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Questions tagged [inequality]

Questions on proving, manipulating and applying inequalities.

7
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0answers
50 views

How big must the union of a group's Sylow p-subgroups be?

For various orders $n$ it's a common exercise to prove that a finite group $G$ of order $n$ can't be simple by using the Sylow theorems to show that there is some prime $p \mid n$ such that the number ...
1
vote
4answers
44 views

Show that the following inequality holds when $x>0$

$\require{cancel}$ Show that the following inequality holds for $x>0$ $$1+\frac{x}{2}-\frac{x^2}{8}<\sqrt{x+1}<1+\frac{x}{2}.$$ I proceeded as follows $$\sqrt{x+1}=1+\frac{x}{2}-\frac{...
0
votes
1answer
30 views

Proving $\left( 1-\frac{2}{n} \right )^{\frac {n\ln n}{4}}-\left( 1-\frac{1}{n} \right )^{\frac {2n\ln n}{4}}<0$

As a part of a solution I'm writing I need to prove: $$\left( 1-\frac{2}{n} \right )^{\frac {n\ln n}{4}}-\left( 1-\frac{1}{n} \right )^{\frac {2n\ln n}{4}}<0$$ for large enough $n$. I checked in ...
0
votes
0answers
5 views

Union Bound Proof

I am asked to prove that $Pr(X_{i} \geq k) \leq$ $n \choose k$ $(1/n)^k$, which using Markov I have found to be $Pr(X_{i} \geq k) \leq 1/kn$. Now I need to use Boole's inequality to show that $1/kn \...
3
votes
1answer
39 views

Show that $x_{n+1} = \frac{x_n + 1}{n+1}$ is decreasing starting from some $n_0$ and find $n_0$.

Given $n\in \mathbb N$ and: $$ \begin{cases} x_{n+1} = \frac{x_n + 1}{n+1} \\ x_1 = -10 \end{cases} $$ Show that $x_n$ is decreasing starting from some index $n_0$. Find $n_0$. I've tried to ...
-2
votes
1answer
31 views

Proving $(1-a)(1-b)(1-c)(1-d) > 1-a-b-c-d$

Prove the following inequality for $a,b,c,d \in (0,1)$: $$(1-a)(1-b)(1-c)(1-d) > 1-a-b-c-d$$ I have a problem. I don't know if my idea is good $a=b=c=d $ $(1-a)^4 > 1- 4a $ So, this is ...
1
vote
4answers
90 views

The $1997$ IIT JEE problem

Let $S$ be a square of unit area. Consider any quadrilateral whose $4$ vertices lie on each side square $S$. Let the length of the sides of this quadrilateral be $a,b,c,d$. Then prove that $$2 \leq a^...
0
votes
2answers
16 views

Sum of fractions Inequality [duplicate]

Prove that: $$ {a\over b} + {b \over c} + {c \over a} \ge 3 $$ Assuming $a, b, c > 0$. I was able to prove that this is true: $$ {a\over b} + {b \over a} \ge 2 $$ by just rearranging it to get: ...
0
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0answers
28 views

An inequality with constraints.

I came across a result in a control theory book (without proof), which states that: Given two variables $x,z \in \mathbb{R}$ and four parameters $c_{1}, c_{2}, k_{1}, k_{2}$ with $c_{1}, c_{2} > 0$...
-2
votes
2answers
72 views

Inequality that I believe can be conquered with Cauchy Schwarz I.E

There are ten real numbers $x_0, . . . , x_9$ with $x_0 = 0, x_9 = 9$. What is the smallest possible value of the expression $$(x_1 − x_0)/1 +(x_2 − x_1)/2+(x_3 − x_2)/3+ · · · +(x_9 − x_8)/9$$? I ...
1
vote
0answers
12 views

a bound on $\|z\|_p$ with high probability for generalized Gaussian vector

Let $f(x)$ be the pdf of the generalized Gaussian distribution(GGD), which is given by \begin{align} f(x)=\frac{v}{2\sigma\Gamma(\frac{1}{v})}\exp\left(-\left[\frac{|x|}{\sigma}\right]^{v}\right),~x\...
2
votes
1answer
46 views

Solving a system of equations and inequalities

How would one proceed with proving that there exists none $(x,y,z)$ such that all the following hold : $$x+2y+7z = 0$$ $$\mathbb{P}(x + 3y + 9z \geq 0) = 1$$ $$\mathbb{P}(x + y + 5z \geq 0) = 1$$ $$\...
1
vote
4answers
29 views

Solution of $\log_{\frac12}(x) > 4$

Find all x for: $\log_{\frac12}(x) > 4$ My solution: $\log_{\frac12}(x) > \log_{\frac12}(\frac{1}2^4)$ $x > (\frac{1}2^4)$ $x>\frac{1}{16}$ This is my solution but this is wrong. The ...
1
vote
0answers
17 views

Fourier transform of $\int_{x}^{y} \frac{d}{dt}(f\ast g)(t) dt$?

Let $f, g\in \mathcal{S}(\mathbb R)$. Then the convolution of two Schwartz class function is again Schwartz class function, that is, $f\ast g \in \mathcal{S}(\mathbb R).$ It is also known that the ...
0
votes
4answers
66 views

Proof that $\left|\arctan (x)-\frac{π}{4}-\frac{(x-1)}{2}\right| \leq \frac{(x-1)^2}{2}$

I'm trying to prove that, for every $x \geq 1$: $$\left|\arctan (x)-\frac{π}{4}-\frac{(x-1)}{2}\right| \leq \frac{(x-1)^2}{2}.$$ I could do it graphically on $\Bbb R$, but how to make a formal ...
2
votes
3answers
51 views

Prove $a < 1/a < b < 1/b$ implies $a < -1$

I've solved a simple proof from Velleman's How to Prove It (p. 107, Q. 8) but I think my proof is suboptimal and was wondering if there's a better way I could prove it. The prompt is as follows: ...
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0answers
33 views

A Question from Inequality [on hold]

LET , n₁+n₂+n₃+n₄.........nₙ = Q also n₁,n₂,n₃.........nₙ have two prime factors given that every : n = Kβ and Q = 2 ω Find the minimum value of K₁²a₂a₃a₄.....aₙ+ K₂²a₁a₃a₄....aₙ+ ....and so on ...
0
votes
2answers
40 views

Facing a problem in solving for $x$ from $-2x-1>49$

Solve for $x$, $$-2x-1>49\tag1$$ $$-2x>49+1\tag2$$ $$x>50\div (-2)\tag3$$ $$x>-25\tag4$$ But my teacher says it is wrong!! How? I have followed all the steps in my calculation!!
3
votes
3answers
66 views

Is there a $\gamma>1$ such that $\frac{\binom{4p}{2p-1}}{\binom{4p}{0}+\binom{4p}{1}+\cdots+\binom{4p}{p-1}}>\gamma^{p}$

In a proof of the Larman-Rogers conjecture (there is $\gamma>1$ such that $\chi(\mathbb{R}^{d})>\gamma^d) $ they used that there is a $\gamma>1$ such that $\frac{\binom{4p}{2p-1}}{\binom{4p}{...
0
votes
1answer
24 views

Please present a sharp upper bound or a tight upper bound for $a^{1/q}-b^{1/q}$ ($a\geq b\geq 0$ and $q>1$).

Please present a sharp upper bound or a tight upper bound for $a^{1/q}-b^{1/q},\;$ ($a\geq b\geq 0$ and $q>1$).
2
votes
1answer
49 views

Showing that $\left(1+\frac{\varepsilon^2}{17n}\right)^n-1\leq \frac{\varepsilon^2}{16}$ when $0\leq \varepsilon\leq 1$

Question Show that $\left(1+\frac{\varepsilon^2}{17n}\right)^n-1\leq \frac{\varepsilon^2}{16}$ when $0\leq \varepsilon\leq 1$. This inequality appeared in the middle of an argument I was reading and ...
0
votes
1answer
22 views

Trace inequality with matrix square-roots

Suppose I have symmetric matrices $A \in \mathbb{R}^{n \times n}$ and $B \in \mathbb{R}^{n \times n}$ which are both positive definite. I am wondering if I one can bound ${\rm tr}\left(A - B \right)$ ...
0
votes
2answers
36 views

Calculating the time at which 2 airplanes will be a specific distance apart from each other.

I am trying to find the time at which 2 airplanes could potentially collide with each other. When I say collide, I mean the circular region around an airplane overlaps with the circular region around ...
-1
votes
1answer
41 views

inequality $1-\frac{2}{(n+1)^2}\leq\sum_{k=1}^n\frac4{k(k+1)(k+2)}\leq1-\frac{2}{(n+2)^2}$ [on hold]

How to prove the inequality $$1-\frac2{(n+1)^2}\leq\sum_{k=1}^n\frac4{k(k+1)(k+2)}\leq1-\frac2{(n+2)^2}$$ I tried to solve it using integration but failed, should I use integration to solve this ...
-2
votes
0answers
31 views

Prove that the series is Cauchy

Prove that $\ 2/5 +2/5^2 +2/5^3 +.....+ 2/5^n$ is Cauchy. Assuming $n$ is greater than or equal to $m$, it follows that $\ 2/5^{m+1} + 2/5^{m+2} + ...+2/{5^n}$ is less that or equal to 2/(m+1) + 2/(m+...
2
votes
1answer
29 views

Sum of squares related with integers

Let $x_1,x_2,x_3,...,x_{19}$ be positive integers satisfying $\sum\limits_{i=1}^{19}x_i=2020$ and $x_i \geq 2$ for $i=1,2,...,19$. Find the smallest value of $$P=\sum\limits_{i=1}^{19} x_i^2.$$ I've ...
2
votes
4answers
37 views

Prove that $\sqrt[n]{x}-1 \le \frac{x-1}{n}$

how do you prove this inequality? $\sqrt[n]{x}-1 \le \frac{x-1}{n}$ It looks to me as if Bernoulli's inequality would be useful. How about the following $\frac{x-1}{n} \geq \sqrt[n]{x}-1$ From here ...
0
votes
2answers
24 views

Show that there always exist a $n \in \mathbb {N}$ with $a,b \in \mathbb {R}$ and $b>1$ such that $b^n>a$.

I am not allowed to use limits. I have to prove it by exploiting the Bernoulli inequality. If $b\geq a$, then there is nothing to show. If $b <a$ and if $b\geq2$ then I can say $$a <[a]+1 <...
0
votes
1answer
15 views

Inequality with fraction and n-th root

Prove that $$ p(\sqrt[p]{n+1}-1)< \frac{1}{\sqrt[p]{1}}+\frac{1}{\sqrt[p]{2^{p-1}}}+...+\frac{1}{\sqrt[p]{n^{p-1}}}< p\sqrt[p]{n} \quad p\in \mathbb{N},p\ge 2 $$ I used AM-GM to prove it. ...
0
votes
1answer
47 views

Prove that if $a+b+c=1$ then $\frac{ab}{c+1}+\frac{ac}{b+1}+\frac{bc}{a+1} \le \frac{1}4$ $a,b,c$ are positive real numbers [duplicate]

I simplified the inequality , but I didn't get anything that may seem useful. I got $$^\sum_{cyc} 4a^2b+8ab+4ab^2\le abc+ab+ac+ab+a+b+c+1$$
3
votes
3answers
27 views

Binomial inequality : $\binom{n}{k} \leq n^k$

$\forall n \in \mathbb N$ and $k \in [[0,n]]$, show that : $$\binom{n}{k} \leq n^k$$ I already showed that : $\frac{1}{n^k}\binom{n}{k}$$\leq$$\frac{1}{(n+1)^k}\binom{n+1}{k}$
0
votes
3answers
37 views

How to solve this quadratic inequality?

My question: Find the values of q for which the quadratic equation $qx^2-4qx+5-q=0$ will have no real roots. Does this mean the discriminant has to be less than 0 in order to get no real roots? My ...
0
votes
0answers
25 views

Doubt in law of reciprocal

Law of reciprocal: If both sides of inequality have same sign, while taking its reciprocal the sign of inequality gets reversed. Thus, a >b> 0 → 1/a < 1/b But if both sides of inequality have ...
0
votes
0answers
14 views

Geometric average less or equal to arithmetic one [duplicate]

Let $\ $ ${y_1\cdot...\cdot y_n} \in \mathbb{R}$ $\ $ be positive $\quad$ Prove: $\sqrt[n]{y_1\cdot...\cdot y_n}$ $\le$ $\frac{y_1+...+y_n}{n}$ I have tried to find this by searching keywords like ...
1
vote
3answers
49 views

How do you prove the follwing $|\sqrt[n]{x}-\sqrt[n]{y}| \le \sqrt[n]{|x-y|}$

how do you prove this inequality? $|\sqrt[n]{x}-\sqrt[n]{y}| \le \sqrt[n]{|x-y|}$ At first glance I thought the triangle inequality would be useful but it's not form what I see. If I raise everything ...
0
votes
1answer
34 views

Metric space problem (Triangle inequality)

Let $X = \mathbb{R}^+$ $f\colon X\times X \to [0,\infty)$ (Triangle inequality) $$\frac{|x-y|}{(1+x)(1+y)}\leq\frac{|x-z|}{(1+x)(1+z)}+\frac{|z-y|}{(1+z)(1+y)}$$ I only known than $|x-y|=|x-z+z-y|=...
-3
votes
4answers
34 views

I have a problem in solving this inequality $\sqrt{2x-3}<x-1$ [on hold]

What I can't understand is where do I start from ? $$\sqrt{2x-3}<x-1$$
6
votes
2answers
208 views

On an expected value inequality.

Given $X$ a random variable that takes values on all of $\mathbb{R}$ with associated probability density function $f$ is it true that for all $r > 0$ $$E \left[ \int_{X-r}^{X+r} f(x) dx \right] \...
5
votes
2answers
93 views

Prove $\frac{x+1}{x-1}\ln x \geq 2$

I'm trying to get back to maths after quite a pause, and I've solved a question in a way that does not satisfy me. I'm pretty sure there is a nicer way to go and I'd like your opinion. $\forall x \...
-1
votes
1answer
34 views

Show that $a^2b^2+c^2d^2 -2abcd \geq 0 $ with $a,b,c,d \in \mathbb{R}$

Is it possible to Show that without argueing with Limits. I want to prove the Schwarz-Inequality with induction, if I could show this then I could prove the inductionstep.
0
votes
1answer
43 views

Show that for $a \ne 1$, $a > 0$ the sequence $\{x_n\} = n(1-a^{1\over n})$ is increasing

I'm having difficulties with the following problem: Let: $$ \begin{cases} x_n = n(1-a^{1\over n})\\ a > 0 \\ a \ne 1 \\ n \in \mathbb N \end{cases} $$ Show that $\{x_n\}$ is an increasing ...
0
votes
2answers
22 views

Inequality problem (may be) involving means

If n is a positive integer then how can I prove that $$2^n>1+n \sqrt{2^{n-1}}$$ .Any hint may help.My textbook mentions this problem in category of A.M. ,G.M. , H.M. inequalities.So please give ...
4
votes
1answer
58 views

If $x>\sqrt{xy}>y$, then $x>y>0$.

I am trying to prove the following: If $x>\sqrt{xy}>y$, then show that $x>y>0$. My argument is as follows: We only need to show $y>0$. Suppose $y<0$. Then, for $\sqrt{xy}$ to be ...
2
votes
1answer
35 views

Inequality on Hilbert Space

The problem is a sort of Cauchy-Schwarz inequality: Let $(x_n)_{n\in\mathbb{N}}$ be a sequence in a Hilbert Space $H$, and $(c_n)_{n\in\mathbb{N}}\in l^2(\mathbb{N}).$ Let also $F$ be a finite ...
2
votes
1answer
32 views

If $u_1=4$ and $u_{n+1}=\frac{3u_n+5}{u_n-1}$, then $u_n$ approaches $a=5$ as $n\to\infty$. Show that, if $u_n>a$, then $u_{n+1}<u_n$.

The sequence of positives numbers $u_1,u_2,u_3...$ is such that $u_1=4$ and $u_{n+1}=\dfrac{3u_n+5}{u_n-1}$ for all $n\ge1$. Given that $u_n\rightarrow a$ as $n\rightarrow\infty$, find the value of $...
0
votes
0answers
21 views

Inequality from proof of theorem on stochastic processes . Theorem from book by Skorokhod Theory of stochastic processes

Page from book Need help to prove inequality in red frame. Looks like it should be used inequality involving supremums above , but I don't see how.
0
votes
1answer
42 views

Trace inequality for a symmetric matrix

Let S be a symmetric positive n×n matrix and $$B \in M_n(R)$$ a triangularizable matrix with spectrum in $$[0, 1]$$ Prove the inequality: $$tr(BS)\ge tr(B)\det(S)$$
0
votes
1answer
26 views

Inequality involving convolutions

We have well known inequality for convolution: $\|g\ast f\|_{X} \leq \|g\|_{Y} \|f\|_{Z}$ where $Y, X, Z$ are suitable Lebesgue spaces, e.g., see Young's inequality. Let $f:\mathbb R^{2}\to \mathbb ...
0
votes
1answer
27 views

Show that $x_n = \left(1+{x\over n}\right)^{n+k}$ is decreasing for $0<x<k$; $n, k\in \mathbb N$

I've been working on some classical proofs of the sequences in the form $\left(1+{x\over n}\right)^p$. So: Let $n,k \in \mathbb N$ and: $$ \begin{cases} x_n = \left(1+{x\over n}\right)^{n+k}\\ 0 ...
0
votes
1answer
42 views

Equality without calculator [duplicate]

$ e^\pi$ and $\pi^e $ Which one is greater, How do I proceed without a calculator, I do not have any idea how to solve , using AM GM concept