Questions on proving and manipulating inequalities.

learn more… | top users | synonyms (1)

0
votes
1answer
9 views

Given that a,b,c are distinct positive real numbers, prove that (a + b +c)( 1/a + 1/b + 1/c)>9

This is how I tried doing it: Say p= a + b + c and q=1/a + 1/b + 1/c, Now if I use AM>GM for p and q I get: ((p+q)/2)> (pq)^{1/2} ((a + b +c)( 1/a + 1/b + 1/c))^{1/2} < (a+1/a + b+1/b + c+1/c)/2 ...
1
vote
0answers
24 views

How to prove/dispute the following log inequality?

I was wondering if the following inequality is true: $$\forall x,N\in \mathbb N^+: \lceil \log_2\left(\lfloor\frac{N}{x}+1\rfloor\right)\rceil\leq \lceil\log_2 (N+1)\rceil - \lfloor\log_2 (x)\rfloor ...
1
vote
0answers
12 views

Existence of solution for linear matrix inequality?

Suppose $x$ is a $n\times1$ column vector. How to know whether the following matrix inequality has solution or not? $$Ax\leq B$$ where $A$ is a $m\times n$ matrix and $B$ is a $n\times 1$ column ...
2
votes
1answer
32 views

Integral inequality $\int_0^x{f(t)^3 dt \leq \left( \int_0^x f(t) dt\right)^2} :\forall x>0$

Let $f(0) = 0$ and $0<f'(x)\leq1$ for all $x \geq0$, then prove: $$\int_0^x{f(t)^3 dt \leq \left( \int_0^x f(t) dt\right)^2} :\forall x>0$$ The hint I was given was "differentiate, factor and ...
0
votes
2answers
34 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 ...
1
vote
1answer
33 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
34 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
votes
0answers
30 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
61 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
votes
2answers
55 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
votes
0answers
50 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
votes
0answers
21 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
18 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
12 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
24 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
votes
0answers
19 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
votes
1answer
30 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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
986 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
votes
3answers
75 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
votes
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
votes
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
vote
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
78 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
63 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
29 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
50 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
43 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 ...