Questions on proving and manipulating inequalities.

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0
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2answers
31 views

Show$\:\frac{1}{\left|x^2+x+1\right|}\:\ge \:\frac{1}{x^2-\left|x\right|-1}$

This is the answer I can come up with. I get the complete opposite of what I'm supposed to get. My mistake is probably in the first part, could anyone help me out? $$\left|x^2+x+1\right|\:\ge ...
0
votes
1answer
23 views

Extending Minkowsky inequality to double summation?

I know the Minkowski inequality for sequences as follows : $$\left(\sum_{k=1}^n|x_k+y_k|^p\right)^{1/p} \leq \left(\sum_{k=1}^n|x_k|^p\right)^{1/p}+\left(\sum_{k=1}^n|y_k|^p\right)^{1/p}$$ Now say we ...
0
votes
0answers
33 views

Prove that for all a,b,c > 0 [duplicate]

Prove that for all $a,b,c > 0$ that $ \dfrac{a+b+c}{\sqrt[3]{abc}} + \dfrac{8abc}{(a+b)(b+c)(c+a)} \geq 4 $ My attempt: I thought this was very easy but the second part I am getting $\le 1$ ...
0
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0answers
26 views

Four real numbers p,q,r,s satisfy $p+q+r+s = 9$ and $p^2 + q^2 + r^s + s^s = 21$. [duplicate]

Four real numbers p,q,r,s satisfy $ p+q+r+s = 9 $ and $ p^2 + q^2 + r^2 + s^2 = 21 $. Prove that there is a permutation $ (a,b,c,d) $ of $ (p,q,r,s) $ such that $ ab-cd \geq 2 $. My attempt I tried ...
4
votes
1answer
56 views

Let a,b,c be positive real numbers numbers such that $ a^2 + b^2 + c^2 = 3 $

Let a,b,c be positive real numbers numbers such that $ a^2 + b^2 + c^2 = 3 $. Prove that $ (a+b+c)(a/b + b/c + c/a) >= 9 $ My Attempt I tried AM-GM on the symmetric expression so the a+b+c >= ...
4
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2answers
47 views

Prove that if $a$,$b$,$c$ are non-negative real numbers such that $a+b+c =3$, then $abc(a^2 + b^2 + c^2)\leq 3$

Prove that if $ a,b,c $ are non-negative real numbers such that $a+b+c = 3$, then $$ abc(a^2 + b^2 + c^2) \le 3 $$ My attempt : I tried AM-GM inequality, tried to convert it to $a+b+c$, but I think ...
0
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0answers
46 views

A lower bound for $\log\left( \frac{a+x^2}{b+x^2}\right)$

I am looking for a tight lower bound for $$f(x)=\log\left( \frac{a+x^2}{b+x^2} \right)$$ $x>0$ and $1<b<<a$. I didn't check for convexity analytically, but I plotted this function ...
0
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0answers
19 views

Find $\min x^TAy+b^Tx+c^Ty$ subject to $1^Tx=1^Ty=1,x\ge 0,y\ge 0$

The problem seems to be easy but I can't find a solution :( Problem: Given $A\in\mathbb{R}^{m\times n}, A\ge 0, b\in\mathbb{R}^{m}, c\in\mathbb{R}^{n}$. Minimize $f(x,y) = x^TAy+b^Tx+c^Ty$ subject to ...
1
vote
0answers
17 views

Leading up to Young's Inequality

I am trying to prove Young's Inequality by considering the function $$h(u) = \frac{u^p}{p} + \frac{C^q}{qu^q}$$ for $C,u>0$ and $p,q >1$. We also require $$\frac{1}{p}+\frac{1}{q}=1$$ so that ...
1
vote
0answers
10 views

Kneser Inequality in multivariables

Based on the Kneser Inequality ("Polynomials and Polynomial Inequalities", p. 260) one has $\Vert q \Vert_{[-1, 1]} \Vert r \Vert_{[-1, 1]} \leq C(n, m) \Vert q r \Vert_{[-1, 1]}$ where all norms ...
1
vote
0answers
22 views

Values satisfying the inequality [on hold]

if $ 1-\cos x=\frac {\sqrt3}{2} |x| +a$ has no solution then find the complete set of values of $'a'$.Here is the question i got struck.
0
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0answers
18 views

Basic inequality problem [on hold]

Here is my problem if $ 16-x^2> |x-a|$ is to be satisfied by atleast one negative value of $x$, then i have to find complete set of values of $'a'$ .Please provide me hint to solve this ...
0
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1answer
29 views

How to solve Inequality with factorials

Im reading a book in Numerial analysis and I have the following which I dont understand involving inequalities and factorials, What i have is the following: $$\frac{1}{(2n+1)!(2n+1)} \leq 5*10^{-9}$$ ...
3
votes
3answers
166 views

Inequalities proven by real analysis or induction.

Let $t\in [-1,1]$. Prove that $(1+t)^p+(1-t)^p\ge2$ when $p\ge 1$ and that $(1+t)^p+(1-t)^p \le 2$ where $0 \le p\le 1$. I am not sure how I should solve it. I tried induction at first and it was ...
0
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0answers
11 views

Inequality involving expectations of vector/matrix norms

I'm reading a paper and trying to understand the proof of a lemma regarding expectations of norms of random vectors. The author's notation does not distinguish between vector and matrix norms, nor ...
2
votes
1answer
39 views

Help with fraction inequality

Let $a,b,c$ be three numbers such that: $a,b\in (0.5,1)$ $c \in (0.25,0.5)$ $c < 0.5a$ $c > 0.5b$ $a + b < 1 + c$ Let $$f(a,b,c) = \frac{1+c}{a+\frac{bc}{c+0.5b}}$$ What is the ...
2
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1answer
30 views

Inequality $(a+b)^2 + (a+b+4c)^2\ge \frac{kabc}{a+b+c}$ for $a,b,c \in\mathbb{R}$

Find biggest constans k such that $(a+b)^2 + (a+b+4c)^2\ge \frac{kabc}{a+b+c}$ is true for any $a,b,c \in\mathbb{R}$ Could you check up my solution? I'm not sure it's ok - $(a+b)^2 + (a+b+4c)^2 \ge ...
0
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1answer
38 views

How does the Euler Totient Function apply here?

How many positive integers $< 2013$ are divisible by $2$ Can I somehow use Euler's Totient function to find this?
2
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1answer
48 views

Inequality $x^4+y^4+(x^2+1)(y^2+1)\ge x^3(1+y) +y^3(1+x)+x+y$ for $x,y \in\mathbb{R}$

Prove for $x,y \in\mathbb{R}$ that such inequality exists ; $x^4+y^4+(x^2+1)(y^2+1)\ge x^3(1+y) +y^3(1+x)+x+y$ And here is what I realised ; because $(x^2+1)(y^2+1) >=1$ and $x^4+y^4 \ge 0$ ...
0
votes
1answer
20 views

summation inequality with logarithms

show: $$\sum_{i=1}^n \log_{2}\,i = O(n\log n)$$ Proof by induction: $$\sum_{i=1}^n \log\,i \le n\log n$$ $$\text{Test for n=1: }\sum_{i=1}^1 \log_{2}\,i \le 1\log 1$$ $$0 \le 0\text{ true for ...
-5
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0answers
37 views

Inequality please help [on hold]

Adam is running marathon . He has complete 10 mile in 90 minute . What should his average split be in order to complete the race less than 4 hours
0
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2answers
27 views

proof by induction summation inequality

show by induction that: $$\sum_{i=1}^n i^2 = O(n^3)$$ what I have so far: $$\sum_{i=1}^n i^2 <= n^3$$ base case: for n=1 $$\sum_{i=1}^1 i^2 <= 1^3$$ ...
6
votes
4answers
982 views

Slick proof of exponential inequality

Today I saw that using taylor series, one can show that $e^x+e^{-x}\leq 2e^{x^2/2}$. Is there a slick proof using some sort of Jensen-type inequality or integral bound?
0
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3answers
74 views

Cannot follow proof that $n! \leq en(n/e)^n$

prove that $n! \leq en(n/e)^n$ skip proof for base (n=1)... Assume it holds for $n-1$, verify for $n$. We have $n! = n* (n-1)! \leq n * e(n-1)(\frac{n-1}{e})^{n-1} $ by inductive assumption. we ...
-1
votes
1answer
19 views

question on proving inequalities [on hold]

If I need to prove $t(x) \ge0 $, for all $ x>0$ and I prove that $t(x) \gt 0 $, for all $ x>0$ does that make for a proof or is it wrong?
0
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4answers
35 views

How to study positivity of $x\sqrt{4-x^2}-4\arcsin({\frac x2})$

I have to study where the function is positive/negative. What's the method to solve the inequality $x\sqrt{4-x^2}-4\arcsin({\frac x2})>0$ ?
0
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2answers
30 views

Olympiad minimum question, minimal value

If the numbers $A, B, C$ are such that the expression $\sqrt{A-B} + \sqrt{(B+3)^2} + C^2 - 4C + 4$ is as small as possible, then $A+B+C$ is? I thought start with, $A > B > C$ without loss of ...
3
votes
2answers
69 views

Prove, inequality ,positive numbers

$$\frac{a}{e+a+b}+\frac{b}{a+b+c}+\frac{c}{b+c+d}+\frac{d}{c+d+e}+\frac{e}{d+e+a}<2$$ Prove that for positive numbers $a,b,c,d,e$ there is such inequality
-1
votes
3answers
23 views

How can we make this expression small? [on hold]

How can we make the following expression small: $$(bx-ay)^2+(cx-az)^2+(cy-bz)^2+(ay-bx)^2+(az-cx)^2+(bz-cy)^2$$, where $a,b,c,x,y,z$ are nonnegative reals? Note: I'm not looking for an exact answer, ...
1
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0answers
12 views

Saturation of the Babenko–Beckner inequality

The Babenko-Beckner inequality $|| \mathcal F f ||_q \geq C(q,p)||f||_p$ is a well-established theorem. It relates the $q$-Norm of a Fourier transform $\mathcal F f$ of a function $f$ to its ...
-1
votes
1answer
22 views

Proving that a system of equalities and inequalities is inconsistent (Vol. 4)

I am studying sign pattern matrices and I have to prove this statement. Prove that the system $$\begin{cases} a,b,d,e,f,g,h,i>0 \\ -a+e-i>0 \\ -ae+ai+bd-ei+fh>0 \\ aei−hfa-bdi−gbf>0 \\ ...
1
vote
1answer
57 views

Find the value of $ ( ab + bc + ca )^2 $

If $a,b,c$ are real numbers which satisfy $a^2+b^2+ab = 9$ $b^2+c^2+bc = 16$ $c^2+a^2+ca = 25$ find the value of $ ( ab + bc + ca )^2 $
2
votes
1answer
27 views

Counterexample Poincaré Inequality for $H_0^1$ in 2D

Is there any counterexample to the Poincaré inequality $$\int_\Omega|f|^2dx\leq C(\Omega)\int_\Omega|\nabla f|^2dx $$ for $f\in H_0^1(\Omega)$, $C(\Omega)>0$ and $\Omega\subset\mathbb{R}^2$? I ...
1
vote
3answers
86 views

If $G(x)=P[X\geq x]$ then $X\geq c$ is equivalent to $G(X)\leq G(c)$ $P$-almost surely

Suppose $[\Omega,\mathcal{F},P]$ denotes a probability triplet and $X:\Omega\to\mathbb{R}$ is a real-valued random variable. Define $$ G(x)=P[X\geq x]. $$ Claim: for any constant $c$, the event ...
1
vote
5answers
77 views

If $a^2+b^2+c^2=1$ then prove the following.

If $a^2+b^2+c^2=1$, prove that $\frac{-1}{2}\le\ ab+bc+ca\le 1$. I was able to prove that $ ab+bc+ca\le 1$. But I am unable to gain an equation to prove that $ \frac{-1}{2}\le\ ab+bc+ca$ . Thanks in ...
-4
votes
2answers
33 views

Prove and disprove the following inequality.

Prove: $ 0 \le a \lt b$ implies $ 0 \le a^2 \lt b^2 $ and $0 \le \sqrt{a^3} \lt \sqrt{b^3}$. Now show that the statement is false if the hypothesis $a \ge 0$ or $a \lt 0$ is removed. EDIT: Someone ...
3
votes
0answers
61 views

Modified Doob's $L^1$ inequality

Let $X_n$ be a non-negative submartingale. Show that for all $\lambda >0$ $$ P(\sup_{k\leq n} X_n \geq 2\lambda) \leq \frac{1}{\lambda} \int_{X_n \geq \lambda} X_n dP$$ In Doob's weak $L^1$ ...
0
votes
1answer
15 views

Showing that $\Re z \le |\Re z| \le |z|$ and $\Im z \le |\Im z| \le |z|$

What I'm wanting to show is that $$\Re (z) \le |\Re (z)| \le |z|$$ and also $$\Im(z)\le |\Im(z)| \le |z|$$ So what I've done so far is to consider $$z=x+iy$$ Using the above $z$ I also said that $$\Re ...
-1
votes
1answer
16 views

Prove that if a is not 0, then |x|>c>0 implies |(1/a)-(1/x)|<(|a-x|/(c|a|))

For a,c, and x in the reals, prove that if a is not 0, then |x|>c>0 implies |(1/a)-(1/x)|<(|a-x|/(c|a|)). I'm trying to practice these kinds of questions, and any help or suggestions are greatly ...
-1
votes
2answers
28 views

Prove that if $y>1$, then $\forall M\in\mathbb{R}$, there exists an $N$ in the natural numbers s.t. $n\geq N$ implies $y^n>M$. [on hold]

For $y\in\mathbb{R}$, prove that if $y>1$, then $\forall M\in\mathbb{R}$, $\exists N\in\mathbb{N}$ such that $$ n≥N \implies y^n>M. $$ I'm not used to proving these kinds of questions so any ...
-2
votes
0answers
16 views

How would I get Maple to display all integer solutions to this system of inequalities? [on hold]

I need to find all the integer solutions satisfying: $$20+x\geq0;\space2x+5y\geq;\space-x-2y\geq0.$$ I'm not sure which Maple functions would work and whatnot. A guy can only google this stuff for so ...
0
votes
2answers
29 views

Spivak's Calculus, chapter 1 problem 19 (inequalities)

I'm having trouble with problem 1-19 in Spivak's Calculus. I have to prove that if $|x-x_0| < \frac{\epsilon}{2} $ and $ |y-y_0| < \frac{\epsilon}{2} $ then $ |(x-y)-(x_0-y_0)| < \epsilon $. ...
0
votes
2answers
28 views

Show using inequality of means that $a\cdot n \cdot \frac{1}{n} \le a^2n^2+\frac{1}{n^2}$

Show using inequality of means that for $a>0$ and $n\in\mathbb{N}$: $$a\cdot n \cdot \frac{1}{n} \le a^2n^2+\frac{1}{n^2}$$ I'm sure it's not that complicated, but I'm probably missing ...
0
votes
1answer
50 views

Suppose $f(x,y)=c(x^{m}+y^{n})-dx^{a}y^{b}+sin(x+y^{2})$, show that f is greater than

Suppose $f(x,y)=c(x^{m}+y^{n})-dx^{a}y^{b}+\sin(x+y^{2})$, where m,n are positive even integers, a,b are positive integers, c,d are positive real numbers and $\frac{a}{m}+\frac{b}{n}<1$. Show that ...
2
votes
1answer
49 views

How to show without calculator that $\left\lfloor\, \log_{10}{999^{999}}\right\rfloor =\left\lfloor\, \log_{10}{999^{999}}+\log_{10}2\right\rfloor$

By wolfram alpha, I get $\left\lfloor\, \log_{10}{999^{999}}\right\rfloor =\left\lfloor\, \log_{10}{999^{999}}+\log_{10}2\right\rfloor=2996$. How to prove that $\left\lfloor\, ...
0
votes
1answer
42 views

Which one is greater?

For any $x\in\mathbb{R}^+$, let $x\diamond 1=x$ and $x\diamond (n+1) = x^{x\diamond n}$ for $n\in\mathbb{N}$. For example, $2\diamond 3 = 2^{2^2}=16$. If $t$ be an unique positive real number such ...
-1
votes
0answers
21 views

Can a strict inequality be derived from a weaker one?

Suppose P and Q are two statements, with P being the stronger one. Let us denote the set of statements derived from P and Q be A and B respectively. Then can the strongest statement belonging to A be ...
0
votes
3answers
75 views

Proof of an inequality by induction

Let $n \in \mathbb N^+$. Show that if $x_1, x_2, ... , x_n$ are $n$ real numbers such that $-1 \le x_i \le 0$ for each $1 \le i \le n$, then $$(1 + x_1)(1 + x_2)...(1 + x_n) \ge 1 + x_1 + x_2 + ... + ...
2
votes
2answers
56 views

Show $\lvert\lambda_1a_1 + \lambda_2a_2 + \cdots + \lambda_na_n\rvert < 1$ when $\lvert a_i\rvert < 1$ and $\lambda_i\geq 0$

If $\lvert a_i\rvert < 1$, $\lambda_i\geq 0$ for $i = 1,\ldots,n$ and $\lambda_1 + \lambda_2 + \cdots + \lambda_n = 1$, show that $$ \lvert\lambda_1a_1 + \lambda_2a_2 + \cdots + ...
0
votes
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, ...