Questions on proving and manipulating inequalities.

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0
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1answer
143 views

Help with Inequality involving absolute values of trig

I am trying to wrap my ahead around the following problem: Prove that for all $x,y$ in $\Bbb R$ $ |\sin(x) - \sin(y)| \leq |x-y|$ And prove that for $x,y$ in $R$ $|\cos(x) - \cos(y)| \leq |x - y|$ ...
3
votes
2answers
190 views

Prove that $\log^25 + \log^27 > \log12$.

Prove that $\log^25 + \log^27 > \log12$. What I tried so far: $\log^25 + \log^27 > \log3 + \log4$ $(\log5 + \log7)^2 - 2 \cdot \log5 \cdot\log7 > \log3 + \log4$ But it seems that I'm not ...
1
vote
1answer
131 views

How prove this distributions inequality $cov(\theta_{i},\theta_{j})\ge 0$?

Question: let random variable $\theta$ has dendity $f_{\phi}(\phi)$,and the random vector $\theta=(\theta_{1},\theta_{2},\cdots,\theta_{n})$,such $\theta_{i}|\phi$ are all independent from each ...
2
votes
1answer
38 views

Problem on inequality

Prove that, $E|X|^p < \infty $ iff $\sum_{k=1}^{\infty}k^{p-1}P\{|X| \geq k\} < \infty$. Where E is the expectation and P is the usual probability measure. There was a mistake one it's correct.
1
vote
3answers
80 views

Proving that a statement about $<$ is valid

I need to do assignment for my homework, in which I need to prove that the following statement is valid. $$ (s<t \text{ and } t<u)\implies(s<u) $$ I need to do this assignment using the laws ...
1
vote
2answers
145 views

How find this maximum of $P_{1}+P_{n}$

Question $n$ students attend a test of $m$ problems where $m, n \ge 2$. The scoring rule for each problem is: If $x$ students answer a problem incorrectly, then a correct answer worth $x$ points ...
1
vote
0answers
58 views

SOLVED — Equal or Equivalent Inequalities?

Suppose I have two inequalities: $2ab \leq a^{2} + b^{2}$ $0 \leq a^{2} + b^{2} - 2ab$ Now, obviously these inequalities are just rearrangements of one another. The question I have is: do I say ...
3
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0answers
75 views

A challenging non homogenous fractional inequality.

The following problem is a challenging generalization of several difficult inequalities, where none of the usual methods used in inequalities seems to work. I would like to know if someone has a ...
0
votes
1answer
55 views

Solving inequality. Did I do it right?

Solve the following inequality: $$5(y-2)-3(y+4)\ge2y-20$$ I made calclations and I found: $$ 0\ge2 $$ What does it mean? Are my calculations right? Here's how I did it: $$ ...
3
votes
3answers
248 views

Proving that $n!≤((n+1)/2)^n$ by induction

I'm new to inequalities in mathematical induction and don't know how to proceed further. So far I was able to do this: $V(1): 1≤1 \text{ true}$ $V(n): n!≤((n+1)/2)^n$ $V(n+1): ...
0
votes
1answer
37 views

Visualize the effect of adding another constraint

I have 2 eqns $$ x_1+4x_3\leq4$$ $$ x_2+4x_3\leq4 $$ $$x_1\geq0$$ $$x_2\geq x_3\geq0$$ By drawing geometrical figure I have vertices whose co-ordinates is $(0,0,0) , (4,0,0) (0,0,1) ,(0,4,0) ...
14
votes
2answers
355 views

How prove this inequality $|x\sin{\frac{1}{x}}-y\sin{\frac{1}{y}}|<2\sqrt{|x-y|}$

for any real numbers $x,y\neq 0$,show that $$|x\sin{\dfrac{1}{x}}-y\sin{\dfrac{1}{y}}|<2\sqrt{|x-y|}$$ This problem is from when I deal this problem How prove this analysis function ...
2
votes
2answers
68 views

Induction $(1+\frac{1}{x^n})(1+\frac{1}{y^n}) \geq (1+2^n)^2$

How to prove this inequality using Induction (or any simpler method): Let (x,y) be real positive numbers, so that x+y=1; and n an integer: Prove this: $\begin{align}(1+\frac{1}{x^n})(1+\frac{1}{y^n}) ...
2
votes
3answers
56 views

inequality question on real numbers #2

Expirimentally it seems that $$\sum_{1\leq j\leq \lfloor n/8 \rfloor}\left(\frac{\pi en}{4j}\right )^{j/2}<2^{c_0n}$$ where $c_0=0.6$ and large $n$. Is there any proof? Thank you.
1
vote
0answers
74 views

Correct constant for Khintchine-Kahane inequality?

I'm reading this paper http://jmlr.org/papers/volume13/kloft12a/kloft12a.pdf In Lemma 1, it says the constant for the Khintchine-Kahane inequality is q*. But if so, I don't think the authors can get ...
0
votes
1answer
35 views

Inequality and absolute value: $p + |k| \gt |p| + k$

Here is the problem I am confused about. The given relation is: $p + |k| \gt |p| + k$ It is not mentioned whether $p$ and $k$ are integers. I need to determine whether $p$ and $k$ are ...
1
vote
3answers
91 views

Maximum Of Two Variables' Formula

$x$,$y$ are real numbers satisfying $(x-1)^{2}+4y^{2}=4$ find the maximum of $xy$ and justify it without calculus. Does there exist a tricky solution using elementary inequalities (AM-GM or ...
0
votes
1answer
76 views

Proof of $|\int _a ^b \mathbf f | \leq \int _a ^b |\mathbf f|$

Let $\mathbf f:[a,b]\rightarrow \mathbb R ^n$ be continuous. I'm trying to prove the following fact: $$|\int _a ^b \mathbf f| \leq \int _a ^b |\mathbf f|,$$ where: $$|\mathbf x|^2=\sum _{i=1} ^n x_i ...
7
votes
1answer
162 views

About $e^{\pi}\gt {\pi}^e, \ e^{e^{\pi}}\lt {\pi}^{{\pi}^{e}},e^{{\pi}^{{\pi}^e}}\gt {\pi}^{e^{e^{\pi}}}$ and their generalization

Let us define a sequence $\{(a_n,b_n)\}$ as $$(a_1,b_1)=(e,\pi),\ \ (a_{2n},b_{2n})=(e^{b_{2n-1}},{\pi}^{a_{2n-1}}),\ \ (a_{2n+1},b_{2n+1})=(e^{a_{2n}},{\pi}^{b_{2n}})$$ for $n=1,2,3,\cdots$. Then, ...
13
votes
1answer
363 views

Prove: $3(a^4+b^4+c^4)+48\ge 8(a^2b+b^2c+c^2a)$

Let $a, b, c$ - real numbers. Prove that $3(a^4+b^4+c^4)+48\ge8(a^2b+b^2c+c^2a)$
0
votes
3answers
68 views

Does $A>B$ imply $A^B<B^A$?

Does $A>B$ imply $A^B<B^A$? A naive doubt, but I cannot find a proof. Does the property always hold true? $A>B$ does not necessarily imply that $A^B<B^A$. How do we know if $A^B<B^A$ ...
2
votes
1answer
150 views

$\sum_{k_1+k_2+\cdots+k_N=n,\ k_i\ge0\in\mathbb Z}\frac1{\prod_{j=1}^{N}\{(N-1)k_j+1\}}\le 1$ is true for any $n,N\in\mathbb N$?

Is the following true for any $n,N\in\mathbb N$? $$\sum_{k_1+k_2+\cdots+k_N=n,\ k_i\ge0\in\mathbb Z}\frac1{\prod_{j=1}^{N}\{(N-1)k_j+1\}}\le 1$$ Motivation : I've known the $N=3$ case. ...
0
votes
1answer
36 views

Does this inequality hold in general : Please verify

Let $r>1$, $i<j$ where $r$ is a real number and $i$ and $j$ are positive integers. Whether this inequality holds in general: ...
11
votes
6answers
608 views

$x,y,z$ positive real numbers , $x+y+z=3$ $\implies x^4y^4z^4(x^3+y^3+z^3)≤3$

If $x,y,z$ are positive real numbers with $x+y+z=3$ then how to prove (without using calculus) that $\space$ $x^4y^4z^4(x^3+y^3+z^3)≤3$ ?
1
vote
5answers
950 views

Finding the possible values of x

The lengths of a triangle are 5cm, 7cm and x cm. What are the possible values of x? I don't know where to start with this question as i don't thin enough information has been given. With an answer ...
1
vote
2answers
113 views

Solve this Inequality!

One of the examples to solve in the book is: $$ -3<4-x<2 \quad\text{and}\quad -1 \leq x-5 \leq 2 $$ What I've done so far: I've solved each inequality getting to $7>x>2$ and $4 \leq x ...
1
vote
3answers
153 views

Proof by Math Induction

I have 3 math induction proofs I have been struggling with for a while. I understand how to do summation proofs but these ones, I can't find a general pattern to solve. Please help. 1) $D(n) = ...
5
votes
1answer
107 views

Good upper bound for $(1-x)^r$

The Bernoulli's inequality gives a lower bound on numbers of the form $(1-x)^r$: $$(1-x)^r\geq 1-rx$$ for integer $r\geq 0$ and real number $0<x<1$. Is there a corresponding upper bound for ...
1
vote
2answers
38 views

finding range of function of three variables

Three real numbers $x$, $y$, $z$ satisfy the following conditions. $x^{2}+y^{2}+z^{2}=1~$, $~y+z=1$ Find the range of $~x^{3}+y^{3}+z^{3}~$ without calculus. I solved this problem only with ...
6
votes
6answers
232 views

Inequality, what is wrong with $\log(-1) = - \log(-1)$?

Can anyone tell me what is wrong with the following line of argument: $$ \log(-1) = \log(-1) - \log(1) = - \bigg( \log(1) - \log(-1) \bigg) = - \log \Big( \frac{1}{-1} \Big) = - \log(-1) $$ ...
3
votes
3answers
138 views

How prove this $a+b\le 1+\sqrt{2}$

let $0<c\le b\le 1\le a$, and such $a^2+b^2+c^2=3$, show that $a+b\le 1+\sqrt{2}$ My try: let $ c^2=3-(a^2+b^2)\le b$
0
votes
1answer
43 views

A basic question on the choice of $\alpha$ in the proof of Cauchy-Schwarz inequality

I see in a book the proof of Cauchy Schwarz inequality for two vectors $u$ and $v$ goes as follows : The vector $u$ is written as $u=\alpha v + (u-\alpha v)$. Then it finds the $\alpha$ for which the ...
1
vote
1answer
56 views

Bounding the integral of the tails of a random variable.

I found an argument like this in a book, but I couldn't understand how we got this bound. Suppose $X_n$ is a sequence of random variables. For some $\delta > 0$ and all $n \geq 1$, $$ \int_{|X_n| ...
4
votes
4answers
208 views

Prove that $|\cos(\sin(x_1)) - \cos(\sin(x_2))| \leq |x_1 - x_2|, \forall x_1, x_2 \in \mathbb R$.

I asked this question without any limitation on methods that might be used. I believe it's turned out to be interesting to see a variety of different approaches. It turns out that the aim of the ...
0
votes
1answer
259 views

Prove that the Q function is bounded such that $Q(x)\le\frac{1}{2x^2}$

Prove that the gaussian Q function is bounded on the top by $\frac{1}{2x^2 }$, i.e. $Q(x)\le\frac{1}{2x^2}$ for $x\ge0$, using the chebyshev inequality and the nakagami-m distribution with m=0.5(that ...
1
vote
0answers
46 views

Ineqality regarding LCM of $1, 2, \ldots, n$

While going through F. Beukers proof of irrationality of $\zeta(3)$ I found the inequality $d_{n} < 3^{n}$ for all sufficiently large values of $n$ where $d_{n}$ denotes the LCM of all the numbers ...
1
vote
2answers
54 views

Logarithm problem : Prove that $log_{3^2} \frac{1}{2} > 0$

Logarithm problem : Prove that $log_{3^2} \frac{1}{2} > 0$ My approach : $log_{3^2} \frac{1}{2} > 0$ $\Rightarrow \frac{1}{2} log_3 \frac{1}{2} >0$ $\Rightarrow \frac{1}{2} [ log_3 1 ...
1
vote
1answer
122 views

Chebyshev inequality and $Q$-Function

Prove that the Gaussian $Q$ function is bounded on the top by $1/2x^2$, i.e. $Q(x)\le 1/2x^2$. for $x\ge 0$ using the Chebyshev inequality and the Nakagami $m$ distribution with $m=0.5$ that reduces ...
2
votes
1answer
160 views

Concentration inequality for the median

Most concentration inequalities talk about deviation of the sample mean from the population mean. Is there a result bounding the probability of deviation of the sample median from the median of the ...
1
vote
1answer
63 views

Does this inequality hold for $r>1$?

Does this inequality hold? Let $r>1$ and $i<j$ where $i$ and $j$ are positive integers, then ...
45
votes
2answers
2k views

Fastest way to check if $x^y > y^x$?

What is the fastest way to check if $x^y > y^x$ if I were writing a computer program to do that? The issue is that $x$ and $y$ can be very large.
5
votes
1answer
73 views

Given $x,y,z >0$, $1/x+1/y+1/z = 4$, prove that $ 1/(2x+y+z)+1/(x+2y+z) +1/(x+y+2z) \le 1$

Given $x,y,z >0$, $1/x+1/y+1/z = 4$, prove that $$ 1/(2x+y+z)+1/(x+2y+z) +1/(x+y+2z) \le 1 .$$ Any hints or direction will be appreciated.
1
vote
0answers
56 views

Prove that the length of this curve decreases as one of its parameters increases

The following is the problem statement of one of my assignment questions. Consider the $\partial_t (t,s) = K(t, s)N(t, s)$ for all $t \geq 0$, and for all $s \in [0, 1]$, where $T(t, s) = ...
0
votes
4answers
128 views

How do I find the range of vales of $K$ satisfying the inequality $ k^2 - 9k + 16$?

the range of vales of K satisfying the inequality $k^2 - 9k + 16>0$ im quite confused, could you also explain how?
1
vote
4answers
113 views

Let $a,b,c >0$ and $a+b+c \le 1$. Prove the inequality

Let $a,b,c >0$ and $a+b+c \le 1$. Prove: $\sqrt{a^2+1/a^2}+\sqrt{b^2+1/b^2}+\sqrt{c^2+1/c^2} \ge \sqrt{82}$
0
votes
3answers
189 views

How find this maximum of this complex numbers of $x,y$

let $x,y$ be complex numbers,such that $|x|=|y|=1$. Can anyone help me to find the maximum value of the following expression $$|1+x|+|1+xy|+|1+xy^2|+\cdots+|1+xy^{2013}|-1007|1+y|$$ my try: ...
3
votes
1answer
146 views

How find this inequality$\sqrt{\left(\frac{x}{y}+\frac{y}{z}+\frac{z}{x}\right)\left(\frac{y}{x}+\frac{z}{y}+\frac{x}{z}\right)}+1$

let $x,y,z>0$,show that $$\sqrt{\left(\dfrac{x}{y}+\dfrac{y}{z}+\dfrac{z}{x}\right)\left(\dfrac{y}{x}+\dfrac{z}{y}+\dfrac{x}{z}\right)}+1\ge 2\sqrt[3]{\dfrac{(x^2+yz)(y^2+xz)(z^2+xy)}{x^2y^2z^2}}$$ ...
0
votes
1answer
68 views

What's the fastest way to solve this inequality?

$|x^2-3x+2|>|x|+1$ Thanks in advance.
3
votes
2answers
107 views

How prove this inequality $|n\sqrt{2009}-m|>\dfrac{1}{kn}$

if $k>\sqrt{2009}+\sqrt{2010}$,show that for any positive integer numbers $m,n$, have $$|n\sqrt{2009}-m|>\dfrac{1}{kn}$$ My try: $$\Longleftrightarrow(n\sqrt{2009}-m)^2k^2n^2>1$$ ...
1
vote
1answer
72 views

Tail probability of the $\chi^2$ distribution

Ho to prove that $$ \int_{2s\epsilon^{-2}}^{\infty}\frac{1}{\Gamma(d/2)2^{d/2}}x^{d/2-1}e^{-x/2}dx \leq const.\epsilon^{-d}\exp(-\epsilon^{-2}s) $$ holds for $\epsilon >0$ sufficiently small? Here ...