Questions on proving, manipulating and applying inequalities.

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3
votes
1answer
337 views

Solve an inequality using Cauchy-Schwarz Inequality

Le $a,b,c,d \in \mathbb{R^{+}}$. Using Cauchy-Schwarz Inequality prove that the following inequality holds: $$\frac{1}{\frac{1}{a+c} + \frac{1}{b+d}} \ge \frac{1}{\frac 1a + \frac 1b} + \frac{1}{\...
-1
votes
1answer
40 views

Ordering of real numbers compatible with n-th powers/reciprocal powers (induction)

I have to use induction to prove that $$0 \leq a < b \implies 0 \leq a^n < b^n$$ for all natural n. Also (perhaps very similarly) that $$0 \leq a < b \implies 0 \leq a^{1/n} < b^{1/n}.$$ ...
-1
votes
1answer
89 views

Maximize the sum of 3 numbers subject to 6 linear inequalities

(PUMaC 2006 Algebra #10) If $x, y, z$ are real numbers and \begin{alignat*}{9}2x+\ &&y+\ &&z\leq&66\\ x+\ &&2y+\ &&z\leq&60\\ x+\ &&y+\ &&2z\...
1
vote
1answer
53 views

Proving that a system of equalities and inequalities is inconsistent (Vol. 3)

I am studying sign pattern matrices and this is (hopefully!) the last of the systems that I have to prove inconsistent. Prove that the system $$\begin{cases} a,b,d,e,f,g,h,i>0 \\ -a+e-i=0 \\ -...
1
vote
1answer
68 views

Proving that a system of equalities and inequalities is inconsistent

Prove that the system $a,b,d,e,f,g,h,i>0$ $ae+ai−bd+ei−fh=0$ $aei−hfa-bdi−gbf=0$ is inconsistent. I tried using some standard techniques such as factoring, or multiplying an equality and ...
0
votes
1answer
20 views

Inequalities and arithmetic operations

I'm reading a paper with some math involved, and on a demonstration the author makes this assumptions: $a/b < e/f$ and $c/d < e/f$ And after that and without stating anything else it ...
1
vote
1answer
37 views

How can I prove this inequality using Cauchy's inequality?

Cauchy's inequality is given by: for real numbers, $a_1,...,a_n$, $b_1,...,b_n$, $(a_1^2,...,a_n^2)^{1/2}(b_1^2,...,b_n^2)^{1/2} \geq |a_1b_1+a_2b_2+...+a_nb_n|$. Assuming this, prove that $\sum_{k=1}^...
3
votes
4answers
86 views

Find all $x$ for which $x+3^x<4$

Find all $x$ for which $x+3^x<4$ I'm stuck at this one...how does one solve for $x$? I've tried: $x+3^x<4$ $3^x<4-x$ $x<\log_3({4-x})$ But I don't know where to go from there. If I ...
3
votes
2answers
106 views

An upper bound for an average exponentiated weighted sum of a vector from $\{-1,1\}^n$

Suppose $\mathcal{S}=\{\mathbf{x}:\mathbf{x}\in\{-1,1\}^n\}$, that is, $\mathcal{S}$ contains all $2^n$ vectors of length $n$ containing -1 and 1. I am interested in the following average: $$A_{f(n)}...
1
vote
1answer
39 views

Inequality and Intuitionistic logic

$x, y \in \mathbb{R}$ Is the proposition $x \leq y \Rightarrow x=y \lor x<y$ true in intuitionistic logic ? And what about $x \leq y \Rightarrow \lnot(\lnot(x=y \lor x<y))$ (with $\lnot$ the ...
6
votes
3answers
191 views

Why is $ \frac{a^2}{a+b}+\frac{d^2}{a+d}+\frac{b^2}{b+c}+\frac{c^2}{c+d} \geq 0.5 $ with $a+b+c+d = 1$?

For positive real numbers $a,b,c,d>0$ it seems to be true that: if $$a+b+c+d = 1$$ then $$ \frac{a^2}{a+b}+\frac{d^2}{a+d}+\frac{b^2}{b+c}+\frac{c^2}{c+d} \geq 0.5 $$ I can't think of a way ...
2
votes
2answers
35 views

How to go about solving this inequality question?

$\cos(3x-\pi/3) \leq (1/2).$ Here is what I have done so far... Let $3x-\pi/3 = X$. So I need to solve $\cos(X) \leq 1/2$. Which is all $X$ from $\pi/3$ to $5\pi/3$, so-- $\pi/3 \leq X \leq 5\pi/3 ...
1
vote
1answer
60 views

How should I think when combining multiple inequalities?

When reading/writing papers, I have always find it not obvious when two or more inequalities are combined. For example, taken from my current research $$\text{Pr}(X \le ab) \le -a (1-p)^{-N} (1 - (1-...
2
votes
5answers
129 views

$e^{x} > 1$ and $0 < e^{x} < 1$

So $$\exp(x) := \sum_{n=0}^{\infty} \frac {x^n} {n!}$$ How to prove that $\exp(x) > 1$ when $x > 0$ and moreover $\exp(x) < 1$ when $x<0$ Is it possible with induction? Or must I use ...
1
vote
4answers
108 views

If $0<a<b$, prove that $a<\sqrt{ab}<\frac{a+b}{2}<b$ [duplicate]

If $0<a<b$, prove that $a<\sqrt{ab}<\frac{a+b}{2}<b$ So far I've got: $a<b$ $a^2<ba$ $a<\sqrt{ab}$ And: $a<b$ $a+b<2b$ $\frac{a+b}{2}<b$ So I need to prove ...
1
vote
4answers
128 views

Demonstration of the inequality of Cauchy-Schwarz

After the demonstration of the inequality of Cauchy-Schwarz make by my professor, I still don't understand some steps of the demonstration. To prove this inequality, my professor use the induction ...
0
votes
1answer
61 views

Setting bound for an infinite expected value

Say $X=2^Z$ and $Z$ is a geometric random variable with $p=1/2$. It follows that, $E[X] = \infty$ So setting the upper bound by the markov inequality, $$P(X \geq t) \leq \frac{E[X]}{t} = \frac{\...
0
votes
0answers
19 views

The conditions on marginals that guarantees a certain class of measures

$x$ and $y$ are $m\times n$ matrices. $a, B,C$ are $m\times 1$ matrices. $b, A$ are $n\times 1$ matrices. $$\sum a_i=1, 0\leq a_i\leq 1, \forall i$$ $$\sum b_j=1, 0\leq b_j\leq 1, \forall j$$ ...
1
vote
1answer
69 views

Bernoulli's inequality variation

To prove: $(1+a_1)(1+a_2)\ldots(1+a_n)\geq\dfrac{2^n}{n+1}(1+a_1+a_2+\ldots+a_n)$ when $a_i\geq1$ This seems to be based on Bernoulli's Inequality (which can be proved by induction). Trying the ...
1
vote
2answers
196 views

Inequality $(n!)^2\le \left[\frac{(n+1)(n+2)}{6}\right]^n$

Prove that $$ (n!)^2\le \left[\frac{(n+1)(n+2)}{6}\right]^n $$ holds for all $n\in\mathbb{Z^+}$. I tried induction but there's no obvious way to go from $n$ to $n+1$.
0
votes
2answers
258 views

Suppose that $a$ and $b$ are nonzero real numbers. Prove that if $a<\frac1a<b<\frac1b$ then $a<-1$

Suppose that $a$ and $b$ are nonzero real numbers. Prove that if $a<\frac1a<b<\frac1b$ then $a<-1$ I'm stuck on this one. Where does one begin?
5
votes
2answers
75 views

Prove the inequality $a^3+2 \geq a^2+2 \sqrt{a}$

Prove the inequality $a^3+2 \geq a^2+2 \sqrt{a},a \geq 0.$ One way to do it is using the formula $$ a^3+2 - a^2-2 \sqrt{a}=(\sqrt{a}-1)^2(1+(a+1)(\sqrt{a}+1)^2) \geq 0. $$ But I hope there is a ...
5
votes
3answers
322 views

which is bigger $2^{n^{1.001}}$ or $n!$

for a big enough n, How to detrmine which is bigger? $2^{n^{1.001}}$ or $n!$ I have tried to make a series: $a_n = \frac{2^{n^{1.001}}}{n!}$ and then try finding the limit of $\frac{a_{n+1}}{a_n}$ ...
4
votes
3answers
172 views

Proving that if $xy + yz + zx \geq \frac{1}{\sqrt{x^2+y^2+z^2}}$, then $x+y+z\geq \sqrt{3}$

If $x, y, z$ are positive real numbers such that $$xy + yz + zx \geq \frac{1}{\sqrt{x^2+y^2+z^2}},$$ then prove that $x+y+z\geq \sqrt{3}$.
0
votes
0answers
73 views

Inequality with cumulative probability function of binomial distribution

Prove that $F(k; 2k+1, p) > F(k-1; 2k-1, p)$ where $p < 1/2$. Here $F$ is the cumulative probability function of binomial distribution Intuitively the inequality is obvious as expected value ...
3
votes
2answers
80 views

find the range of values

Let $x,y,z$ be positive real numbers where $$ \frac{1}{3} \leq xy + yz + zx \leq 3. $$ Determine the range of values for $xyz$ and $x+y+z$. I found this question on the British Mathematical Olympiad ...
3
votes
1answer
102 views

Prove that for any positve real

Prove that for any positive real numbers $x,y,z$ such that $xyz \geq 1$ $$\frac{x^5-x^2}{x^5+y^2+z^2} + \frac{y^5-y^2}{y^5+z^2+x^2} +\frac{z^5-z^2}{z^5+x^2+y^2} \geq 0.$$ This problem is from the $...
3
votes
2answers
82 views

inequality symbols question (beginning algebra)

Please help me with this problem: "In each of the following exercises $x$ and $y$ represent any two whole numbers. As you know, for these numbers exactly one of the statements $x < y, x = y$, or $...
3
votes
0answers
183 views

Prove $\displaystyle\int_{0}^{1} \left|\frac {f^{''}(x)}{f(x)}\right|\, dx \geq 4$ [closed]

I find an interesting theorem,but have no idea to prove it. $f(x) \in C^2[0,1]$ and $f(0)=f(1)=0$ , $f(x) \not = 0 \ \ , x\in (0,1) $ Prove that if $\displaystyle\int_{0}^{1} \left|\frac {f^{''...
7
votes
3answers
161 views

How to find the minimum $m$ for a given $n$ in this inequality?

For a given $n \in \Bbb N$, how do you find the minimum $m \in \Bbb N$ which satisfies the inequality below? $$3^{3^{3^{3^{\unicode{x22F0}^{3}}}}} (m \text{ times}) > 9^{9^{9^{9^{\unicode{x22F0}^{...
2
votes
1answer
66 views

Holder type inequality

If $A$ is a symmetric and positive semidefinite matrix is it true that $$\sum_{i,j=1}^n A^{i,j}x^iy^j \leq \sqrt{\left(\sum_{i,j=1}^n A^{i,j}x^ix^j\right)\left(\sum_{i,j=1}^n A^{i,j}y^iy^j\right)},$$ ...
0
votes
2answers
61 views

How many solutions does the equation have $x_1 + x_2 +x_3 = 8$ have with restrictions $x_1 \leq 2$ and $x_2 \leq 3$ (for all nonnegative numbers)

How many solutions does the equation have $x_1 + x_2 + x_3 = 8$ have with restrictions $x_1 \leq 2$ and $x_2 \leq 3$ (for all nonnegative numbers)? I seem to be stuck on the multiple conditions that ...
3
votes
3answers
137 views

Elementary Proof that $x^x \geq x!$

Is there an elementary proof that $x^x \geq x!$ for natural numbers $x$? I am not looking for a heuristic argument such as the one that there are $x$ terms in $x^x$ and $x!$ and since almost every ...
0
votes
0answers
26 views

Help with proof of logarithmic inequality

I have a fairly simple question about the logarithm. I want to show that $|log(1+x)|\leq K*|x|$ for a $K\in R$ that we choose such that this holds. My question is: How can we prove this? My idea ...
0
votes
1answer
62 views

Binomial coefficients bounded by entropy exponential

So I'm trying to prove that for $\frac{1}{2}< x \leq 1$ we have $$\sum_{\lceil nx \rceil}^{n}{n \choose k} \leq 2^{nh(x)}$$ I've managed to prove that $$\sum_{0}^{\lfloor nx \rfloor}{ n\choose k}\...
1
vote
2answers
89 views

How can I prove that $x-{x^2\over2}<\ln(1+x)$

How can I prove that $$\displaystyle x-\frac {x^2} 2 < \ln(1+x)$$ for any $x>0$ I think it's somehow related to Taylor expansion of natural logarithm, when: $$\displaystyle \ln(1+x)=\color{red}...
5
votes
1answer
124 views

Inequalites of triangle side with $abc = 1$

Let $a,b,c$ be the sides of a triangle with $abc=1$. Prove that $$ \frac{\sqrt{b+c−a}}{a} + \frac{\sqrt{c+a-b}}{b} + \frac{\sqrt{a+b−c}}{c} \ge a+b+c $$
2
votes
6answers
89 views

Solving $\dfrac{x+2}{x}>0$

I want to find values of $x$ such that $\dfrac{x+2}{x}>0$ : $1+\dfrac{2}{x}=\dfrac{x+2}{x}>0 \implies \dfrac{2}{x}>-1 \implies \dfrac{1}{x}>\frac{-1}{2} \implies x<-2 $. But by ...
2
votes
0answers
205 views

Hölder's inequality and log convexity of $L^{p}$ norm

Hölder's inequality of $L^{p}(X,\mu)$ $\left\Vert fg \right\Vert_{r} \leq \left\Vert f \right\Vert_{p} \left\Vert g \right\Vert_{q}$ where $0<p,q,r\leq \infty$ and $\frac{1}{p}+\frac{1}{q}=\frac{...
0
votes
1answer
30 views

Show that $\frac c {1+c} \le \frac a {1+a} + \frac b {1+b}$ , for $c \le a+b$ and $a,b,c \ge 0$

Show that $\frac c {1+c} \le \frac a {1+a} + \frac b {1+b}$ , for $c \le a+b$ and $a,b,c \ge 0$ So need to show $\frac c {1+c} \le \frac {a+b+2ab} {1+a+b+ab}$ We have $\frac c {1+c} \le \frac {a+b} {...
2
votes
1answer
44 views

Given $x,y,z>0$: $\frac{2}{3x+2y+z+1}+\frac{2}{3x+2z+y+1}=(x+y)(x+z)$. Find Minimum Value Of: $P=\frac{2(x+3)^2+y^2+z^2-16}{2x^2+y^2+z^2}$

Given $x,y,z>0$: $\frac{2}{3x+2y+z+1}+\frac{2}{3x+2z+y+1}=(x+y)(x+z)$ $(1)$ Find Minimum Value Of: $P=\frac{2(x+3)^2+y^2+z^2-16}{2x^2+y^2+z^2}$ I found $2x+y+z\geq 2$ from (1) but it not work :( ...
0
votes
0answers
42 views

$\sup$ of a $C^s$ smooth function.

I want to prove that for a function $F\in C^k(\mathbb{R}^n)$ which vanishes at zero, and a function $u\in H^k(\mathbb{R}^n)$we get: $$\left\| \int_{r=0}^1 F'(ru)(\cdot)dr \right\|_{L^{\infty}} \leq \...
-2
votes
2answers
230 views

Inequality $\prod\limits_{r=1}^{\infty}(1+(\frac{1}{2})^r)<\frac 52$ [closed]

Prove this inequality. $\prod\limits_{r=1}^{\infty}\left(1+\left(\frac{1}{2}\right)^r\right)<\dfrac 52$ I have tried to prove it using induction but it is not coming.
3
votes
2answers
89 views

Prove the triangle inequality is valid for the norm $\lVert f \rVert =(\int_0^1 \lvert f(x) \rvert ^2 dx)^{1/2}$

I.e, prove $\lVert f+g \rVert\ \le \lVert f \rVert + \lVert g \rVert$ for all $f,g$ in $C^\infty [0,1]$, $$\lVert f \rVert =(\int_0^1 \lvert f(x) \rvert ^2 dx)^{1/2}$$ I think we're supposed to use ...
2
votes
4answers
217 views

asymptotically sharp upper and lower bound for for arctan [closed]

How do I prove that $\frac{\pi}{2}-\frac{1}{x}<\arctan(x)<\frac{\pi}{2}-\frac{1}{x}+\frac{1}{3x^3}$ for all $x>0$?
3
votes
1answer
121 views

Please help me to prove the following inequality

Let $a_i\geq0$ for all $i\in\{1,2,...,n\}$ with $\sum_{i=1}^na_i=1$. Suppose $x_1,x_2,...,x_n$ are non-negative reals. Show that $(\sum_{i=1}^na_i^2x_i^2)(\sum_{i=1}^n\dfrac{1}{x_i^2})\leq\dfrac{(z_{\...
0
votes
3answers
51 views

Prove inequality of 2 functions for every $ x \gt 1 $

Prove that for every $ x \gt 1$ exists: $$ 2x^3 + x^2-􀀀2x < x^4+5x^2-5 $$ I got this on my calculus course at college, can I get some suggestions? I'm really breaking my head on this, and ...
3
votes
1answer
164 views

Confusion about order of operations with point-in-tetrahedron formula

I am not a math student, but I am attempting to roll my own GJK-based hit detection function. It would seem that most of the Internet that I've searchedd through chooses to either ignore or obfuscate ...
0
votes
1answer
103 views

The $55$-th IMO, problem $1$

Let $a_0<a_1<a_2<\dots$ be an infinite sequence of positive integers. Prove that there exists a unique integer $n\ge1$ such that $$a_n<\frac{a_0+a_1+\dots+a_n}{n}\le a_{n+1}$$ This is ...
0
votes
0answers
30 views

Why does from this inequality follow the identity of the Limit Superior?

Let $\gamma_{g,n}$ be the number of ways of putting down g $\ell$'s down on the discrete interval $[0,n-1]$ with the $\ell$'s separated by at least two $0$'s and let $\gamma_n=\sum_{g=0}^n\gamma_{n,g}$...