Questions on proving, manipulating and applying inequalities.

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2
votes
1answer
45 views

A conditional inequality which itself implies a sharper version of it [duplicate]

Problem: Given that $m, n$ are positive integers such that $\sqrt{7} -\frac{m}{n} > 0$. Then show that $\sqrt{7}-\frac{m}{n} > \frac{1}{mn}$. I have failed to do this fascinating problem. My ...
2
votes
0answers
24 views

Poincaré-like inequality

Let $\Omega\subset\mathbb{R}^3$ be an open bounded set. Let $\partial\Omega=\Gamma^1\cup\Gamma^2$, with $\Gamma^1\cap\Gamma^2=\emptyset$. We denote as $\Gamma^1_j$, $j=1,\dots,p_{\Gamma^1}+1$, the ...
1
vote
2answers
68 views

An inequality involving $\frac{x^3+y^3+z^3}{(x+y+z)(x^2+y^2+z^2)}$

$$\frac{x^3+y^3+z^3}{(x+y+z)(x^2+y^2+z^2)}$$ Let $(x, y, z)$ be non-negative real numbers such that $x^2+y^2+z^2=2(xy+yz+zx)$. Question: Find the maximum value of the expression above. ...
5
votes
2answers
294 views

Proof of this inequality

I have a finite sequence of positive numbers $(a_i)_1^n$ for which: $a_1>a_n$, $a_j\geq a_{j+1}\geq\cdots\geq a_n$ for some $j\in\{2,\ldots,n-1\}$, $a_1>a_2>\cdots>a_{j-1}$, $a_j\geq ...
0
votes
3answers
53 views

Proof by mathematical induction that $2^n < (n+2)!$ for all $n\ge 0$

I have been trying to get this.. For hours. Prove by M.I. that $2^n < (n+2)!$ for $n\ge0$ Here is what I am doing: Base case checks out at $n=0$ Make assumption for: $n=k$ Want to prove: ...
2
votes
2answers
83 views

Prove that $a^b b^c c^a \geq a^c b^a c^b$

I want to prove that $$a^b b^c c^a \geq a^c b^a c^b$$ a, b, c are integers and $0 <a\leq b\leq c$ I have tried using Muirhead's Inequality, but didn't work. Any hints please?
4
votes
1answer
54 views

$p$-norm inequality

Let $p \ge 2$ and $q$ such that$${1\over p} + {1\over q} = 1.$$Is it true that there exists a constant $c$ such that for all $x$, $y$ such that $\|x\|_q \le 1$ and $\|y\|_q \ge 1$$$\left\|x - ...
2
votes
2answers
41 views

How to show using proof by induction: $\sqrt[n]{n!} \leqslant \frac{n+1}{2}, n \in \mathbb{Z}^+$

I'm having quite a few problems with the following proof by induction question: $$\sqrt[n]{n!} \leqslant \frac{n+1}{2}, n \in \mathbb{Z}^+$$ I manage to do the easy parts of the base step ($n=1$) ...
-4
votes
7answers
117 views

Prove that $\frac{1-\sqrt{1-x^2}}{x}\le1$ [closed]

What are different ways to prove that: $$\frac{1-\sqrt{1-x^2}}{x}\le1$$ for $0<x<1$ Thanks!
0
votes
2answers
41 views

How to Show that $1+\left(\left\lceil\dfrac{x}{n}\right\rceil -1\right)n\leq x$?

For $x$ integer in $\{1,\ldots,n^2\}$. I would like to show that $$ 1+\left(\left\lceil\dfrac{x}{n}\right\rceil-1\right)n \leq x. $$ I start by the property of the ceiling function which gives me $$ ...
2
votes
0answers
34 views

$\mathbb{P}[X_1(k^\ast)] \leq \left( \frac{e}{k^\ast} \right)^{k^\ast} \frac{1}{1-e/k^\ast} \leq n^{-2}$ inequality is used to prove the theorem

In the book Randomized Algorithms from Motwani and Raghavan, it is stated in page 44 that $$\mathbb{P}[X_1(k^\ast)] \leq \left( \frac{e}{k^\ast} \right)^{k^\ast} \frac{1}{1-e/k^\ast} \leq n^{-2}.$$ ...
0
votes
0answers
40 views

Explanation of this integral identity in the proof of Wirtinger's inequality from Hardy-Littlewood-Polya

I report the following excerpt from the book "Inequalities" by Hardy-Littlewood-Polya, page 184, where Wirtinger inequality is proven using variational methods. I'm trying to understand what is the ...
0
votes
0answers
12 views

Model linearly: What products to make, how much to make and in what plants to make them?

A company wants to make 3 new products for the upcoming week. We are given that: Each product can be made in 1 of 2 plants. At most 2 of the 3 new products should be chosen to be made. Only 1 of ...
0
votes
1answer
30 views

Any advice on how to tackle this inequality? ($x^{a}e^{x+c}\leq b$ )

How might one go about solving the inequality: $x^{a}e^{x+c}\leq b$ where $a,b,c$ are arbitrary constants ($b\geq 0$ and $a\neq0$) for $x$. My first place would be to try and get all of the ...
1
vote
6answers
79 views

What are some good books on algebraic inequalities?

By algebraic inequalities I mean inequalities like Cauchy's inequality, the AM-GM inequality etc. I need it for the International Mathematics Olympiad (IMO), so I hope I can find some books that ...
5
votes
4answers
140 views

How do I show that $\frac a{1 - a^2} + \frac b{1 - b^2} + \frac c{1 - c^2} \ge \frac {3 \sqrt 3}2$

For $0 \lt a, b, c \lt 1$, if $ab + bc + ca = 1$, show that $$\frac a{1 - a^2} + \frac b{1 - b^2} + \frac c{1 - c^2} \ge \frac {3 \sqrt 3}2.$$ I want to use trigonometric substitution: For the ...
0
votes
1answer
9 views

Name for inequality about sums of exponents with same base

Is there a name for the following inequality regarding sums of exponents which share a base? $$\text{For all integers $b \geq 2$, $n \geq 1$,} \\ \sum_{i=0}^{n-1}{b^i} < b^n$$
1
vote
0answers
29 views

Model linearly: Determine amount of units for production

A company produces 2 products in a week. Let $x_i$ denote the number of units of product $i$ to produce. Each product requires liters of Chemical X to make. Info is given below: \begin{array}{|c|c|} ...
2
votes
2answers
32 views

Expectation inequality

How can I prove that $$ \mathbb{E} [X^2] \geq \mathbb{E} [|X|]^2 $$ This resembles a lot the Cauchy-Schwarz inequality but I'm unable ti prove it with the usual method (i.e. when there are two random ...
2
votes
1answer
71 views

How to prove $({\log_2 x})^{n+1} \le x^n$

I want to show that $({\log_2 x})^{n+1} \le x^n$ when $n \ge 1$ and $x \ge 1$. I know that ${\log_2 x}$ can be shown to be $\lt x$ with: $x \lt 2^x$ $\log_2 x \lt x$ and obviously adding the same ...
3
votes
1answer
41 views

A sharp upper bound on discrete Young's inequality for sum with $f$ and $f^{-1}$

Problem: $f$ is a strictly monotonic and continuous function on $[0, 1]$, such that $f(0)=0$ and $f(1)=1$. Then prove that $f(\frac{1}{10})+f(\frac{2}{10})+\cdots ...
0
votes
0answers
43 views

Inequalities on the surface of a sphere

Suppose $S$ is the surface of the sphere $x^2+y^2+z^2=a^2$ above the $xy$ plane. Can we determine the following inequalities on $S$ without sketching the surface (as that's quite hard to ...
1
vote
0answers
37 views

$4$ or more type $2$ implies $3$ or less type $1$

I'm having difficulties with the logic with the last part of the reformulation part of the problem below. Let $x_i$ be the the number of ships of type $i$ to purchase. For $4a:$ (the ...
0
votes
1answer
25 views

A Question on Positive Definite Matrix and Sesquilinear Form

Assume that $A$ is a symmetric positive definite matrix. For any vectors $x\in \mathbb{R}^n$ and $y\in \mathbb{R}^n$, if the inner product $x^T y\geq0$, then $x^T Ay\geq0$. I guess the assertion is ...
4
votes
0answers
97 views

Prove with some AM-GM inequality?

I have proved the following inequality: Let $a,b,c>0$ $$\dfrac{(a+\sqrt{ab}+\sqrt[3]{abc})}{3}\le \sqrt[3]{a\cdot\dfrac{a+b}{2}\cdot\dfrac{a+b+c}{3}}$$ My solution ...
0
votes
2answers
47 views

Proof of analogue of the Cauchy-Schwarz inequality to vector

Cauchy-Schwarz inequality can be writen: $$ \left|\sum_{i=1}^nx_i\cdot y_i\right|^2 \le \sum_{i=1}^n|x_i|^2\sum_{i=1}^n|y_i|^2,\qquad \forall x_i,y_i \in \Bbb R $$ My question is, if $x_i,y_i \in ...
1
vote
2answers
27 views

An inequality to find the range of an unknown coefficient

Find the range of $a$ such that $$a(x_1^2+x_2^2+x_3^2)+2x_1x_2+2x_2x_3+2x_1x_3 \geq 0, x_i\in \mathbb{R}$$ I tried to use Cauchy Inequality but it seems not...
0
votes
0answers
6 views

Under what conditions the following inequality holds?

I have the following the inequality holds? $A+B>w^{1/m}A+w^{1-1/m}B$ where $A>0, B>0$, $w$ and $m$ are positive integer number satisfying $1\leq w,m<\infty$. My question is under what ...
1
vote
2answers
111 views

Question about an inequality which seems right but not easy to prove

The origin problem is as follows: let $a,b,c,d$ are positive real numbers,and $a+b+c+d=4$ prove:$$\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{d}+\frac{d^2}{a}\geq ...
0
votes
0answers
22 views

maximizing a ratio of sums

Consider 3 n-uplets of real positive values : $a_i$ such that $\forall i , \frac{1}{X} \leq a_i \leq X$ where $X>1$ $b_i$ such that $\forall i , \frac{1}{Y} \leq b_i \leq Y$ where $Y>1$ and ...
0
votes
1answer
40 views

Least positive integer such that $\cos^k \left(\frac{\pi}{2k}\right)\geq \frac{99}{100}$

I wrote a program to figure this out and found $k=123$. Writing $f(k)$ as the function, I showed that $f$ is strictly increasing on the positive integers, and Wolfram agrees that ...
1
vote
1answer
27 views

Alternative Form of Bernoulli’s Inequality

Why is the following Bernoulli inequality true (I found it here: http://www.lkozma.net/inequalities_cheat_sheet/ineq.pdf): $(1+x)^r \leq 1 + (2^r-1)x$ for $x \in [0,1]$ and $r \in \mathbb{R} - ...
0
votes
0answers
22 views

Showing how a dot product is less than 0

Considering a function $f\in C^1(\mathbb{R}^N)$ with $g=\nabla f(x)$, I want to prove a result about optimizing it. However the part I am struggling with I think can be reduced to simple matrix dot ...
1
vote
2answers
72 views

Is there integral or series for $\sqrt{10}-\frac{4^4}{3^4}$ (to prove the inequality)?

Both of these numbers are bad approximations for $\pi$, but they turn out to be much closer together: $$\sqrt{10}-\frac{4^4}{3^4}=0.00178$$ Since there is a lot of questions here about integrals and ...
0
votes
4answers
52 views

Showing $\nu_1^2 + \nu_1\nu_2 + \nu_2^2+\nu_2\nu_3 + \nu_3^2 > 0$ for $\nu_1, \nu_2, \nu_3$ not all $0$

Let $\nu_1, \nu_2, \nu_3 \in \mathbb{R}$ not all be zero. I wish to show $$\nu_1^2 + \nu_1\nu_2 + \nu_2^2+\nu_2\nu_3 + \nu_3^2 > 0\text{.}$$ Wolfram seems to suggest splitting this into cases, ...
2
votes
0answers
26 views

Equality in Hardy's inequality via Hölder's

I'm working on Exercise 3.14 in Rudin's Real and Complex Analysis. I was able to answer part (a): that for real $p$ satisfying $1<p<\infty$, for every function $f$ in $L^p(0,\infty)$, when $F$ ...
0
votes
0answers
10 views

Need to understand the trick to convert expression into Chernoff inequality form

In a proof we are trying to use Chernoff bound $P(X \geq (1 + \delta)\mu) \leq exp(- \frac{\delta^2 \mu }{3})$ Thus we say $P(h'> (\frac{\alpha_1 n^{2/3}}{h})E[h']) \leq P(h'> (1 + ...
0
votes
1answer
18 views

Solving $-\sqrt{1-x^2} \le y \le 0$ for $x \in [0,1]$

I want to solve $-\sqrt{1-x^2} \le y \le 0$ for $x \in [0, 1]$. I'm interesting in solving this for $x$ by any method. For now I can only do it by sketching the graph. Also I'd be grateful if ...
1
vote
0answers
9 views

On a criterion for almost perfect numbers using the abundancy index

Let $\sigma(x)$ denote the sum of the divisors of $x$, and let $I(x) = \sigma(x)/x$ be the abundancy index of $x$. A number $y$ is said to be almost perfect if $\sigma(y) = 2y - 1$. In a preprint ...
5
votes
0answers
59 views

Prove that $e^x|\int_x^{x+1}\sin(e^t)dt|\le 2$ [duplicate]

Prove that $e^x|\int_x^{x+1}\sin(e^t)dt|\le 2$. Use mean value theorem $$\int_x^{x+1}\sin(e^t)dt=\sin(e^\xi)$$ And we have $$|\sin(e^\xi)|\le\frac{2}{e^x}$$ where $\xi\in(x,x+1)$ I stuck here. ...
8
votes
2answers
255 views

To prove this log inequality (middle school)

This following Problem is from a book introduce Telescopic Sums,the book introduce the idea is use identities to write the sum in the form $$\sum_{k=2}^{n}[F(k)-F(k-1)]$$ then cansel out terms to ...
0
votes
1answer
24 views

How to deal with this in equality?

In an equation like this $|x + 3| = 2$, I know I can do something like $x + 3 = 2$ or $x + 3 = -2$ and solve for $x$ from there on. How do I solve it in this situation? $|x + 3| > 2$. Can I apply ...
0
votes
0answers
27 views

Inequality for general polynomials.

This may be a bit too general for anyone to give much, but nonetheless, I'd like to see if anyone has any insight. What conditions would be necessary for the following to hold: ...
0
votes
1answer
33 views

Probability of at least one triangle in Erdos-Renyi graph

Cross-posting here as I didn't get a satisfactory solution on cv. This is a well-known problem in random graph theory, where we show that if $X$ is the number of triangles in $G(V,E,p)$ with ...
2
votes
1answer
23 views

Perfect numbers of the form $12m+1$ and $\sum_{d\mid n}\frac{1}{\phi(d)}$, where $\phi(m)$ is Euler's totient function

If there are no mistakes combining Exercise 9 a) (Chapter 3, page 71) and Exercise (Chapter2, page 47) of Apostol's Introduction to Analytic Number Theory we can prove easily Lemma. If $n$ is a ...
0
votes
1answer
36 views

The sum of the first $ m nth $ powers.

I want to show this inequality $$\frac{m^{n+1}}{n+1} < 1^n + 2^n + \dots + m^n <\frac{\left(m+1\right)^{n+1}}{n+1}$$ Where $ m $ and $ n $ are positive integers. I have a hint that ...
0
votes
2answers
35 views

For what values of $k,n$ does the inequality $(4-k)n\ge (2-k)k$ holds?

Let $n, k \in \mathbb{N}$ such that $k\le n$ for what values of $k$ and $n$ is $(4-k)n\ge (2-k)k$ and for what values $(4-k)n < (2-k)k$ if $k=1,2,3,4$ and $n\ge k$ the first inequality holds ...
0
votes
0answers
54 views

Prove $(a-b)^2+(b-c)^2+(c-a)^2 \geq a^2+b^2+c^2-3\sqrt[3]{a^2b^2c^2}$ [duplicate]

For nonnegative real numbers $a,b$ and $c$, prove $$(a-b)^2+(b-c)^2+(c-a)^2 \geq a^2+b^2+c^2-3\sqrt[3]{a^2b^2c^2}$$ It is clear that the inequality is equivalent to $a^2+b^2+c^2 + ...
1
vote
3answers
54 views

Bounds for integral over $x^{-x}$

Let $a,b>0$ and consider $$ F(a,b) := \int_a^\infty x^{-b x}\, dx = \int_a^\infty \exp(- b x \log(x) ) \, dx $$ For sufficiently large $a$ it holds $$F(a,b) \leq \int_a^\infty \exp(-b x) \, dx = ...
4
votes
1answer
98 views

How do I prove that $\left | \sum_{j=1}^n a_j \right |^2 + \left | \sum_{j=1}^n (-1)^j a_j \right |^2 \le (n+2) \sum_{j=1}^n a_j^2$?

For any $a_j \in \Bbb R, \, j = 1, 2, \cdots, n$, one has the bound $$\left | \sum_{j = 1}^n a_j \right |^2 + \left | \sum_{j = 1}^n (-1)^j a_j \right |^2 \le (n + 2) \sum_{j =1}^n a_j^2.$$ This is ...