Questions on proving and manipulating inequalities.

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0
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1answer
42 views

Which is bigger, the number of neurons in the brain or the all the stars in the observable universe?

In other words, is 100 billion larger than $10^{22}$ or ....? Are there also other interesting comparisons of systems with large number of members? i.e. the sand on the beach, the atoms in the air, ...
3
votes
1answer
42 views

Name of Inequality

Let $x_i, y_i$ be complex numbers for all $i$. Is there a name for the following inequality? $$\left| \sum_{i=1}^n x_i \right| \leq \sum_{j=1}^n |x_j| $$ In particular, is it a special case of this ...
0
votes
1answer
18 views

Prove that $E(t)$ satisfies the following differential inequality.

It is given that $u_t=2u_{xx}-3u ,\hspace{0.3cm} u_x(0,t)=0=u_x(1,t)$ Use the Young's inequality to show that the energy $$E[u,u_x](t):=\frac{1}{2}\int_0^1(|u|^2+|u_x|^2)dx$$ satisfies the ...
-1
votes
0answers
18 views

Exponential estimate/inequality

I have a vector $x=(x_1,\dots, x_n)\in \mathbb{R}^n$ and some variance $\sigma^2 >0$. I know that the following inequality is wrong (but I present it because it would make world nicer in my view) ...
1
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0answers
26 views

Is there a name for the inequality $\min(a+b,c+d) - \min(a,c) \ge \min(b,d)$?

Is there a name for the inequality $$\min(a+b,c+d) - \min(a,c) \ge \min(b,d)$$? And does anyone have any nice examples or applications, especially with an economic flavor? The transposed multivariate ...
3
votes
2answers
53 views

Show that $x^2 + y^2 + 1 \le \sqrt{(x^3 + y + 1)(y^3 + x + 1)}$

For $x, y \ge 0$ prove that: $$x^2 + y^2 + 1 \le \sqrt{(x^3 + y + 1)(y^3 + x + 1)}$$ What I think would apply is the AM-GM Inequality, so first, $$(x^2 + y^2 + 1)^2 \le (x^3 + y + 1)(y^3 + x + ...
3
votes
1answer
31 views

Symmetric and homogeneous three variable inequality with radicals.

While trying to solve a problem, I got the following inequality which appears correct, but I cannot prove. For positive $x, y, z$, $$\sum_{cyc} \frac{x}{y^2+z^2} \ge \sum_{cyc} ...
3
votes
1answer
39 views

Finding the lowest upper bound of product of two number using Young's inequality

Young's inequality for product can be stated as follows: $ab \leq \frac{1}{p}a^p + \frac{1}{q}b^q$ where a and b are nonnegative real numbers and p and q are positive real numbers such that 1/p + 1/q ...
1
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2answers
38 views

Prove the inequality $x \le x+(1-x) \sin^2(x) \le 1$ for $x \in (0,1)$ by using derivative

The problem: show that $x \le x+(1-x) \sin^2(x) \le 1$ for $x \in (0,1)$ I tried to solve it with the derivative and the inequality $\sin(x) \le x$ for $x>0$ thanks for helpers
2
votes
2answers
224 views

Is this logically valid?

$$1+\frac{1}{2}+\frac{1}{3}.....+\frac{1}{n-1} > ln(n)$$ and so, necessarily, $$1+\frac{1}{2}+\frac{1}{3}.....+\frac{1}{n-1}+\frac{1}{n} = 1+\frac{1}{2}+\frac{1}{3}.....+\frac{1}{n} > ln(n)$$ ...
3
votes
0answers
119 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} + ...
-1
votes
1answer
24 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
74 views

Maximize $x + y + z$

(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+\ ...
-5
votes
0answers
41 views

Find integers which satisfy the following inequality [closed]

Let $a_1$, $a_2$, ..., $a_n$ be $n$ real numbers whose squares sum to $1$. Prove that for any integer $k \geq 2$, there exist $n$ integers $x_1$, $x_2$, ..., $x_n$, each with absolute value $\leq k-1$ ...
1
vote
1answer
28 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
33 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
16 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
23 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 ...
3
votes
4answers
70 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
32 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: ...
0
votes
0answers
19 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
55 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
33 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
34 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 - ...
2
votes
5answers
126 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 ...
0
votes
4answers
57 views

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

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
80 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
0answers
36 views

Trigonometry graph sketch [closed]

Sketch the graph of $y=|\tan x|$ for $0 \leq x \leq 360^{0}$ showing clearly the position of the asymptotes. Solve the inequality $|\tan x| < 1$ for $0 \leq x \leq 360^{0}$.
-4
votes
1answer
29 views

Which is true: $|x| - |y| < |x-y|$ OR $|x-y| < |x| - |y|$ [closed]

Which is true: (for rational numbers) $$ \lvert x \rvert - \lvert y \rvert < \lvert x - y \rvert $$ or $$ \lvert x-y \rvert < \lvert x \rvert - \lvert y \rvert $$ ? thanks in advance
0
votes
1answer
31 views

what are the properties of the definite integral that are related to inequalities? [closed]

what are the properties of the definite integral that are related to inequalities? I've been searching the internet and asking teachers regarding this seemingly implausible connection, but haven't ...
-2
votes
0answers
30 views

Maxima and minima of 2 variable function with conditions

Let $a=2001$. Consider the set $A$ of all pairs of integers $(m,n)$ with $n\not=0$ such that $m<2a$ $2n\mid 2am-m^2+n^2$ $n^2-m^2+2mn\le 2a(n-m)$ For $(m,n)\in A$, let $$f(m,n)=\frac ...
0
votes
1answer
36 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} = ...
0
votes
0answers
16 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
19 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 ...
0
votes
2answers
136 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
33 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?
4
votes
2answers
62 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 ...
7
votes
3answers
152 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 $\frac{a_{n+1}}{a_n}$ ...
3
votes
3answers
80 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
22 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 ...
3
votes
2answers
58 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
92 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 ...
2
votes
2answers
68 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
124 views

Prove $\int_{0}^{1} |\frac {f^{''}(x)}{f(x)}| dx \geq 4$ [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 $\int_{0}^{1} \bigl|\frac ...
3
votes
0answers
68 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}) > ...
2
votes
1answer
43 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
33 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
118 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
23 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
23 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 ...