Questions on proving and manipulating inequalities.

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3
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2answers
74 views

$\int_0^1 f(x)^2\le 1$ and $\int_0^1 f'(x)^2\le 1$ $\Rightarrow$ $\left|f(x)\right|\le \sqrt3$

Let $f:[0, 1]\rightarrow \mathbb{R}$ be a function that is continous on $[0,1]$ and derivable on $(0, 1)$. If $\int_0^1 f(x)^2\le 1$ and $\int_0^1 f'(x)^2\le 1$, show that $\left|f(x)\right|\le ...
4
votes
0answers
31 views

Finite Series Inequality [duplicate]

For each $n=1,2,{\dots}$ and $x\in(0,{\pi})$, prove that the series $$S_n(x)=\sum_{k=1}^{n} \frac{\sin(kx)}{k}>0$$
2
votes
1answer
53 views

Bell's inequality

Let $\xi, \eta, \zeta$ be random variables such that $|\xi|, |\eta|, |\zeta| \le 1$. I need to prove such inequality: $|\mathbb{E}(\zeta \xi)-\mathbb{E}(\zeta \eta)| \le 1 - \mathbb{E}(\xi \eta)$ ...
1
vote
4answers
145 views

Proving a logarithm inequality

$$\frac{1}{n+1}< \log(1+ 1/n)$$ Any ideas? I tried estimating the difference between $1/n$ and the logarithm and comparing with $1/n-1/(n+1)$ but I miserably failed.
3
votes
1answer
126 views

Extra help on inequality

Someone very helpfully provided an answer to an inequality. See Hard Olympiad Inequality However I don't get part of their answer. How did they get the last factorization??? Thanks so much for any ...
0
votes
1answer
25 views

A not clear chain of inequalities

Look at the following excerpt from the book "J.Talbot, D.Welsh - Complexity and Cryptography": First of all I don't understand the chain of inequalities. Moreover if in the last term ...
0
votes
1answer
60 views

Find $a$ satisfy an inequality for all $t>0$

Let we think about 2-d orthogonal coordinate $xOy$. set point $A(a,a)$, where $a \in \Bbb{R}$. and $P$ is a point in the function $$y=\frac{1}{x}$$. if $$|PA|\geq 2\sqrt{2}$$. find all $a\in \Bbb{R}$ ...
6
votes
1answer
120 views

Hard Olympiad Inequality

Let x,y,z be positive real numbers such that $xy+xz+yz=1$. Prove that $$\sqrt{x^3+x}+ \sqrt{y^3+y}+ \sqrt{z^3+z} \geq 2 \cdot \sqrt{x+y+z}$$. I tried to square expand homogenize then majorize. But I ...
1
vote
3answers
83 views

if $abc=1$, then $a^2+b^2+c^2\ge a+b+c$

This is supposed to be an application of AM-GM inequality. if $abc=1$, then the following holds true: $a^2+b^2+c^2\ge a+b+c$ First of all, $a^2+b^2+c^2\ge 3$ by a direct application of ...
1
vote
2answers
34 views

Inequality with trigonometric functions

Find all values for $a$ such that the following inequality holds: $$\sin^6x + \cos^6x + a\sin x \cos x \ge 0$$ To be fair, I didn't manage to get anything helpful wiht my calculations. I tried to ...
0
votes
1answer
20 views

why does this inequality hold with expectations of supremums

I'm reading a proof on criterion for a class to be Glivenko-Cantelli and I don't see why this holds? $$E \sup_{g\in G} \left|E\left[ \frac{1}{n}\sum_{i=1}^n(g(X'_i)-g(X_i))\big|X_1^n\right]\right| ...
1
vote
1answer
63 views

Number of solutions for 2 equations involving 4 variables

Given that $a, b, c, d$ are positive integers, What are the number of solutions for the given 2 equations, $\mathbf{ad - bc > 0}$ $\mathbf{a + d = n }$ where, $n$ is a given positive integer.
2
votes
2answers
49 views

Help on this inequality

If a,b,c are positive numbers, prove the inequality $$ \frac{1}{a(1+b)}+\frac{1}{b(1+c)}+\frac{1}{c(1+a)} ≥ \frac{3}{1+abc} $$
0
votes
2answers
44 views

Inequality $e^x-1+e^x(1-\cos \pi y+\sin \pi y)<1$

Find a ranges of $x,y\in \mathbb R^+$ in which the following inequality is verified $$e^x-1+e^x(1-\cos \pi y+\sin \pi y)<1$$ My approach: $$2-\cos \pi y+\sin \pi y<\frac{2}{e^x}\leq 2$$ Then ...
0
votes
3answers
29 views

Wrong way to use MVT to prove inequality

I am asked to show that $|\sin x - \sin y| \leq |x-y|$ using the mean value theorem. What I have done seems 'fishy'. I defined $h(x) = |x-y|-|\sin x - \sin y| $. Then $h'(x) = 1 - |\cos x| \geq 0$ ...
1
vote
0answers
26 views

Why are inequalities not accepted in GCSE Statistics Questions?

I was helping somebody with GCSE maths work, and they were doing a GCSE past paper, and they did a statistics question where there is a rubbish question (the typical no time scale and overlapping ...
2
votes
3answers
55 views

How to understand Cauchy's proof of AM-GM inequality(the last step)

The AM-GM inequality: $$a_1a_2\cdots a_n\leq\left(\frac{a_1+\cdots + a_n}{n}\right)^n$$ the trivial case: $a_1a_2 \leq \left(\frac{a_1+a_2}{2}\right)^2 $ is self-evident. then cauchy use this fact ...
0
votes
1answer
19 views

Linear Inequalities - Allocation Problem

The problem at hand can be summarized as follows: we have to allocate a ressource to $n$ production units. The allocation to production unit $i$ is $x_i$. Each of the production unit will produce at ...
1
vote
1answer
24 views

Prove triangle inequality of vector norm

I am trying to show that $||x+y||_p \leq ||x||_p + ||y||_p$ where $p$ is an integer larger than 1, but not infinity (I proved those cases already), and $||x||_p = (\sum_{i=1}^n |x_i|^p)^{\frac{1}{p}}$ ...
1
vote
0answers
43 views

Don't understand a PDE argument involving $L^p$ norms and inequalities

I'm reading this paper. I do not understand how the author proves Corollary 5.12 for the case $p=2$. He addresses everything except $p=2$ but claims it holds for all $p$. Can someone help me to see ...
1
vote
0answers
21 views

Proving an elementary inequality of real vectors related to the p-Laplacian [duplicate]

How would you prove the following inequality? $$ \left|\left|a\right|^{p-2}a-\left|b\right|^{p-2}b\right|\leq C_p\left|a-b\right|^{p-1} $$ where $a,b\in\mathbb{R}^{n}$ and $C_p>0$ is some constant ...
0
votes
1answer
19 views

Arccos and inequalities?

There is something I don't understand with arccos and inequalities. Suppose I have this inequality $cos(x) ≤ \frac{1}{2}$ Having $x = 90$, satisfies this since $cos(90) = 0$. Then since arccos is ...
1
vote
1answer
31 views

Solving inequalities with absolute values on both sides

I need to find the solution sets for the following inequalities: $$|3+2x|\leq|4-x|$$$$|2x-1|+|1-x|\geq3$$ After a bit of tinkering with the first one, I think the solution set is $[-7, \frac13]$, ...
0
votes
0answers
13 views

Does the inequality $\int_{\Omega}(-\Delta)^{\frac 12}G(w(x))(u(x)-C)^+ \geq 0$ hold? If not, can we bound it from above in a particular way?

Let $G$ be a locally Lipschitz function such that $G(0)=0=G'(0)$ and $G$ is also increasing. I want to know if $$\int_{\Omega}(-\Delta)^{\frac 12}(G(w(x))(u(x)-C)^+ \geq 0$$ where $C$ is a constant. ...
0
votes
2answers
37 views

Can one compare $\left(\sum_{i=1}^n \frac1 x_i\right)^{-1}$ with $\min x_i, x_i>0 \forall x_i=1,..,n$

Let $x_1,\dots, x_n \in \mathbb{R^+}$. Is there any nice way of comparing $$ \left(\sum_{i=1}^n \frac1 x_i\right)^{-1} $$ and $$ \min_{i\in\{1,\dots,n\} }x_i $$ If one has $n$ assets uncorrelated ...
1
vote
1answer
34 views

Upper bound for ${n \choose cn}$

Is it true that for any $0<c<1/2$ and sufficiently large $n'$, there exists a $d <2$ such that ${n \choose cn} < d^n$ for all $n>n'$? Clearly we have to assume $cn$ is an integer. I ...
0
votes
1answer
41 views

Binomial coefficient - first two terms, proof of inequality

I've seen the following and I'm not sure whether it is true or not, and if yes, why it holds. $(1-p)^x \geq 1-p x$ for $p\in (0,1)$ and $x>0$. Do I need some additional Information to prove ...
1
vote
3answers
46 views

Proof of triangle inequality for $d(x,y)=\sqrt{\lvert x-y\rvert}$

There is this problem that says: show that $d(x,y)=\sqrt{\lvert x-y\rvert}$ is a distance function on $\mathbb{R}$, and I am unable to proof the triangle inequality for this? any suggestion I look ...
2
votes
1answer
186 views

if $a_n=\frac{a_{[\frac{n}{2}]}}{2}+\frac{a_{[\frac{n}{3}]}}{3}+\ldots+\frac{a_{[\frac{n}{n}]}}{n}$,then $a_{2n}<2a_{n}$

Question: Consider the following sequence : $$a_1=1 ; a_n=\frac{a_{[\frac{n}{2}]}}{2}+\frac{a_{[\frac{n}{3}]}}{3}+\ldots+\frac{a_{[\frac{n}{n}]}}{n}$$. Prove that: $$a_{2n}< 2a_{n } (\forall ...
0
votes
1answer
20 views

How to simplifying and solving this polynomial?

I have a problem with simplifying the polynomial. In the first time, I see that this polynomial is quite simple, but when I'm trying, I realized that this polynomial isn't as easy as I saw. Here is ...
1
vote
1answer
29 views

How find this range of the fucntion $f(a,b)=\left(\frac{1}{a}+\frac{1}{b}+1\right)(a-3b+15)$

let $a,b>0$,and such $a+2b=3$,find range of the follow function $$f(a,b)=\left(\dfrac{1}{a}+\dfrac{1}{b}+1\right)(a-3b+15)$$ My idea: since ...
0
votes
1answer
27 views

If a, b and c are the lengths of the sides of a triangle, prove that $\frac a{b+c-a} + \frac b{c+a-b} + \frac c{a+b-c} \ge 3$

If a,b,c are sides of a triangle, prove:$\sqrt{a+b-c} + \sqrt{b+c-a} + \sqrt{c+a-b} \le \sqrt a + \sqrt b + \sqrt c$
6
votes
1answer
35 views

How prove this inequality $\frac{1}{(a+1)^2+\sqrt{2(b^4+1)}}+\frac{1}{(b+1)^2+\sqrt{2(c^4+1)}}+\frac{1}{(c+1)^2+\sqrt{2(a^4+1)}}\le\frac{1}{2}$

let $a,b,c>0$,and such $abc=1$, show that $$\dfrac{1}{(a+1)^2+\sqrt{2(b^4+1)}}+\dfrac{1}{(b+1)^2+\sqrt{2(c^4+1)}}+\dfrac{1}{(c+1)^2+\sqrt{2(a^4+1)}}\le\dfrac{1}{2}$$ My idea : Use Cauchy-Schwarz ...
0
votes
0answers
23 views

Compound inequalities with separate variables

So I've got to find the set of elements for which the following inequality holds true: $$x-2<12\leq6-5x$$ I've only been taught to solve these kinds of inequalities with one x in the middle, so I ...
0
votes
1answer
10 views

Inequalities with cubic polynomials

So I was checking my friend's notes and found this: Find the interval for which $x^3-x^2-x+1>0$ is true. We haven't been taught to factor cubic polynomials (the rest of the exercises are with ...
1
vote
2answers
22 views

Inequality doubt with taylor expansion

Can I prove that $\forall x>0$ $$e^{x/(1+x)} < 1+x$$ Showing that $e^{x/(1+x)} = 1+x-\frac{x^2}{2}+o(x^2)$ and so $-\frac{x^2}{2}+o(x^2)<0$ for all $x>0$? How i can be sure that $o(x^2)$ ...
0
votes
1answer
18 views

Exercise on a series

Prove the following inequality: $$\sum_{k=m+1}^{\infty} \frac{1}{k!}< \frac{1}{m*m!} \forall m\in \mathbb{N^+}$$ My strategy of attack was to set up an inequality like ...
10
votes
1answer
159 views

How prove $\frac{1}{a_{1}}+\frac{1}{a_{2}}+\frac{1}{a_{3}}+\cdots+\frac{1}{a_{n}}<2$

Let $$A=\{a_{1},a_{2},\ldots,a_{n}\}\subset N$$ Suppose that for any two distinct subsets $B, C\subseteq A$, we have $$\sum_{x\in B}x\neq \sum_{x\in C}x$$ Then show that ...
6
votes
1answer
70 views

Show that $0 < \frac{y-x}{1+xy} < 1$.

Let $E$ a finite set of real, with at least 5 elements. I remember that my teacher proved that there exist two of its elements $x<y$, such that $0 < \dfrac{y-x}{1+xy} < 1$. Unfortunately I ...
1
vote
2answers
43 views

An exponential/polynomial inequality

Prove that there is at least $1$ real number $a>0$ with the property $$a^x\ge x^a $$ for any $x>0$.
1
vote
1answer
50 views

Prove that $a^n - b^n + c^n - d^n \ge (a-b+c-d)^n$

Following on from an earlier question, and in search of a conceptual insight, I asked myself: Given real numbers $a \ge b \ge c \ge d \ge 0 \tag{1}$ Prove that $a^n - b^n + c^n - d^n \ge (a - b + c ...
7
votes
1answer
219 views

A $2\times2$ Matrix inequality

$M,N$ are $2\times2$ real matrices, and $MN=NM$. Then, for any three real numbers $x,y,z$, we have $$4xz\det(xM^2+yMN+zN^2)\geq(4xz-y^2)\big(x\det(M)-z\det(N)\big)^2 $$ some thought: 1). ...
1
vote
1answer
14 views

Proving an inequality on $\sum_{1\leq i,j \leq n} \langle c_i ,c_j \rangle \times \langle l_i ,l_j \rangle$

This is a question that stumped me during an exam I took today. Let $c_1,...,c_n,l_1,...,l_n$ be vectors of $\mathbb R^n$ and $\langle .,.\rangle$ denote the dot product. Prove that ...
1
vote
1answer
34 views

Triangle inequality frobenius norm

I'm trying to show that the frobenius norm is a norm. however it appears as if triangle inequality isnt met. $$||A+B||_F = \sqrt{\sum_{i,j=1}^n |a_{ij}+b_{ij}|^2} \leq \sqrt{\sum_{i,j=1}^n ...
5
votes
1answer
46 views

How prove this $\frac{a_{1}}{a_{2}}+\frac{a_{2}}{a_{3}}+\cdots+\frac{a_{n}}{a_{n+1}}>\frac{n}{2}-\frac{1}{3}$

let $$a_{n}=2^n-1$$ show that $$\dfrac{a_{1}}{a_{2}}+\dfrac{a_{2}}{a_{3}}+\cdots+\dfrac{a_{n}}{a_{n+1}}>\dfrac{n}{2}-\dfrac{1}{3}$$ My idea : since ...
5
votes
0answers
85 views

Very hard inequality: $\frac{a}{\sqrt{a+pb}}+\frac{b}{\sqrt{b+pc}}+\frac{c}{\sqrt{c+pa}} \le k_p \sqrt{a+b+c}.$

Given $p>0$. Find the smallest real number $k_p$ such that the following inequality holds for any non-negative reals $a,b,c$: $$\frac{a}{\sqrt{a+pb}}+\frac{b}{\sqrt{b+pc}}+\frac{c}{\sqrt{c+pa}} \le ...
1
vote
1answer
57 views

prove that $\sqrt{4-a^2}+\sqrt{4-b^2}+\sqrt{4-c^2}+(\sqrt{3}-1)(|a-1|+|b-1|+|c-1|)\ge 3\sqrt{3}$ if $a+b+c=3$

$a,b,c\in[0,2]$ observation by triangle inequality $|a|+|b|\ge |a+b| $ $|a-1|+|b-1|+|c-1|\ge |a+b+c-3|$ but $a+b+c=3$ hence ...
2
votes
1answer
38 views

Show inequality of complex number: $|\frac{a+b}{1+a\bar{b}}|<1$

Suppose $a,b\in\mathbb{C},|a|<1,|b|<1$, how to see $\displaystyle\left|\frac{a+b}{1+a\bar{b}}\right|<1$?
0
votes
0answers
26 views

Double sum, find upper bound

I have a double sum $$\sum_{i=1}^{\log(n)} \sum_{j=\log(n) - i}^{\log(n)} \left(\frac{1}{2}\right)^i$$ And I'd like to show it's $\mathcal{O}(1)$ i.e. there is a constant $c$ that is an upper bound of ...
1
vote
4answers
49 views

How do I solve this quadratic inequality with numbers on both sides?

I thought I'd teach myself some A-Level Maths at home and I'm stuck on a problem I got from mymaths. Problem is mymaths don't bother providing answers, and the tutorial section didn't show me how to ...