Questions on proving and manipulating inequalities.

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Taylor Polynomials: Estimating accuracy of an approximation f(x) ≈ Tn(x)

$f(x) = \sqrt{x},\space\space\space\space a = 4,\space\space\space\space n = 2,\space\space\space\space 4 \le x \le 4.7$ I approximated f by a Taylor polynomial with degree 2 at the number 4. ...
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6answers
336 views

how to determine without calculator,bigger between these two numbers

how can you determine who is bigger between these two numbers: (without calculator): $\left(\frac{1}{2}\right)^{\frac{1}{3}}$ , $\left(\frac{1}{3}\right)^{\frac{1}{2}}$
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0answers
22 views

Find a family of linear inequalities for which the unit ball is a solution set

Find a family of linear inequalities for which the unit ball $A = \{x \in \mathbb R^n \ | \ \|x\|_2 \le 1\}$ is a solution set Would it just be $x_1^2 + ... + x_n^2 \le 1$?
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1answer
45 views

For which positive integers $n$ does $P(n)$ fail to hold?

Let $n$ be a natural number and let $z$ be a complex number. Consider the following proposition: $P(n)$: If $\cos (nz)$ is bounded above by one in absolute value, then $\cos z$ ...
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10 views

On a certain “obvious” implication concerning odd perfect numbers

Let $N = {q^k}{n^2}$ be an odd perfect number given in Eulerian form (i.e., $q$ is prime with $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n) = 1$). ($\sigma(x)$ gives the sum of the divisors of $x$, ...
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0answers
13 views

Find an Upper bound of absolute value (triangle equality application)

Given the functions f(x) and g(x), how can I find a bound for the absolute value \begin{equation} \|f(x)-g(x)-2\| \end{equation} is it correct to say $\|f(x)-g(x)-2\|\leq ...
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2answers
38 views

Can this inequality be solved with Mean value theorem

As my sub-assignment I have to solve inequality: $$ \ln\left(\frac{1}{x} + 1\right) -\frac{1}{x + 1} > 0 $$ If I understood MVT correctly, I should set $g(x)=\ln\left(\frac{1}{x} + 1\right) ...
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polynomial ineq Miklos Schweitzer 2014 [on hold]

Let $n \ge 1$ be a fixed integer. Calculate the distance $inf_{p,f} max_{0\le x \le1}$ $|f(x)-p(x)|$, where $p$ runs over polynomialsof degree less than $n$ with real coefficients and $f$ runs over ...
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1answer
25 views

Poincare' s inequality for vectorfields on the sphere

Let $\mathbb{S}^2$ be the standard 2-sphere, and let $V$ be $\mathcal{C}^1$ vectorfield on it. I'd like to understand if it is true that there exists $C > 0$ such that, for all such $V$, we have $$ ...
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3answers
54 views

Proving $x^2 - y^2 + z^2 \gt (x - y + z)^2$ [on hold]

Prove that $$x^2 - y^2 + z^2 > (x - y + z)^2$$ where: $x < y <z$ for all natural numbers. Thank for help.
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25 views

Prove $ \sin\ (\alpha'+\beta'+\gamma')\cos\ \beta \geq \sin\ \alpha' \cos\ \gamma+ \sin\ (\beta'+ \gamma')\cos\ (\alpha+\beta )$

${\bf Question}$ : Let $$ 0< \alpha,\ \beta,\ \gamma,\ \alpha + \beta + \gamma \leq \pi,\ \alpha':=r\alpha,\ \beta':=r\beta,\ \gamma':=r\gamma,\ 0< r< 1 $$ Prove $$ \sin\ ...
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2answers
62 views

Using mathematical induction to prove $\frac{1}1+\frac{1}4+\frac{1}9+\cdots+\frac{1}{n^2}<\frac{4n}{2n+1}$

This induction problem is giving me a pretty hard time: $$\frac{1}1+\frac{1}4+\frac{1}9+\cdots+\frac{1}{n^2}<\frac{4n}{2n+1}$$ I am struggling because my math teacher explained us that in ...
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2answers
30 views

$\forall x,y\in \mathbb{R}\quad |\sqrt{|x|}-\sqrt{|y|}|\leq\sqrt{|x-y|}\leq\sqrt{|x|}+\sqrt{|y|}$.

$\forall x,y\in \mathbb{R}\quad |\sqrt{|x|}-\sqrt{|y|}|\leq\sqrt{|x-y|}\leq\sqrt{|x|}+\sqrt{|y|}$. i tired we want to prove for all $x,y\in \mathbb{R}\quad ...
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1answer
59 views

$\forall\ x,y,z\in \mathbb{R}$ Show that: $|x+y|+|y+z|+|x+z|\leq |x+y+z|+|x|+|y|+|z|$

$\forall\ x,y,z\in \mathbb{R}$ Show that: $$|x+y|+|y+z|+|x+z|\leq |x+y+z|+|x|+|y|+|z|$$ i tired, i notice that $x,y,z$ plays a symmetrical role in the inequality notice also that ...
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0answers
14 views

Can we find some of those variables verifying this inequality

Let us consider $7$ variables verifying these inequalities: $c>2$ , $x<y$ , $z>w$ , $t^{c}>t$ , $t>x$ , $z<B$ , $w<B$ My question is: Can we find some of those variables ...
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0answers
7 views

Trying to track data reduction using two variables

I want to run a contest with 16 groups to see who can reduce their data usage to a specific level. They must reduce their personal data usage to under 1200 MB and their group average to under 900 MB. ...
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2answers
31 views

how to prove $a+b-ab \le 1$ if $a,b \in [0,1]$?

Given: $0 \le a \le 1$ $0 \le b \le 1$ Prove: $a + b - ab \le 1$
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1answer
11 views

General or specific property? $(1-p)^{-x^2} = x^2 p + \mathcal{O}(p^2)$

As told in the title, I found this equality: $$(1-p)^{-x^2} = x^2 p + \mathcal{O}(p^2)$$ and wonder whether this is true in general or whether it does only hold in the context I've seen it. It comes ...
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1answer
52 views

Inequality: $(a^3+3b^2+5)(b^3+3c^2+5)(c^3+3a^2+5) \ge 27(a+b+c)^3$

Proving inequality for positive real $a,b,c > 0$: $$ (a^3+3b^2+5)(b^3+3c^2+5)(c^3+3a^2+5) \ge 27(a+b+c)^3$$
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2answers
41 views

Prove the inequality: $\frac{a}{c+a-b}+\frac{b}{a+b-c}+\frac{c}{b+c-a}\ge{3}$

Prove the inequality: $\frac{a}{c+a-b}+\frac{b}{a+b-c}+\frac{c}{b+c-a}\ge{3}$ Where $a,b,c$ are sides of a triangle. It is clear that $c+a-b$ is positive but how to use it?
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1answer
38 views

Consider $f(x) = \frac{2x^3-1+\sin x}{x^2-3}$. Show that $f (x) < 2x$ for most negative values of $x$.

Consider $$f(x) = \frac{2x^3-1+\sin x}{x^2-3}$$ Show that $f (x) < 2x$ for most negative values of $x$. How do I start this/ what concepts does this questions test?
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1answer
60 views

Pretty lower bound on the gamma function

According to http://functions.wolfram.com/06.05.29.0006.01, for every $x\geq 2$ it is $$ \left( \frac{x}{e}\right)^{x-1} \leq \Gamma(x) \leq \left( \frac{x}{2}\right)^{x-1}, $$ where $\Gamma$ is the ...
12
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5answers
245 views

How to prove $n < \left(1+\frac{1}{\sqrt{n}}\right)^n$

I want to know how to prove the following inequality. For $n = 1, 2, 3, \ldots $ $$ n < \left(1+\frac{1}{\sqrt{n}} \right)^n $$ I tried with math induction but I failed.
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1answer
38 views

Sum of trigonometric functions

Is the following inequality true? $$\left|\sum_{i=1}^{n}\left(\cos(x_i) \prod_{j\neq i}\sin(x_j)\right)\right|\le 1$$ I tried to count the extremes but it didn't work.
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Finding maximum value of a 3-variable function using inequality.

Let $a, b, c$ be positive real numbers satisfying $a^2 +b^2+c^2=14$. Find the maximum value of $f(a,b,c)=\frac{4(a+c)}{a^2+3c^2+28}+\frac{4a}{a^2+bc+7}+\frac{5}{(a+b)^2}-\frac{3}{a(b+c)}$
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2answers
22 views

Inequalities with arctan

I don't understand how to solve inequalities with arctan, such as: $$\arctan\left(\frac{1}{x^2-1}\right)\ge \frac{\pi}{4} $$ If someone could solve this and give me a very brief explanation of what ...
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2answers
23 views

Logarithmic inequality: $\log_{1/3}^2(x^2-3x+2) - \log_{1/3}(x-1)>\log_{1/3}(x-2) +6$

I need help solving this: $$\log_{1/3}^2(x^2-3x+2) - \log_{1/3}(x-1)>\log_{1/3}(x-2) +6$$ So far I could not make sense of this, because I don't understand how to handle $\log^2$ or the $+6$ at ...
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2answers
24 views

Solving Inequalities with the use of their properties and cases [on hold]

Solve following inequality $$\dfrac4x + 3 \gt \dfrac2x + 1$$ and then graph the solution set on real number line.
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1answer
44 views

$sup_{x,y\in R}{(\cos{x^2}+\cos{y^2}-\cos{xy})}-\inf{(\cos{x^2}+\cos{y^2}-\cos{(xy)})}=6$

let $x,y\in R$,prove or disprove $$sup_{x,y\in R}{(\cos{x^2}+\cos{y^2}-\cos{xy})}-\inf_{x,y\in R}{(\cos{x^2}+\cos{y^2}-\cos{(xy)})}=6$$ I think we must show that ...
2
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1answer
39 views

Prove $\liminf a_n + \liminf b_n\le \liminf (a_n +b_n) $ [duplicate]

$a_n$ and $b_n$ are two bounden sequences Prove $$\liminf(a_n) + \liminf(b_n)\le \liminf(a_n +b_n)$$ Should I use $$\inf(a+b) = \inf(a) + \inf(b)$$ and i could not come up with how to proceed from ...
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3answers
45 views

Show that $2(a^3+b^3+c^3)>a^2(b+c)+b^2(c+a)+c^2(a+b)>6abc$

If $a,b,c$ are positive real numbers, not all equal, then prove that $$2(a^3+b^3+c^3)>a^2(b+c)+b^2(c+a)+c^2(a+b)>6abc$$ How can I show this?
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1answer
114 views

Is $\frac{1}{e^\gamma\log x} \prod\limits_{p < x,p\,\text{prime}} \frac{p}{p-1}<1+ \prod\limits_{p<x,p\,\text{prime}}\frac{1}{p^{n+1}-1}?$

Let $n$ be an initially arbitrarily large variable, but always decreasing (and more specifically non-increasing) to exactly $1$ when $p$ is the largest prime in the product. Then, denoting with ...
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1answer
36 views

An inequality about Hermitian matrices

Say one knows the following statement, That for any Hermitian matrix $H$ with eigenvalues $\lambda_1 \geq \lambda_2 ..\geq \lambda_n$ one has, that in any basis, for any positive integers $1 \leq i_1 ...
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2answers
37 views

Maximising a sum - closed form?

As a follow up to this question, I am wondering the following: Suppose $\sum_{i=1}^n x_i=0,\;\sum_{i=1}^n x_i^2=1$. Is it there a closed form for $\max \sum_{i=1}^n x_ix_{i+1}?$ ($x_{n+1}=x_1$) For ...
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6answers
121 views

Irrational number inequality : $1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}>\sqrt{3}$

it is easy and simple I know but still do not know how to show it (obviously without simply calculating the sum but manipulation on numbers is key here. ...
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0answers
15 views

Prove $|\log F(v)|\leq |\log F(0)|+|v|+|v|^2$ for $F$ is the standard normal CDF

Suppose that $F$ is the CDF of a standard normal distribution. Hayashi (2000) claims that the following is true $$ |\log F(v)|\leq |\log F(0)|+|v|+|v|^2\quad\text{for all}\quad v. $$ How does one ...
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1answer
30 views

Proving Cauchy-Schwarz related proof using induction

So the first thing I was asked to prove was this: If $a_1,a_2,...,a_n$ and $b_a,b_2,...,b_n$ are real numbers, use induction to show. ...
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0answers
33 views

Inequality about squareroots [duplicate]

If $a,b\geq 0$ show that $\left| \sqrt{a}-\sqrt{b}\right|\leq\sqrt{\left|a-b\right|}$. WLOG we can assume that $a\geq b$. If one of them is $0$ this is trivial. So assume none of them is $0$. Now, ...
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1answer
15 views

Inequality involving different diameter average

I have found an assertion in a scientific book (Hinds, Aerosol Technology, 2nd Edition, 1998, p. 83-84) that claims: Given the general form [here for grouped data] for the diameter of an average ...
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1answer
45 views

Maximisation problem

I am trying the following question: If$$a+b+c+d=0,\;a^2+b^2+c^2+d^2=1$$ Then what is the maximum value of $ab+bc+cd+da?$ By the rearrangement inequality I can get $ab+bc+cd+da\leq 1$ but I am ...
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0answers
34 views

On the average length of the Steiner net for $n$ randomly chosen points in the unit square

$n$ points are randomly chosen in the unit square with respect to the uniform measure. What is the average length $L$ of the associated Steiner net (tree of minimum length through each of the $n$ ...
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2answers
80 views

What's the name of this strange inequality?

There is an inequality: $$\sqrt[n]{\prod_{i = 1}^{n}{(a_i+b_i)}} \geq \sqrt[n]{\prod_{i = 1}^{n}{a_i}} + \sqrt[n]{\prod_{i = 1}^{n}{b_i}}$$ which I even don't know its name. I'd like to have an ask ...
2
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0answers
66 views

Prove, using the method of mathematical induction that the following holds true

For natural numbers $n\ge1$ show the following inequality using induction. $$n^{1/n}\le 1+\sqrt{\frac{2}{n}}$$
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2answers
43 views

Inequality about sum of finitely many real numbers

Suppose that $a_1, a_2, ..., a_n$ are non-negative real numbers. Put $S = a_1 + a_2 + ... + a_n$. If $S < 1$, show that $$ 1+S\leq(1+a_1)...(1+a_n)\leq\dfrac{1}{1-S}.$$ I tried induction on $n$ ...
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1answer
16 views

Unclear Application of Cauchy's Inequality

I was looking for a solution to a problem (both found here), where I came across the following ($a, b, c > 0$): Applying Cauchy's inequality, we get $(\frac{c}{a+2b} + \frac{a}{b+2c} + ...
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1answer
20 views

upper bound on this matrix norm

What would be the upper bound on the 2-norm (or any norm) of the following matrix product ? Please consider the smallest upper bound. $\|\left(I+BA^T\right)\left(I+AA^T\right)^{-1}\|< ?$ where A ...
2
votes
1answer
36 views

An elementary inequality about $n$-th roots

I want to show that for each $m,n\in\Bbb{N}$, $$\large{ \dfrac{1}{\sqrt[n]{1+m}}+\dfrac{1}{\sqrt[m]{1+n}}\geq 1}.$$ I tried induction but it doesn't work. Tried to apply the Bernoulli inequality ...
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0answers
23 views

Euclid geometry-triangle inequality [on hold]

How to prove that the sum of the diagonals of a convex pentagon are larger than the scope of the pentagon?(using that a+b>c in triangle?
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1answer
25 views

$a^{x}\left( y-z\right) +a^{y}\left( z-x\right) +a^{z}\left( x-y\right) >0$

If a is any positive number except $1$ , and $ x, y, z,$ are REALS no two of which are equal, then $a^{x}\left( y-z\right) +a^{y}\left( z-x\right) +a^{z}\left( x-y\right) >0$. It is quite easy ...
2
votes
4answers
60 views

Inequality $ \vert \sqrt{a}-\sqrt{b} \vert \leq \sqrt{ \vert a -b \vert } $

I have the following inequality on my class notes that I haven't been able to prove, I was even wondering if it is actually true: $$ \forall a,b \in \mathbb{R}^{\ge0} \left( \left| \sqrt{a}-\sqrt{b} ...