Questions on proving, manipulating and applying inequalities.

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0
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1answer
28 views

Inequality in the limit

Given that we have the following conditions: $f = O(\delta)$, $g = O(\delta^2)$, $f > 0, \delta > 0$, can we conclude that as $\delta \to 0^+$, $f+g>0$?
2
votes
5answers
143 views

How would you prove inequality $2^n \gt n^{10}$ using induction

For the base case I can put a number such as $100$ for $n$ so $2^{100}\gt 100^{10}$. Ok so now the induction hyp: $2^{n+1} > (n+1)^{10}$ for $n \gt 101.$ where do I go from here? Also do I have ...
0
votes
1answer
47 views

How can Use Gronwall for this PDE?

I'm trying to prove this. First I tried to multiply the equation by $\phi(x,t)$ and use the Gronwall Lemma, but it didn't work. Can anyone help? Here's the problem: Given a smooth field $u:\; \...
2
votes
1answer
52 views

How to take irrational numbers and cases out of this inequality’s proof

Consider the following simple inequality in rational numbers $x,r$ : $$\text{If } x\geq 0,\ r\geq 0 \text{ and } r^2\geq 2,\ x^3-6x+4r\geq 0.\tag{1}$$ If one is allowed to use irrational numbers ...
1
vote
1answer
42 views

Proving inequalities hold for all values

Show that the inequality $$\frac st + \frac tr + \frac rs \ge 3$$ holds for all positive $r,s$ and $t$.
10
votes
1answer
162 views

How are inequalities from IMO built?

I notice that there are lots of apparently difficult inequalities in IMO. Are there some techniques to manipulate well-known inequalities in order to built a difficult exercise? What are the main ...
1
vote
1answer
104 views

Triangle Inequality Property for the Euclidean Metric

I've read in many of my books that the triangle inequality for a metric space of the Euclidean Metric is defined as: $$d(x,y) \leq d(x,z) + d(z,y)$$ But when I look up the proof, to help me ...
6
votes
3answers
78 views

Bound for $\sum_{k=1}^\infty\left(\frac{1}{2^k+k^2}\right)$

I found for the series: $$S=\sum_{k=1}^\infty\left(\dfrac{1}{2^k+k^2}\right)$$ a bound: $$S\le\dfrac{\pi^2}{6+\pi^2}$$ which is in good agreement with the approximate value of $S$ calculated with ...
1
vote
2answers
60 views

A Doubt on Stolarsky's theorem .

Recently, While I was solving some problems, I saw a question which was symmetric . I thought of applying Stolarsky's inequality, but it was not homogeneous. My question is, Can I normalize it to ...
1
vote
4answers
59 views

Is $\max\left(\frac{A}{B},\frac{C}{D}\right) \ge \frac{\max(A,C)}{\max(B,D)}$

Is $\max\left(\frac{A}{B},\frac{C}{D}\right) \ge \frac{\max(A,C)}{\max(B,D)}$? Given that $A,B,C,D>0$. What about $\frac{A}{B},\frac{C}{D}>1$. Is there a better bound for the left ...
0
votes
1answer
88 views

How logarithms affect given condition

I am working with long productcs of probabilies and in order of avoinding underflow I am using the addition of (negative) logarithms. P(A) =-log(P(a1) + -log(P(a2)+.... In the end I get a positiv ...
1
vote
0answers
55 views

Integral Hölder bound

I was wondering if it is possible to find the following bound or if not, find a counterexample of it. Let $f\in C_0^1$ (compactly supported continously differentiable, in particular $\alpha$-Hölder ...
0
votes
1answer
132 views

Graph Theory inequality

I've been trying to prove the following inequality If G is r-regular graph and $\kappa (G)=1 $, then $\lambda (G)\leq \left \lfloor \frac{r}{2} \right \rfloor$ I've tried manipulating the Whitney ...
3
votes
3answers
150 views

How to prove which of two numbers written as powers is bigger?

Prove which number is larger: a) $10^{100!}$ or $10^{10^{100}}$ b) $e^\pi$ or $\pi^e$ I know we all know how to plug these into the calculator and check, but how someone mathematically prove which ...
1
vote
2answers
78 views

$ \cos {A} \cos {B} \cos {C} \leq \frac{1}{8} $

In an acute triangle with angles $ A, B $ and $ C $, show that $ \cos {A} \cdot \cos {B} \cdot \cos {C} \leq \dfrac{1}{8} $ I could start a semi-proof by using limits: as $ A \to 0 , \; \cos {A} \...
2
votes
1answer
96 views

If $a\le b$ and $l,m\ge 1$, then $|l+e^{i\gamma}|(a+mb)\leq (l+m)|a+e^{i\gamma}b|$

Let $a, b$ be any two positive real numbers such that $a\geq lb$ where $l\geq 1.$ Suppose $\gamma $ is any real such that $0\leq \gamma\leq 2\pi.$ Is it true that $$|l+e^{i\gamma}|(a+mb)\leq (l+m)|...
4
votes
2answers
103 views

Given $a+b+c=4$ find $\max(ab+ac+bc)$

$a+b+c = 4$. What is the maximum value of $ab+ac+bc$? Could this be solved by a simple application of Jensen's inequality? If so, I am unsure what to choose for $f(x)$. If $ab+ac+bc$ is treated as a ...
1
vote
2answers
80 views

Relationship between Difference of Two Numbers and Their Square Roots

Is there a relationship between the difference of two numbers and the difference of their square roots? For example, can we say that ${| \sqrt x - \sqrt y|\leq |x - y|}$ when ${ x, y \geq 1 }$, but ...
1
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4answers
79 views

Real Between Rationals

Let $x$ be a real number. Show that, for any $\varepsilon>0$, there exist two rationals $q$ and $q'$ such that $q<x<q'$ and $|q-q'|<\varepsilon$ How should I approach this prove?
7
votes
1answer
179 views

An Inequality Invollving The Riemann Zeta Function

I'm having trouble proving the following inequality for $2<r<3$: $$(1+2^{-r})\frac{(3^r+1)^2}{3^{2r}+1}>\frac{\zeta(r)}{\zeta(2r)}.$$ I can easily plot the graph, and the inequality clearly ...
0
votes
0answers
51 views

Proving inequality equation

Let $a_1, a_2,....,a_n$ be positive numbers such that $\sum_{i=1}^n a_i = 1$ Then for any vector $(x_1,x_2,...x_n) \ge (0,0,...,0)$ I want to show that $$x_1^{a_1}*x_2^{a_2}*...*x_n^{a_n} \le a_1*...
0
votes
1answer
57 views

A counting problem on the integer lattice

Let $K$ be a subset of the integer lattice $\mathbb Z^2$such that it contains elements of the form $k=(k_1,k_2) $ where $k_1,k_2$ are integers and $k_2\neq 0$. Find $m$, an integer if possible, such ...
9
votes
1answer
109 views

How prove this $(x_{1}+x_{2})(x_{2}+x_{3})\cdot (x_{n}+x_{1})\ge 2^n-n^2+n\sum_{i=1}^{n}x_{i}$

Let $x_{i}>0,i=1,2,\ldots,n$ and such $\prod_{i=1}^{n}x_{i}=x_{1}x_{2}\cdots x_{n}=1$. Show that this inequality $$(x_{1}+x_{2})(x_{2}+x_{3})\cdots (x_{n}+x_{1})\ge 2^n-n^2+n\sum_{i=1}^{n}x_{i}$...
2
votes
1answer
69 views

If $ab+bc+ca=1$, then $\frac{((a+b)^2+1)}{(c^2+2)}+\frac{((b+c)^2+1)}{(a^2+2)}+\frac{((c+a)^2+1)}{(b^2+2)} \geq 3$

Let $\displaystyle a, b, c> 0, ab+bc+ca=1$. Prove that the following inequality holds: $$\frac{((a+b)^2+1)}{(c^2+2)}+\frac{((b+c)^2+1)}{(a^2+2)}+\frac{((c+a)^2+1)}{(b^2+2)} \geq 3.$$ I tried ...
3
votes
2answers
112 views

Let $f$ be a holomorphic in $D(0,1)$, with Re$\,f(z) >0$ and $f(0)=1.$ Then $\lvert\, f'(0)\rvert\leq 2$

Let $f:D(0,1) \to \mathbb{C}$ be a holomorphic function, such that $$ \mathrm{Re} \,f(z) >0\quad \text{and}\quad f(0)=1. $$ How to prove $\lvert\, f'(0)\rvert\leq 2 \ ?$ This is now a self-...
2
votes
1answer
37 views

Does a maximum value exist for this expression?

Let $x$, $y$, $z$ be positive real numbers and $x + y + z =3$. Does a maximum value exist for this expression? $$\displaystyle E = \frac{x}{2 y+3 z}+\frac{4 y}{5 z + 6 x}+\frac{7 z}{8x+9 y}.$$ I ...
2
votes
3answers
55 views

Show $|\sin(y)y - \sin(x)x| \leq C|y - x|$ for some $C > 0$

Show $|\sin(y)y - \sin(x)x| \leq C|y - x|$ for some $C > 0$. This is one of the steps in a bigger problem I'm trying to solve, and while it first appeared it would be entirely straightforward, I ...
0
votes
1answer
29 views

Show that $d(u,v)=\exp(-\max\{j\ge 0, u_k=v_k \space\mbox{for}\space 0\le k\le j\})$ is a distance over $E=\Bbb{R}^\Bbb{N}$.

Let $E=\Bbb{R}^\Bbb{N}$, $u=(u_k)_{k\in\Bbb{N}}$ and $v=(v_k)_{k\in\Bbb{N}}$. Define $$ d(u,v) = \left\{ \begin{array}{ll} \exp(-V(u,v)) & \mbox{if}\quad u\ne v \\ 0 & ...
0
votes
2answers
90 views

In $ \triangle ABC$ show that $ 1 \lt \cos A + \cos B + \cos C \le \frac 32$

Here is what I did, tell me whether I did correct or not: \begin{align*} y &= \cos A + \cos B + \cos C\\ y &= \cos A + 2\cos\left(\frac{B+C}2\right)\cos\left(\frac {B-C}2\right)\\ y &= \...
10
votes
7answers
280 views

For all real $\theta$ prove that $ \cos(\sin\theta) \gt \sin(\cos\theta)$

How do I prove this? Im not able to even start it. Help please!
0
votes
1answer
23 views

How to use the triangle inequality to get $d(x_k,y)\ge d(x,y)-d(x_k,x)$ (Prove limit of a sequence is unique)

Here, $d$ is the Euclidean distance: And the triangle inequality in terms of $d$ is: I have no idea how to come up with the $d(x_k,y)\ge d(x,y)-d(x_k,x)$ inequality. What I've tried: $\begin{...
3
votes
1answer
127 views

Least value of an Expression?

Find the least value of $\dfrac {3a}{b+c} + \dfrac{4b}{a+c}+ \dfrac{5c}{a+b}$ for positive $a, b, c$. I tried using the Cauchy-Schwarz inequality, but could not proceed after a bad equation which ...
-1
votes
1answer
34 views

Solve for $x$ an inequality with logarithms [closed]

I am trying to solve this equation $\frac{(n(n-1))}{2} + X (2\log n +2) < nX $ I would like to solve it for X ? What should i do ? Thanks
4
votes
1answer
65 views

How can I prove that $a^{2} < b^{2} $implies that $a < b$ in the Real Numbers? [closed]

The answer to my question doesn't seem to exist elsewhere on the internet. I have the sets $ A=\{ a : a\in R: a > 0,\ a^2 < 3\} $ and $ B=\{ b: b\in R: b>0,\ b^2 > 3\} $, and I'm just ...
0
votes
2answers
136 views

Is this true about the open intervals on the real line?

Let $a<b$ and let $m$ be a positive integer such that $$3^{-m} < \frac{b-a}{6}.$$ Then can we find a positive integer $k$ such that the open interval $$\left(\frac{3k+1}{3^m}, \frac{3k+2}{3^m}\...
1
vote
1answer
51 views

Inequality with continued fractions: $\theta_r \geq a_{r+2}\theta_{r+1} + \theta_{r+1}$

I want to prove that the following inequality is true (or that is false, I don not know but I think it is true). $$\theta_r \geq a_{r+2}\theta_{r+1} + \theta_{r+1}.$$ Here the notation is as follow: $\...
1
vote
1answer
71 views

What is the algebra involved in solving the inequality $\sqrt x\le 2$

I would like to know how one would solve $\sqrt x\le 2$ algebraically. How do you get rid of the radical sign? Do you square both sides? Why is this allowed to do in an inequality? I already have ...
0
votes
1answer
136 views

Measurability of the floor function

Let $u(x)=⌊x⌋$, i.e the largest integer not greater than $x$ . Determine $\{u≥a\}$ for all $a\in \mathbb{R}$. Show that $u$ is Borel-measurable. Can anyone help me with this problem?
1
vote
1answer
97 views

Symmetric inequality with three variables and rational functions of them

Let $a_1$, $a_2$, $a_3$ be three different real numbers. Define real numbers $b_1$ $b_2$ $b_3$ as $$b_1 = (1+ \frac{a_1a_2}{a_1-a_2})(1+ \frac{a_1a_3}{a_1-a_3}) $$ $$b_2 = (1+ \frac{a_2a_1}{a_2-a_1})(...
0
votes
1answer
26 views

Proof of inequality containing a function and its integral

I came across a proof that used following equalty, but for me it didn't look that obvious and I was not able to prove it, can you give me a hint (is it even true)? The statement of the inequality was ...
6
votes
1answer
124 views

A proof by strong induction that $a_n\le3^n$ where $a_n=a_{n-1}+a_{n-2}+a_{n-3}$

I am not sure whether this is right. Can anyone verify, whether this proof is valid? Thanks! Define a sequence $\{a_n\}_{n\ge0}$ as follows: $$a_0=1,\qquad,a_1=3,\qquad,a_2=9,\qquad,a_n=a_{n-...
1
vote
1answer
36 views

An inequality problem..

Given any $a_{ij}, b_{ij}$ real numbers, for $i,j=1\ldots n$. How can I show that $$ \sum_{i=1}^n\sum_{j=1}^n\Big(\sum_{k=1}^n a_{ik} b_{kj}\Big)^2 \leq \Big( \sum_{i=1}^n\sum_{j=1}^n a_{ij}^2 \Big)\...
2
votes
3answers
75 views

$a > b+1 \Rightarrow a>x>b$?

If I have $a,b \in \mathbb R$ such that $$a > b+1 $$ It is assured that $\exists\space x \in \mathbb Z: a>x>b$ Does this property have some special name? How can this be proved? This idea ...
0
votes
0answers
46 views

Relationship for $\log(A_1+A_2+\cdots+\cdots+A_n)$

It's a very well know fact that $$ \log\left(\prod_{i=1}^n A_i\right)=\sum^{n}_{i=1}\log(A_i) $$ Can we say anything about $$ \log\left(\sum_{i=1}^n A_i\right)=\text{ ????} $$ My question is ...
2
votes
8answers
135 views

Proof that $|\sqrt{x}-\sqrt{y}| \leq \sqrt{|x-y|},\quad x,y \geq 0$

Any hints on how I can prove the inequality: $$|\sqrt{x}-\sqrt{y}| \leq \sqrt{|x-y|},\quad x,y \geq 0$$ Thank you.
2
votes
1answer
35 views

Is $d(i,j) = 1-\textrm{corr}(i,j)$ a metric?

I need to make sure that this function is a metric: $d(i,j) = 1-\textrm{corr}(i,j)$ where $\textrm{corr}(x,y)$ is the Pearson correlation coefficient which ranges from $[-1,1]$. With this scaling I ...
2
votes
3answers
45 views

Inequalities and $x^2$

I would just like to clarify something in regards to inequality and how x^2 would affect it. Why is it that if I have the inequality: $x^2(x+5)(x-6)>0$, for example, I can simply divide out $x^2$? ...
1
vote
1answer
82 views

Calc question about inequality and electric circuit theory [duplicate]

In electric circuit theory, the combined resistance $R$ of two resistors $R_1 > 0$ and $R_2 > 0$ connected in parallel obeys $$\frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2}$$ Show that $$R < \...
0
votes
1answer
65 views

Prove this inequality by math induction

$$\sum \limits_{k=1}^{n-1} k^p < \frac{ n^{p+1}}{p+1} < \sum\limits_{k=1}^n k^p $$ I know how to prove it by using Riemann Sum, but it I was thinking if there is anyway to do it by mathematical ...
0
votes
2answers
40 views

If $z'\le az+b$ then $z(t)\le z_0+bt$

If $z$ satisfies; $z'\le az+b$, $\ z(0)=z_0>0$ with constants $a,b$ why is true that $z(t)\le z_0+bt$, if $a=0$ It is clear that it can't be justified only by integrating. We had only Gronwall ...