Questions on proving and manipulating inequalities.

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2
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2answers
33 views

Show that $a^2 + b^2 + c^2 \geq ab + bc + ca$ for all positive integers $a$, $b$, and $c$

Show that $a^2 + b^2 + c^2 \geq ab + bc + ca$ for all positive integers $a$, $b$, and $c$. I am not sure how to approach this problem. Should I divide this problem into multiple cases based on ...
1
vote
1answer
19 views

Prove that for all $m$, there exist some $k$, such that $(m-n)^2 > m^2$ for all $n>k$

I have a problem where I need to prove: $\forall m \in \mathbb{N}:\exists m \in \mathbb{N} ∋(m−n)^2>m^2~∀n>k$ My thought was since it is only "there exists some k.." can I not say: if $k = ...
1
vote
2answers
336 views

Find $\limsup$ and $\liminf$ of a sequence and prove $\liminf a_n \leq \limsup a_n$.

I have a question of finding lim sup and lim inf of $a_n=\frac{1}{n} + (-1)^n$ and prove $\liminf a_n \leq \limsup a_n.$ So the work below is what I did for the first part. $a_{odd\ n} = ...
1
vote
1answer
220 views

limit inf /sup if $x_n\leq y_n$

I have just 2 problems : 1) Find the $\limsup x_n$ and $\liminf x_n$ where $x_n= e^{-n}$. 2) Let $x_n\leq y_n$ for every $n\in\mathbb{N}$ . Show that $\liminf x_n\leq\liminf y_n$ and $\limsup ...
2
votes
2answers
2k views

Proofs with limit superior and limit inferior: $\liminf a_n \leq \limsup a_n$

I am stuck on proofs with subsequences. I do not really have a strategy or starting point with subsequences. NOTE: subsequential limits are limits of subsequences Prove: $a_n$ is bounded $\implies ...
3
votes
1answer
42 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 ...
2
votes
1answer
42 views

Proof Involving Generalized Mean

Let $x=(x_1,...,x_n) \in \mathbb R^n$ and $$g(p)=\sqrt[p]{\frac{1}{n}\sum_{k=1}^{n} |x_k|^p)}$$ Using Hölder's inequality, show that $g(p)$ is increasing on $(0,\infty)$. For a sequence with ...
0
votes
1answer
17 views

Writting an inequality to represent a situation (Help needed) [on hold]

In the course of the day, Annalise will drive her car to work and later use the city rail system to get around her different branch offices. At one rail stop, she needs to ride a bus. Will she stay ...
0
votes
3answers
73 views

If $X=\{x_n\}, Y = \{y_n\}$ be bounded sequences of real numbers. Then, if $x_n \leq y_n~\forall~n$, then $\lim \inf (x_n) \leq \lim \inf(y_n)$ [duplicate]

If $X=\{x_n\}, Y = \{y_n\}$ be bounded sequences of real numbers. Then, if $x_n \leq y_n~\forall~n$, then show that $\lim \inf (x_n) \leq \lim \inf(y_n)$ and $\lim \sup (x_n) \leq \lim \sup (y_n)$ ...
0
votes
1answer
20 views

How to prove the inequality? [on hold]

Set $f(x)=1-(1-\lambda)^x$, where $\lambda \in (0,1)$, show that $f(x)/x \ge f(x)-f(x-1)$ holds for any $x\ge 1$.
2
votes
2answers
67 views

lim sup, lim inf, and inequalities for $a_n \le b_n$ [duplicate]

Suppose we have two sequences ${a_n}$ and ${b_n}$, which satisfies $ a_n \le b_n$ for $n=1,2,3,\ldots$. Do we have the following inequalities to be true? $$\limsup_{n \to \infty} a_n \le \limsup_{n ...
7
votes
4answers
447 views

$X_n\leq Y_n$ implies $\liminf X_n \leq \liminf Y_n$ and $\limsup X_n \leq \limsup Y_n$

Can anyone prove this question? I tried but I didn't get any I idea, so I hope someone can solve it. Let $X_n\leq Y_n$ for each $n\in \Bbb N$. Show that $\liminf X_n \leq \liminf Y_n$ and $\limsup ...
3
votes
4answers
457 views

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

Given that $a,b,c$ are distinct positive real numbers, prove that $(a + b +c)\big( \frac1{a}+ \frac1{b} + \frac1{c}\big)>9$ This is how I tried doing it: Let $p= a + b + c,$ and $q=\frac1{a}+ ...
1
vote
0answers
49 views

A problem of inequality

Let $a_1, a_2, a_3$; $b_1, b_2, b_3$; $c_1, c_2, c_3$; $d_1, d_2, d_3$ be all real numbers. We need to show that $$\begin{align}(a_1b_1c_1d_1 + a_2b_2c_2d_2 &+ a_3b_3c_3d_3)^4\\ &\leq ...
0
votes
1answer
29 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
14 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 ...
1
vote
1answer
34 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 ...
2
votes
1answer
39 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
36 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 ...
4
votes
2answers
58 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
51 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
36 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 $\le 1$ ...
4
votes
1answer
64 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 >= ...
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 ...
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 ...
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.
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 ...
0
votes
0answers
12 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 ...
84
votes
0answers
2k views

How prove this matrix inequality $\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n\det(A^2+B^2+C^2)$?

Question: let matrices $A,B,C\in M_{n}(C)$ be Hermitian and Positive definite matrices, such that:$$A+B+C=I_{n}$$ Show that: $$\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n\det(A^2+B^2+C^2)$$ ...
3
votes
1answer
116 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 ...
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 ...
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: $$\frac{1}{(2n+1)!(2n+1)} \leq 5*10^{-9}$$ ...
1
vote
0answers
37 views

A maximal inequality on distance to median, so called Lévy's inequality?

Problem (Kai Lai Chung, A Course in Probability Theory, section 5.3, ex6) Suppose $X_1,\dotsc,X_n$ are i.i.d. random variables, and $S_j:=X_1+\dotsb+X_j,S_j^0:=S_j-m_0(S_j)$, where $m_0(S_j)$ is a ...
0
votes
1answer
87 views

$4x^{4} + 4y^{3} + 5x^{2} + y + 1\geq 12xy$ for all positive $x,y$

Prove this inequality: $$4x^4 + 4y^3 + 5x^2 + y + 1 \geq 12xy$$ if $x$ and $y$ are real and positive. Please, I am a beginner and have no idea how to solve this, so don't use any strange theorems. ...
0
votes
1answer
35 views

Need help verifying my proof for inequality (I feel it's wrong)

I posted this yesterday Prove the equality that $4x^{4} + 4y^{3} + 5x^{2} + y + 1$ >or equal to $12xy$ if $x$ and $y$ are real and positive and I think I got a solution , Here it goes , this is ...
3
votes
3answers
167 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 ...
6
votes
0answers
81 views
+500

An inequality concerning restricted isometry property

Let $A\in \mathbb{R}^{m\times n}$ be a matrix and let us denote by $A_S$ the submatrix of $A$ with the columns restricted to a set $S\subset [n]:=\{1,2,\cdots, \ n\}$. Then one says that the matrix ...
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 ...
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
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
2answers
56 views

Find the minimum possible value of $x(1-z)+y(1-x)+z(1-y)$

It is given that $$xyz=(1-x)(1-y)(1-z)$$ and $$x, y, z \epsilon (0,1)$$ Find the minimum possible value of the expression: $$x(1-z)+y(1-x)+z(1-y)$$ Using the AM-GM inequality concepts, I can write ...
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 ...
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 + ...
0
votes
4answers
36 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
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 ...
0
votes
1answer
57 views

Curiosity : An inequality involving logarithms

Does exists $\alpha \in R $ and a positive constant C such that $\displaystyle{% \left[\,x\ln\left(\, x\,\right) - x\,\right] -\left[\, y\ln\left(\, y\,\right) - y\,\right]\ \leq\ C\,\left\vert\, ...
-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
6
votes
4answers
990 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?