Questions on proving and manipulating inequalities.

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How to prove that the spectral radius of a linear operator is the infimum over all subordinate norms of the corresponding norm of the operator.

I am trying to understand a proof I have seen of the following theorem: $$\rho(A)=\inf_{\|\cdot\|}\|A\|.$$ I understand that to do this, the idea is to show that 1) $\rho(A)\leq\|A\|$ for any norm, ...
1
vote
1answer
34 views

Strong induction inequality proof

Use strong induction to prove that $$\frac{1}{2^3}+\frac{1}{3^3}+\cdots+\frac{1}{n^3}\leq\frac{5}{8}-\frac{1}{n}$$ $$n\geq2$$ I'm not sure how to go about this. I used base cases n=2, and n=3 but ...
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1answer
28 views
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1answer
18 views

Redefine a discrete compact and convex set

I need to find twice continuously differentiable functions $g_i: \mathbb{R}^2 \rightarrow \mathbb{R}$ $i=1,\ldots,I$ such that the set $\{ 0,1\}^2=\{(0,0), (1,0), (0,1), (1,1) \}$ can be written as ...
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0answers
7 views

Inequality for two operators with functional calculus

Given a sequence of functions $f_n \to f$ in $L^\infty(\mathbb{R}^2)$ and two self-adjoint, unbounded operators $A, B$ is it true that $\|f_n(A,B) - f(A,B)\| \to 0$? With only one operator I can ...
2
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1answer
24 views

How Find this maximum $P=\frac{4}{\sqrt{a^2+b^2+c^2+4}}-\frac{9}{(a+b)\sqrt{(a+2c)(b+2c)}}$

let $a,b,c>0$, find the maximum $$P=\dfrac{4}{\sqrt{a^2+b^2+c^2+4}}-\dfrac{9}{(a+b)\sqrt{(a+2c)(b+2c)}}$$ I think this inequality we can use AM-GM inequality to solve it,and Now first we ...
0
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1answer
21 views

Define a compact and convex set through inequality constraints

I need to find twice continuously differentiable functions $g_i: \mathbb{R}^2 \rightarrow \mathbb{R}$ $i=1,...,I$ such that the set $\{ 0,1\}^2=\{(0,0), (1,0), (0,1), (1,1) \}$ can be written as $\{x ...
4
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3answers
39 views

Help verifying my proof that for any $\epsilon>0$, there exists rational $\frac{j}{k}$ such that $0<\frac{j}{k}<\epsilon$

I'm trying to prove that that for any $\epsilon>0$, there exists rational $\frac{j}{k}$ such that $0<\frac{j}{k}<\epsilon$. Obviously $j,k\in \mathbb{N}$. This is not for homework, it's a ...
1
vote
1answer
11 views

2 Linear equation problems [on hold]

Write objective, constraints and graph for the following two problems: 1.A test offers 2 types of problems. Type A takes 3 Min to solve and B takes 2. You have 20 min to take the test and can only ...
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3answers
36 views

a simple inequality

Is it true that for any real numbers (a,b): $(a - b)^{2} \leq 3a^{2} + 3b^{2}$ Also, if this is true, is there a way to sharpen this bound say $(a - b)^{2} \leq K(a^{2} + b^{2})$, for some $K < ...
2
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2answers
37 views

Prove the inequality.Let a, b and c be nonnegative real numbers.

Let $a$, $b$ and $c$ be nonnegative real numbers. Prove that $a^4+b^4+c^2\ge 8^{½}abc$
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vote
2answers
45 views

Proving that the line integral $\int_{\gamma_{2}} e^{ix^2}\:\mathrm{d}x$ tends to zero

Let $f(z) = e^{iz^2}$ and $\gamma_2 = \{ z : z = Re^{i\theta}, 0 \leq \theta \leq \frac{\pi}{4} \} $. All the sources I have found online, says that the line integral $$ \left| \int_{\gamma_2} ...
4
votes
3answers
81 views

Prove the inequality $({1+\frac{a}b})^n$ + $(1+\frac{b}a)^n$ $\geq$ $2^{n+1}$

Let $a$ and $b$ be positive real numbers and let $n$ be a natural number prove that $$\left({1+\frac ab}\right)^n+\left(1+\frac ba\right)^n\ge2^{n+1}.$$
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0answers
39 views

Which of the following is correct? [on hold]

Let $$X = \frac{1}{1001} + \frac{1}{1002} + \frac{1}{1003} + \cdots + \frac{1}{3001}$$ Then find the correct option. (A) $X < 1$ (B) $X > \frac{3}{2}$ (C) $1 < X < \frac{3}{2}$ (D) ...
2
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0answers
28 views

Integral inequality related to derivation

While trying to understand a proof, i have stumbled upon the following statement: Let $f \in L^p(a,b)$ be a $p$-integrable function. Then the inequality $$\liminf_{s \rightarrow t} \frac{1}{t-s} ...
2
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0answers
29 views

How prove this general inequality $a\left(\frac{\sin{x}}{x}\right)^m+b\left(\frac{\tan{x}}{x}\right)^n>a+b$

if $$m,n<0,a,b>0,a\left[\left(\dfrac{2}{\pi}\right)^m-1\right]\ge b,am\le 2bn$$ show that $$a\left(\dfrac{\sin{x}}{x}\right)^m+b\left(\dfrac{\tan{x}}{x}\right)^n>a+b,\forall ...
2
votes
3answers
47 views

Set of solutions for a binomial inequality

I bumped into the following inequality: $${a-b\choose c}{a\choose c}^{-1} \le \exp\left(-\frac{bc}{a}\right)$$ Playing with it a little bit, trying to bound it asymptotically for large $a$'s, using ...
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1answer
26 views

Prove trace inequality $\mathrm{tr}\{ABCBAD-ABCD-ADCB+CD\} \geq 0$

Let $\mathbf{A}$, $\mathbf{B}$, $\mathbf{C}$, $\mathbf{D}$ be four (generally non-commuting) positive semidefinite matrices of same size. I want to show that (or find a counterexample to) $$ ...
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1answer
20 views

For $1-r<|\theta|<1/2$, $|\frac{2r\sin{2\pi\theta}}{1-2r\cos{2\pi\theta}+r^2}-\frac{\cos\pi\theta}{\sin\pi\theta}|<C\frac{(1-r)^2}{|\theta|^3}$

For $1-r<|\theta|<1/2$ show that $$|\frac{2r\sin{2\pi\theta}}{1-2r\cos{2\pi\theta}+r^2}-\frac{\cos\pi\theta}{\sin\pi\theta}|<C\frac{(1-r)^2}{|\theta|^3}$$ This inequality shows that ...
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1answer
39 views

Very loose bound on sum of first binomials

Let $n\geq k\geq 2$. Is it always true that $$\binom{n}{0}+\binom{n}{1}+\cdots+\binom{n}{k}\leq n^k?$$ The left-hand side is dominated by the term $\dfrac{n^k}{k!}$, so the statement should be true. ...
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1answer
23 views

How to use induction on this type of inequality?

Given $a_1,a_2,\ldots,a_n>0$ and $a_1+a_2+\ldots+a_n<\frac{1}{2}$, prove that $(1+a_1)(1+a_2)\ldots(1+a_n)<2$. Some of you may have already seen this inequality. I was the one who asked ...
6
votes
2answers
102 views

An Inequality Problem with not nice conditions

How to show that $\dfrac{a^3}{a^2+b^2} + \dfrac{b^3}{b^2+c^2} + \dfrac{c^3}{c^2+a^2} \ge \dfrac32$, where $a^2+b^2+c^2=3$, and $a,b,c > 0$ ?
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0answers
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Inequality about ceiling function. How to prove it?

Proposition: Set $n,m\in{\mathbb{Z}}$ with $n\geq{2}$, then $$\lceil{\frac{n}{3}}\rceil\lceil{\frac{n}{3}}\rceil\leq\lceil{\frac{nm}{5}}\rceil$$ I've verified it for small cases (computationally), ...
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3answers
65 views

Is there any book about inequality? [on hold]

I heard there is a book name 'inequality'. But I couldn't find the book. Is there any site or book about inequalities? What i want is collection of inequalities.
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0answers
50 views

Rising Sun Inequality (Dunford-Schwartz maximal inequality)

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be an absolutely integrable function, and let $f^*:\mathbb{R} \rightarrow \mathbb{R}$ be the one-sided signed Hardy-Littlewood maximal function $$f^*(x) := ...
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1answer
36 views

Induction proof that $4^n > 3^n+2^n$ for $n\ge2$

This is a problem with induction and proofs but I'm not sure how to start with proving this one. $$\text{Show that for any $n \geq 2$, $4^n > 3^n+2^n$}$$
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4answers
80 views

Proof that $(1 + x)^n > 1 + nx$ for $x>-1$, $n$ a positive integer [duplicate]

For any positive integer $n$ and real number $x > -1$, show that $(1 + x)^n > 1 + nx$. This is Bernoulli’s inequality but I can't figure out how to start with this. Can someone help? Thanks
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1answer
19 views

Proving elementary inequalities with equations

Assume $b > 0,\ d > 0$. Assume: $$ \frac{a}{b} < \frac{c}{d} $$. Prove that: $$ \frac{a}{b} < \frac{a + c}{b + d} < \frac{c}{d} $$. I would like to find an intuitive way to solve ...
5
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2answers
56 views

How to show $a+b+ad\geq c+d+bc$ given $a\geq c$ and $a+b\geq c+d$?

Let $0\leq a,b,c,d\leq 1$ and $a\geq c$ and $a+b\geq c+d$. Show that $a+b+ad\geq c+d+bc.$ Of course we have $a+b\geq c+d$, but how to relate $ad$ and $bc$?
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2answers
46 views

Inequality with moments

Let $m$ a probability measure, $f$ a positive measurable function (one can assume it is bounded, the existence of the moments is not a problem here). Is $m(f^3) \le m(f^2) m(f)$?
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How prove this inequatity $\frac{x^2}{y}+\frac{y^2}{z}+\frac{z^2}{x}\ge 4+(x-y)^2$

let $x,y,z>0$,and such $$4\le x+y+z\le 5$$ show that $$\dfrac{x^2}{y}+\dfrac{y^2}{z}+\dfrac{z^2}{x}\ge 4+(x-y)^2$$ It seem $\dfrac{x^2}{y}+\dfrac{y^2}{z}+\dfrac{z^2}{x}\ge 4+(x-y)^2$ maybe is ...
6
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2answers
77 views

Prove that positive $x,y$ satisfy $\left(\frac{1}{1+x}\right)^2+\left(\frac{1}{1+y}\right)^2\ge\frac{1}{1+xy}$.

Prove that positive $x,y$ satisfy $$\left(\frac{1}{1+x}\right)^2+\left(\frac{1}{1+y}\right)^2\ge\frac{1}{1+xy}$$ My teacher claims this lemma is often useful. I'm wondering, though: how to prove ...
5
votes
3answers
92 views

Find min of $\frac{\left( \sum kx_k \right) \left( \sum x^2_k \right)}{\left( \sum x_k \right)^3}$

With $n \ge 2$ and $x_1,\ x_2,\ \dots,\ x_n > 0$. Find the minimum of: $$ M = \frac{(x_1 + 2 x_2 + ...+ nx_n)( x^2_1 + x^2_2 +...+x^2_n)} {\left( x_1 + x_2 +...+ x_n \right)^3}$$ For specific ...
5
votes
2answers
100 views

Isoperimetric inequality, isodiametric inequality, hyperplane conjecture… what are the inequalities of this kind known or conjectured?

Question: Which inequalities similar to the famous isoperimetric inequality is known? conjectured? I recently learned about some inequalities which are all similar to the famous isoperimetric ...
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2answers
39 views

Number of solutions for inqeuality

Is there a way we can determine number of solutions for equation $$x*y < d$$ where d is constant and x & y are positive integers greater than 1. I am not interested in actual values, but ...
2
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2answers
38 views

Prove that positive $x,y,z$ satisfy $\sqrt{x+y}+\sqrt{y+z}+\sqrt{z+x}\ge \sqrt{2x}+\sqrt{2y}+\sqrt{2z}$

Prove that positive $x,y,z$ satisfy $$\sqrt{x+y}+\sqrt{y+z}+\sqrt{z+x}\ge \sqrt{2x}+\sqrt{2y}+\sqrt{2z}$$ Actually, this is a part of my solution to another problem, which is as given below: ...
2
votes
1answer
19 views

Inequality with two moduli

I have a question similar to this, find all $x \in \mathbb{R}$ satisfying $\displaystyle 3 < \left| x+1 \right| + \left| x - \frac{1}{2} \right| < 7$ which is rather trivial by distinguishing ...
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1answer
32 views

Show $|xy-x_0y_0|<\epsilon$: necessary assumptions on $x,y$

From Spivak's Calculus, problem 1.21: Prove that if $$|x-x_0| < \min \left( \frac{\epsilon}{2(|y_0|+1)},1 \right)\qquad |y-y_0|< \frac{\epsilon}{2(|x_0|+1)}$$ then $|xy-x_0y_0|<\epsilon$. ...
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1answer
25 views

Is there another way to solve the value field of a parameter of an line.

Assume $P$ is a point in line $x+y=m$, where $m \in \Bbb{R}$. There are two points $A,B$ in circle $$x^2+y^2 = 10$$ such that $PA$ and $PB$ are tangent lines of the above circle. If line: $x+y=m$ has ...
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4answers
116 views

Does $xy\geq x+y$?

I just see the GM-AM inequality. But I would like to compare $xy$ with $x+y$ for any $(x, y)\in\mathbb{R}^2$. It looks like $xy>x+y$ since the first one is multiplication and the second one is ...
2
votes
2answers
45 views

If positive $a,b,c,d$ satisfy $(a^3+b^3)^4=c^3+d^3$, prove that $a^4c+b^4d\ge cd$.

If positive $a,b,c,d$ satisfy $(a^3+b^3)^4=c^3+d^3$, prove that $$a^4c+b^4d\ge cd$$ It kind of seems useful to begin with a division of both sides by $cd$: $$\frac{a^4}{d}+\frac{b^4}{c}\ge1$$ It ...
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1answer
38 views

Solve the inequality…

Can you please show me how can I solve this inequality. I would like to see how it can be done without the graph of the functions. Thank you! $$2\sqrt{(x-1)(x+2)}\ge|x+1|-2$$
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3answers
54 views

How to prove this inequation?

$$ 1+\frac{2}{3n-2}\leqslant \sqrt[n]{3}\leqslant 1+\frac{2}{n}, n\in \mathbb{Z}^{+} $$ How to prove this inequation?
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0answers
11 views

Schur-concave functions, derivative sign help

To establish some inequality I must prove: $$\dfrac{\partial}{\partial ...
0
votes
1answer
48 views

Solve the following inequality…

Can you please verify if I've done this exercise correctly, and if you have a better solution, please, show it to me. Thank you! (The exercise is in the left top corner.)
4
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1answer
138 views
+50

How prove this $|1+x|^a\ge 1+ax+\dfrac{1}{1000}|x|^a$

let $2\le a\le 13,a\in R$,and $x\in R$,show that: $$|1+x|^a\ge 1+ax+\dfrac{1}{1000}|x|^a\tag{1}$$ My try: let $$f(x)=|1+x|^a-1-ax-\dfrac{1}{1000}|x|^a$$ and since if $x>-1$,then ...
1
vote
1answer
29 views

How find this value of $x$ such $\log_{\frac{1}{12}}{(x^2+2x-3)}>x^2+2x-16$

if $$\log_{\frac{1}{12}}{(x^2+2x-3)}>x^2+2x-16$$ Find the value of $x$ My idea: since $$x^2+2x-3>0\Longrightarrow x>1 ,or, x<-3$$ ...
2
votes
3answers
35 views

Find such anti-symmetric matrix $W$ that $A^T WP \geq 0$

$P$ and $A$ are both n-dimensional vectors with non-negative components. $W$ is an $n\times n$ matrix with $W_{ij}=w_i-w_j$, where all $w_k\geq 0$. So $W$ is an anti-symmetric matrix with some ...
3
votes
3answers
43 views

Inequality Exercise in Apostol's Calculus I

Let p and n denote positive integers. Show that: $$n^{p} \lt \frac{(n+1)^{p+1} - n^{p+1}}{p+1} < (n+1)^{p}$$ Attempt at Solution Using the identity $b^{p+1}-a^{p+1} = ...
2
votes
0answers
48 views

Inequality, symmetric function

Let $$F(x,y)=\dfrac{f\left(\frac{x}{x+y}\right)+f\left(\frac{y}{x+y}\right)}{x+y},$$ where $f>0$ is a concave function. Using brute force computation (computer based proof) with ...