Questions on proving, manipulating and applying inequalities.

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1
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1answer
18 views

Prove that $|x-a_1|+|x-a_2|+|x-a_3|\geq a_3 - a_1$, for $a_1<a_2<a_3$, and determine the condition for equality.

I got this question from the first chapter of Courant and John's Introduction to Calculus and Analysis I. The problem is as follows: Prove that $|x-a_1|+|x-a_2|+|x-a_3|\geq a_3 - a_1$, for ...
2
votes
1answer
24 views

Bounding the variance of a sum of independent random variables

Suppose $\{X_i\}_{i=1}^n$ is a sequence of independently distributed random variables that take values in $[0,1]$. Let $\bar{X}_n = \frac{1}{n}\sum_{i=1}^n X_i$ denote the average of the sequence. ...
1
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2answers
38 views

Predicates and Indirectly Proving the last step of Mathematical Induction

Okay to illustrate this problem, I'm going to need to give an example, and go through the steps of Mathematical Induction to show where my question is aimed at. Example : Prove that $$ n^2 \geq 2n + ...
1
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1answer
49 views

How can i prove this inequality without using AM-GM?

How can i prove that if $a_{1}a_{2}a_{3}\cdots a_{n}=1$ then $a_{1}+a_{2}+ \cdots+a_{n}\geq n$ without using AM-GM? I tried this: If $a_{1}a_{2}a_{3}\cdots a_{n}=1$ then assume without loss of ...
0
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1answer
33 views

Taking the limit of $r \to 0$ in $f(t) \leq Cg(r)$ when this inequality holds for all $ r > 0$

Good morning I have proved the following inequalty: $$f(t) \leq Cg(r) \quad \text{for all $t\in [0,T]$ and $r > 0$}$$ for two functions $f$ and $g$, both of which are continuous. Am I allowed to ...
0
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1answer
41 views

Minimum value of $\dfrac{a^3}{b}+2\dfrac{b^2}{c^2}+\dfrac{c}{2a^3}$

Find the minimum value of $\dfrac{a^3}{b}+2\dfrac{b^2}{c^2}+\dfrac{c}{2a^3}$ where $a,b$ and $c$ are positive real numbers. I tried to used the property $a^2+b^2 \geq 2ab$ but I couldn't figure out ...
7
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5answers
656 views

Is this direct proof of an inequality wrong?

My professor graded my proof as a zero, and I'm having a hard time seeing why it would be graded as such. Either he made a mistake while grading or I'm lacking in my understanding. Hopefully someone ...
0
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1answer
49 views

Inequalities - AM-GM

Let $H_n = 1 + 1/2 + 1/3 + ... + 1/n$ Prove that; $H_n + n$ $\geq$ n$(n+1)^\frac{1}{n}$ for $n$ $\leq$ $2$ I have tried writing $H_n + n = 1/2 + 1/3 +...+ 1/n + (n+1)$ but am left with an $n!$ in ...
0
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1answer
53 views

Is true the following statement?

Consider $0<s<1$. Is true the following statement? $(\forall 0<\delta<1)(\exists \gamma>0)$ such that $(\forall x>0)(\exists n\in\mathbb{Z})$ such that $\gamma<s^nx<\delta. ...
1
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2answers
52 views

Cauchy–Schwarz inequality in Complex variables

I have seen various proofs for Cauchy–Schwarz inequality but all of them discuss only of real numbers. Can someone please give the proof for it using complex numbers in simple steps?
2
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1answer
36 views

Does $d(x,y)=\sqrt[3]{|x_1-y_1|+|x_2-y_2|}$ define a metric?

$d: \mathbb{R^2} \times \mathbb{R^2} \rightarrow \mathbb{R}$ where $d(x,y)=\sqrt[3]{|x_1-y_1|+|x_2-y_2|}$ for $x=(x_1, x_2)$ and $y=(y_1, y_2)$ I am trying to determine if $d$ defines a metric on ...
0
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1answer
21 views

Triangle Inequality of the Cartesian product with Max function

Let $(X,d)$ be a metric space. Define $$d'((x,y),(z,w))=max\{d(x,z),d(y,w)\}.$$ I'm trying to prove the triangle inequality for this, but really don't have a clue how. Any tips or suggestions ...
0
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1answer
23 views

How to show $L^a-l^a \ge (L-l)^a$?

Is it possible to show that $L^a-l^a \ge (L-l)^a$ (or the opposite), where $l \in [0,L]$ and $0<a<1$? Thanks a lot!
2
votes
5answers
57 views

Algebraically solving the inequality $\frac{1}{x} - 1 > 0$

$$\frac{1}{x}-1>0$$ $$\therefore \frac{1}{x} > 1$$ $$\therefore 1 > x$$ However, as evident from the graph (as well as common sense), the right answer should be $1>x>0$. ...
1
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2answers
42 views

Prove that $\cot^n\frac{\alpha}{2}+ \cot^n\frac{\beta}{2}+ \cot^n\frac{\gamma}{2}\ge3^\frac{n+2}{2}$

Let $\alpha$,$\beta$ and $\gamma$ angles of some triangle and n natural number, Prove that $$\cot^n\frac{\alpha}{2}+ \cot^n\frac{\beta}{2}+ \cot^n\frac{\gamma}{2}\ge3^\frac{n+2}{2}$$ I've tried ...
2
votes
3answers
51 views

Inequality involving the min function

I'm trying to prove the following inequality: $$ \left|y_{1}\land x_{1}-y_{2}\land x_{2}\right|\leq\left|y_{1}-y_{2}\right|+\left|x_{1}-x_{2}\right|, $$ where $x\land y=\mbox{min}(x,y)$. By ...
0
votes
1answer
54 views

Function inequalities

I want to resolve this inequality, any help? $$\left\lvert\frac {2+\sin (x)}{x+4} \right\rvert<k$$ for $k>0$. (I am sorry for my English. It's not my first language)
0
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2answers
45 views

Is the following inequality right? [closed]

For $1\leq p$, is it true that there exists a constant c such that, for $n\geq0$, $$\left|\sum_{j=0}^{n}A(j)\right|^{p}\leq c\sum_{j=0}^{n}|A(j)|^{p}$$ where $A(j)$ is real. Can $c$ be independent of ...
0
votes
0answers
35 views

Using mathematical induction to prove P(n) [duplicate]

I have the statement $P(n)$: $2^n<(n+1)!$, for $n \geq 2$; $P(2)$: $2^2 < 3!$ which is true I.H P(k): $2^k<(k+1)!$ show that $P(k+1)$: $2^{k+1} <(k+2)!$ Here is my approach: ...
3
votes
1answer
67 views

Finding the value of $1.1^{82}$ using $(1+x)^{82}$ to a certain accuracy

I found this question in a book. How many terms of the Maclaurin expansion of $(1+x)^{82}$ are needed to guarantee finding a value of $1.1^{82}$ to an accuracy of $10^{-6}$? This is how I tried to ...
1
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0answers
11 views

Does there exist an optimal solution $(x^*,y^*)$ to $\max x^TAy$ such that $x^*=y^*$?

Given two positive integers $n \le m$ and non-negative real constants $a_{ijkl} \ (1\le i,k\le n,1\le j,l\le m)$. Let $M$ be the set of $X\in\mathbb{R}^{n\times m}$ satisfying: $X\ge 0$, The sum ...
1
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1answer
55 views

Is it true that $\operatorname{tr}(AA') - (l_1 ^2 + \dots + l_n ^2) > 0$?

I want to know if for a matrix with real entries the next inequality holds: $$\operatorname{tr}(AA') - (|l_1| ^2 + \dots + |l_n| ^2) > 0$$ where $l_1, \dots, l_n$ are the eigenvalues of $A$. I ...
1
vote
1answer
26 views

Seems familiar: find $\max \sum_{i=1}^n\sum_{i=1}^n a_{ij}x_iy_j$ s.t. $x_1+\cdots+x_n=y_1+\cdots+y_n=1$

Given $a_{ij}\ge 0$. Find $$\max \sum_{i=1}^n\sum_{j=1}^n a_{ij}x_iy_j$$ s.t. $x_1+\cdots+x_n=y_1+\cdots+y_n=1, x_i\ge 0, y_i\ge 0\ \forall i$.
1
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1answer
47 views

Why does this integral inequality hold?

The original question was: Let $f$ be a continuous function on $[a,b]$ and suppose that for every $a_1$ and $b_1$ in this interval: $$\int_{a_1}^{b_1}f(x)\text{d}x=0$$ Show that $f(x)=0$. ...
0
votes
1answer
29 views

Binomial inequality problem ${k+n-1 \choose k}\times{k+n+1 \choose k} \leq{k+n \choose k}^2$

Can anyone help we with this problem: Let $a_n={k+n \choose k} $ Prove that $a_{k-1}a_{k+1}\leq a_k^2 $($\forall k$) My first idea was using mathematical induction to proof that for every k element of ...
0
votes
0answers
49 views

Which values of $n$ is this inequality related to prime numbers true for?

Inequality What values of $n$ satisfy the following inequality? $$2(n-2) < Ap_n\prod_{i=3}^n \left(\frac{p_i-2}{p_i}\right)$$ $p$ are prime numbers and the notation $p_i$ indicates the ...
3
votes
0answers
44 views

For which cases division of two decreasing function is decreasing

Let $f$ be a increasing and bijective function defined on $(0,\infty)$. Suppose also that $\lim_{r\to 0}f(r)=0$ and $\lim_{r\to \infty}f(r)=\infty$ and $f(uv)\geq C f(u)f(v)$ for some $C>0$. I want ...
2
votes
1answer
20 views

Asimptotic Inequality of two expressions

$$\left(\log \left(\log \:\left(n\right)\right)\right)^n\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\prod _{k=2}^n\left(\log \:k\right)\:$$ Hi, I need to measure the asymptotic quantity of both expressions each by ...
1
vote
1answer
90 views

Establish the inequality $\frac{{N - \sqrt{N}} \choose \Omega}{N \choose \Omega} \geq (1-\frac{2\Omega}{N})^{\sqrt{N}}$

Establish the inequaity $$ \frac{{N - \sqrt{N}} \choose \Omega}{N \choose \Omega} \geq (1-\frac{2\Omega}{N})^{\sqrt{N}}$$ where $N > \Omega > \sqrt{N}$ and $N$ is a perfect-square. My attempt: ...
0
votes
3answers
39 views

A weird question about the proof of the triangle inequality

Proposition: $|x+y| \leq |x| + |y|$ Proof : $(x+y)^2 = x^2 + 2xy + y^2 $ $\leq |x|^2 + 2|x||y| + |y|^2$ $= (|x| + |y|)^2$ So $(x+y)^2 \leq (|x|+|y|)^2 \Rightarrow \sqrt{(x+y)^2} \leq \sqrt{(|x| ...
1
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1answer
37 views

Conditional Entropy and Gibbs Inequality

We know $$H(X | Y) + H(Y) = H(X, Y)$$ Therefore, $$H(X | Y) \leq H(X, Y) $$ since $$ H(Y) \geq 0$$ If we expand this out, we get $$-\sum_{x,y} {p(x,y) \log p(x | y)} \leq - \sum_{x,y} {p(x,y) \log ...
0
votes
1answer
80 views

Why is Mathematical Induction used to prove solvable inequalities?

As a first year undergrad student I've seen problems where solvable inequalities need to be proven to hold in a specific domain using Mathematical Induction. My question is, if the inequalities are ...
1
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1answer
29 views

If $abcd=1$,where $a,b,c,d$ are positive reals,then find the minimum value of $a^2+b^2+c^2+d^2+ab+ac+ad+bc+bd+cd$.

If $abcd=1$,where $a,b,c,d$ are positive reals,then find the minimum value of $a^2+b^2+c^2+d^2+ab+ac+ad+bc+bd+cd$. Let $E=a^2+b^2+c^2+d^2+ab+ac+ad+bc+bd+cd=(a+b+c+d)^2-(ab+ac+ad+bc+bd+cd)$ I do not ...
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votes
0answers
7 views

An application of Toeplitz determinant

Given a positive real c between 0 and 2 and d is complex. How to show 4 + Re{(c^2)*d} - |d|^2 - 2(c^2) greater or equal to 0 is equivalent to 2d = c^2 + x((4 - c)^2), for some complex x where x is ...
-3
votes
3answers
55 views

if $a,b\in\mathbb{R}$ proof $|a_{1}|+|a_{2}|\geq |a_{1}-a_{2}|\geq ||a_{1}|-|a_{2}||$ [closed]

$a,b\in\mathbb{R}$ ; $|a_{1}|+|a_{2}|\geq |a_{1}-a_{2}|\geq ||a_{1}|-|a_{2}||$ please prove this inequality system...
3
votes
1answer
58 views

In a $\triangle ABC,$ If $\cot A+\cot B+\cot C =\sqrt{3},$ Then prove that $\triangle$ is equilateral. [duplicate]

In a $\triangle ABC,$ If $\cot A+\cot B+\cot C =\sqrt{3},$ Then prove that $\triangle$ is equilateral. $\bf{My\; Try::}$ Using Jensen's Inequality, Let $f(x)=\cot x\;,$ Where $x\in (0,\pi),$ ...
-1
votes
1answer
49 views

Finding a formula for the sum $\frac{n}{1}+…+\frac{n}{logn}$ [closed]

I have the equation below. I want to find an answer: When $n\rightarrow \infty $ $$\frac{n}{1}+\frac{n}{2}+\frac{n}{3}+\frac{n}{4}+....+\frac{n}{logn}=?$$ $$n * ...
1
vote
3answers
132 views

Which prime numbers is this inequality true for?

Inequality What values of $n$ satisfy the following inequality? $$2(n-2) < p_n\prod_{i=3}^n \left(\frac{p_i-1}{p_i}\right)$$ Where $p$ are prime numbers and the notation $p_i$ indicates the ...
0
votes
0answers
21 views

Inequality verification of the ratio of two integrals involving Bessel functions

Given the following integral: $\sigma(k,\theta)=2k^2cos^2\theta\int_0^\infty J_0(2k\tau |sin\theta|) exp(-2s^2k^2\tau^{2H}cos^2\theta)) \tau d\tau$ With the following constraints ...
0
votes
1answer
36 views

Are there any mechanisms to solve a log quadratic inequality?

I have an inequality as follows $$ y \log y - (1-y) \log(1-y) < k$$ where $k$ is a constant. I tried solving the equation using exponential on both sides, but it does not come to a proper form ...
2
votes
1answer
87 views

How do I get this upper bound for Ramsey numbers: $R_k \le \left \lfloor k!e \right \rfloor + 1$?

For every integer $k \ge 2$, $$R_k \le \left \lfloor k!e \right \rfloor + 1$$ where $R_k$ denotes $R(\underbrace{{3, 3, \ldots, 3}}_{k})$.
0
votes
1answer
31 views

Entropy/Variance inequality

The following inequality is sometimes used as a building block to prove log Sobolev inequalities. Does anyone have a simple proof of it? $$ x\log x + y\log y - (x+y)\log \frac{x+y}{2}\leq (\sqrt ...
0
votes
1answer
34 views

Sobolev's inequality implies isoperimetric inequality.

How to show that Sobolev's inequality implies isoperimetric inequality in $\Bbb R^d$? I tried to search it on the internet, but most of the results I got are only about the converse direction. So, ...
0
votes
1answer
32 views

How to find the three largest elements of this matrix?

I have a matrix with positive real numbers $$\mathbf{A}=\begin{pmatrix} a & x & y \\ t & b & z\\ u & v & c\end{pmatrix},$$ where I know that $a,b$ and $c$ are the largest ...
1
vote
1answer
38 views

Let $a,b$ be real numbers with $|a|<1$. If $2ab+b^2 >3$ then prove that $8a^3-6a+b^3>0$.

Let $a,b$ be real numbers with $|a|<1$. If $2ab+b^2 >3$ then prove that $8a^3-6a+b^3>0$. According to me it is false. For $a=0$ ,$ b^2>3$ so $b^3>0$ is false. Am I right?
0
votes
1answer
62 views

Solve $\sin{2x}-\cos{x}<0$

I'm trying to solve the following inequality: $$\sin{2x}-\cos{x}<0$$ $$2\sin{x}\cos{x}-\cos{x}<0$$ $$\cos{x}(2\sin{x}-1)<0$$ The inequality is verified for ...
0
votes
1answer
28 views

Showing multiplication inequality using induction

Use induction to show that: $$\frac{1}{2}\frac{3}{4}\dots \frac{2n-1}{2n} < \frac{1}{\sqrt{3n+1}} $$ for $n > 1$.
2
votes
2answers
24 views

Inequality bound using endpoints?

Currently reading Introduction to Calculus and Analysis by Richard Courant. On page 42 Courant uses the following fact in a larger example explaining why $x^2$ is not uniform continuous (where $a$ and ...
2
votes
5answers
64 views

Prove $\sinh x > x$ for all $x >0$

I did a proof for $\sinh x > x$ for all $x > 0$. But I am not sure if the proof is mathematically valid. I started by showing that $\frac{d}{dx} \sinh x = \cosh x$ and that the limit of $\cosh ...
1
vote
2answers
51 views

Trigonometric inequality - x domain

the following equality is given: $$2 \sqrt2\sin{}x+\sqrt2\cos{x}=\sqrt{-\sin2x}$$ I managed to solve it, but I have a problem with this inequality to define x domain:$$2 ...