the following Hardy inequality if you prove the case $u \in ...
1
vote
3answers
40 views

Proving the convergnce of a sequence

So, I have to prove that the sequence defined as $a_{n+1}=\frac{6(1+a_n)}{7+a_n}$ converges and then find the limit. I have few questions; Do i have to assume that $a_n \geq 0$ or $a_n \leq0$. ...
2
votes
6answers
91 views

Prove $\frac{1}{3}(x+y+z)^2 \geq xy + yz + xz.$

Prove that for nonnegative $x,y,z$ that $\frac{1}{3}(x+y+z)^2 \geq xy + yz + xz.$ I saw this result in a problem but didn't know how to prove it. I tried expanding and collecting to get the ...
0
votes
1answer
31 views

Estimate $|f(x)| \le \frac C{|x|^3}$

Let $$f(x) = \frac{\sin x}x+\frac{\sin(x-1)}{2(x-1)}+\frac{\sin(x+1)}{2(x+1)}.$$ Find the common denominator and use common trigonometric identities to establish that $$|f(x)| \le \frac ...
0
votes
0answers
10 views

Norm and Inner Product Inequality in Hilbert spaces

Let $H$ be a Hilbert space, and suppose that $C \subset H$ is closed, convex and nonempty. Then, for $y_{j}=P_{C}(x_{j})$, $j=1,2$ where $P_{C}$ is the metric projection onto $C$ and $x_{1},x_{2} \in ...
7
votes
1answer
1k views

Proof of $\lim\sup(a_nb_n)\leq \lim\sup(a_n)\limsup(b_n)$

Let $a_n>0$ and $b_n\geq 0$, then $\lim\sup(a_nb_n)\leq \lim\sup(a_n)\limsup(b_n)$ My attempt at a proof is as follows. Let $A_n=\sup\{a_n, a_{n+1},...\}$, $B_n=\sup\{b_n, b_{n+1},...\}$, and ...