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Questions tagged [inequality]

Questions on proving, manipulating and applying inequalities.

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1answer
25 views

Prove that for $|z| < R$, we have $\frac{|z|}{|z^2-n^2|} \leq \frac{R}{n^2-R^2}$

Prove that for $|z| < R$, we have $\frac{|z|}{|z^2-n^2|} \leq \frac{R}{n^2-R^2}$, where $n$ is an integer greater than $R$. This is from page 168, Lang's Complex Analysis. I am having trouble ...
2
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3answers
36 views

System of equations with three variables

Characterize all triples $(a,b,c)$ of positive real numbers such that $$ a^2-ab+bc = b^2-bc+ca = c^2-ca+ab. $$ This is the equality case of the so-called Vasc inequality. I think the answer is that $...
-2
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1answer
19 views

The range of $ab$ if $|a| \le 1$ and $a + b = 1$, $a,b\in\mathbb R$?

The range of $ab$ if $|a| \le 1$ and $a + b = 1$, $a,b\in\mathbb R$ ?
4
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2answers
91 views

Maximize $x_1^3+x_2^3+\cdots + x_n^3$

This is from a Brazilian math contest for college students (OBMU): Given a positive integer $n$, find the maximum value of $$x_1^3+x_2^3+ \cdots + x_n^3$$ where $x_j$ is a real number for all $j \...
1
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0answers
10 views

Holder continuity of the derivate of $|x|^\alpha$ for $\alpha>1$

Suppose $U=B(0,1)$ is an open ball in $\mathbb R^m$ and $\alpha>1$. My question is if $|x|^\alpha\in C^{1,\gamma}(U)$ for some $\gamma\in (0,1]$. I know that $|x|^\alpha\in C^1(U)$ and $$ f_i(x):=\...
2
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6answers
56 views

${x\choose 2}+{n-x\choose 2} \leq {n-1\choose 2}$

In a proof, the author states that it is clear that: Given $x\geq 1$ and $ n-x \geq 1$ and finally also $n\geq 2$ $${x\choose 2}+{n-x\choose 2} \leq {n-1\choose 2}$$ This is not immediately clear ...
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0answers
20 views

Digit Sum Inequality Equation

Let $Q (n)$ be the digit sum of the integers $n$. Now I want to prove that $Q (m + n) ≤ Q (m) + Q (n)$ is valid for all positive integers $m$ and $n$. So we can write: $n = a_n * 10^n + a_{n−1} * ...
1
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1answer
14 views

$\prod_{i=1}^{n}x_i^{p_i}\le p_1x_1+\ldots +p_nx_n$ strict inequality when $p_i \neq p_j$ for $i \neq j$

Let $x_i,p_i \in \mathbb{R}$ and $x_i,p_i>0$. I just showed that for $\sum_{i=1}^{n}p_i=1$ we have the following inequality: $$\prod_{i=1}^{n}x_i^{p_i}\le p_1x_1+\ldots +p_nx_n$$ (I used Jensen's ...
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3answers
69 views

$\frac{|a|}{|b-c|} + \frac{|b|}{|c-a|} + \frac{|c|}{|b-a|} \geq 2$

If $a, b, c$ are distinct real numbers then you demonstrate that: $$ S=\frac{|a|}{|b-c|} + \frac{|b|}{|c-a|} + \frac{|c|}{|b-a|} \geq 2.$$ Using inequality $ |x-y|\leq |x|+|y|$ we showed that $ S &...
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2answers
49 views

Prove using induction that $n^6 < 3^n$,for all $n > 18$

Prove using induction (or using any other elementary precalculus techniques) that $$n^6 < 3^n, \forall n \geq 19.$$ I have no idea how to do this. Writing the induction step, I get that I need to ...
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2answers
19 views

Inequality properties between their multiplications [on hold]

If $A>B>C>D$ then will $$(A/B * A/C * A/D) > (B/A * B/C * B/D) > (C/A * C/B * C/D) > (D/A * D/B * D/C)$$ be always true? If not, in what intervals will it not be true? Obviously $A,...
5
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5answers
127 views

$\frac{1}{15}<(\frac{1}{2}\cdot\frac{3}{4}\cdot\frac{5}{6}\cdot \cdot \cdot \cdot\frac{99}{100})<\frac{1}{10}$.

Show that $$\frac{1}{15}<(\frac{1}{2}\cdot\frac{3}{4}\cdot\frac{5}{6}\cdot \cdot \cdot\cdot\frac{99}{100})<\frac{1}{10}$$ My attempt: This problem is from a text book where is introduced as: ...
1
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4answers
52 views

Let $a>b$ and $ab=1$ show that $\frac{a^2+b^2}{a-b} \geq 2\sqrt{2}$

Let $a>b$ and $ab=1$. Show that $$\frac{a^2+b^2}{a-b} \geq 2\sqrt{2}$$ My attempt: $a-b>0$ $$a^2+b^2\geq 2\sqrt{2}(a-b)$$ $$\frac{a^2+b^2}{2}\geq \sqrt{2}(a-b)$$ By AG inequality we know that $$...
0
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0answers
15 views

Question on estimates

I have question on Theorem $1$ of Evan's book page $329$ or ($6.3$ Regularity). At the part $6$, Evan wrote: We finally combine $(12),(20)$ and $(22),$ to discover $$\int_V |D^h_k Du|^2 dx \...
2
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0answers
30 views

Difficult inequality with three real variables

For any real $e, t, \sigma$ such that \begin{aligned} \label{s} 0&<e<1\,,\\ 0&<t<\pi\,,\qquad\qquad\qquad(1)\\ -\pi/2&\leqslant\sigma\leqslant\pi/2 \end{aligned} the ...
1
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2answers
26 views

Knowing how to order $\{a,b,c,0\}$ implies knowing how to order $\{a,b,c,0,-a,-b,-c\}$?

Suppose I have $4$ real numbers $\{a,b,c,0\}$ and I know that they are all different how to order them from smallest to largest, e.g., I know that $b<a<0<c$ Does this imply that I know how ...
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1answer
41 views

Doubt in proof of AM GM inequality by induction Method

Proof of AM-GM inequality So i was going through the AM-GM inequality and there is particular part i didn't quite understand. My question is IF c1 <1 and ck+1 > 1 then how are they considering c....
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3answers
32 views

Does this inequality make sense? 1 = |1|?

Okay so suppose x = -9. Then we have x < 1 . But 1 = |1| Hence x < |1| Implies -1 < x < 1 But this clearly is not true. Just wondering what the limitations are when using inequalities ...
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2answers
19 views

Probability problem of A given B

How can I prove that $P(A|B)\leq P(A)$? It is conceptually clear, but I want a mathematical proof.
2
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2answers
56 views

One lower bound for $(1+x)\log(1+x)-x$

Problem When studying Chernoff bound, one result is used without proof and reference, which is $$ (1+x)\log(1+x)-x\geq \frac{x^2}{\frac{2}{3}x+2} $$ I am wondering how this is proved. What I Have ...
0
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0answers
40 views

Proof approach for $n! \ge (n/2)^{n/2}$ [duplicate]

I am having trouble proving that $n! \ge \frac{n}{2}^{\frac{n}{2}}$ for all $n \ge 2$. Intuitively, I can see how this is true. However, I am struggling with a formal proof and dont know which way to ...
0
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0answers
24 views

What can we say about the Begaman transform of $f\ast g (t_2)- f\ast g(t_1)$?

Let $f, g\in \mathcal{S}(\mathbb R)$. Then the convolution of two Schwartz class function is again Schwartz class function, that is, $f\ast g \in \mathcal{S}(\mathbb R).$ Now we define $$ H(t)= H(...
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votes
2answers
51 views

Proof of inequality solution

I have to solve this inequality: $$5 ≤ 4|x − 1| + |2 − 3x|$$ and prove its solution with one (or 2 or 3) of this sentences: $$∀x∀y |xy| = |x||y|$$ $$∀x∀y(y ≤ |x| ↔ y ≤ x ∨ y ≤ −x)$$ $$∀x∀y(|x| ≤ ...
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2answers
47 views

How do I solve the inequality $x<x^2-12<4x$?

So first I considered $x < x^2 -12$ so I get $0 < x^2 - x -12$ which is $0<(x+3)(x-4)$ after this I don't know where to go Again, I considered $x^2 - 12< 4x$ which is $x^2 - 4x - 12<...
2
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0answers
41 views

Reconstruction of proof in complex analysis paper

I am reading through this paper of Musin: http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=mzm&paperid=2326&option_lang=eng On the second page (visible in the preview of the ...
1
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0answers
22 views

Best error term in $\sum_{(n,q)=1}\frac{1}{n}$ (harmonic series with coprimality condition)

It is very well known and not difficult to prove that $\displaystyle\sum_{\substack{0<n\leq X\\ \\(n,q)=1}}\frac{1}{n}=\left(\log(X)+\gamma+\sum_{p|q}\frac{\log(p)}{p-1}\right)\frac{\phi(q)}{q}+O\...
0
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2answers
33 views

Find least upper bound of $\{a^{2018} + b^{2018} + c^{2018} | a + b + c = 1, a, b, c > 0\}$

Find least upper bound of $\{a^{2018} + b^{2018} + c^{2018} | a + b + c = 1, a, b, c > 0 \}$. I tried power mean inequality, but could only find greatest lower bound.
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0answers
43 views

What is condition for always be negative/ positive quartic equation?

I have a parametric quartic equation. It is potential of a black hole that I want to always be negative. I thought to make it to two quadratic equation. But it is very difficult to solve. What can I ...
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0answers
23 views

Convexity of $x^{2q}$ implies $\mathbb{E} \big[ \left(X-X^{'}\right)^{2q} \big]\leq 2^{2q-1} \left( \mathbb{E}X^{2q}+ \mathbb{E}X^{'2q} \right) $

This problem is from page 25 of concentration inequalities a nonasymptotic theory of independence $X$ is a random variable and $X^{'}$ is independent copy of $X$ q is integer and $q\geq1$ It is ...
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2answers
64 views

How would I prove ${n^{2n} \gt (2n)!}$ using mathematical induction?

This is what I have done. I checked for $n=1,2$ and $3$ in the first step. I did the assumption for $n=k$ and the claim of $n=k+1$ in the second step, but I don't understand how the step no. $3$ works ...
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0answers
12 views

Inequality involving Weibull distribution

define R(x) as the weibull survival function $R_1(t)=\alpha_1 t^{\beta_1}$ $R_2(t)=\alpha_2 t^{\beta_2}$ with $\alpha_1, \alpha_2 > 0 $ and $\beta_1, \beta_2 >1 $ $\phi_1(t)=\int_{t}^{\infty}...
2
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1answer
30 views

Prove concavity of a particular function

The exercise is the following. Let $f:U\longrightarrow\mathbb{R}$ be semiconcave on the open set $U$, that is there exists a constant $K\geq0$ such that $$ \lambda f(x)+(1-\lambda)f(y)\leq f(\lambda x+...
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votes
1answer
44 views

Refinement of a strong inequality

It's related to this If $a+b+c=abc$ then $\sum\limits_{cyc}\frac{1}{7a+b}\leq\frac{\sqrt3}{8}$ . I make a little refinement wich could be usefull to prove the original one . Let $a,b,c$ be real ...
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4answers
55 views

Is it based on Tchebyshev?

If $a$, $b$, $c$ are positive real numbers such that $a^2 + b^2 + c^2 = 27$. Find the least value of $a^3 + b^3 + c^3$ ? I tried with Tchebyshev inequality on sets $\{a, b, c\}$ and $\{a^2, b^2 , c^2\...
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2answers
52 views

solving natural log inequality

How can I show that $0 \le (\sum_{x=1}^{n}\frac{1}{x})-ln(n) \le 1 - \frac{1}{n}$ Do I raise both sides by $e$ or perhaps take integral of both sides? If so, I'm still not quite sure where to go ...
1
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1answer
67 views

The inequality $\sum_{k=1}^n \frac{1}{k^4} \le 2 - \frac{1}{\sqrt n}$

Prove that for every $n$ we have $$\sum_{k=1}^n \frac{1}{k^4} \le 2 - \dfrac{1}{\sqrt{n}}$$ I've tried induction, but I ended up with polynomials of high degree.
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2answers
26 views

solving integral inequality

How can I show that $0 \le \int_n^{n+1}\frac{1}{n}-\frac{1}{x}dx\le \frac{1}{n}-\frac{1}{n+1}$ I think I need to take the log to solve this, but I'm not quite sure.
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0answers
37 views
+50

Proof of a technical fact in the book of Schapire and Freund on boosting

I am currently looking at Exercise 10.3, Chapter 10 of the book on Boosting by Schapire and Freund. More precisely, in the middle of the exercise they propose to use, without proof, the technical fact ...
0
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1answer
56 views

Question about an inequality.

$$\forall i\in \{1,2,\cdots, k\}, n_i\in\mathbb{N}$$ $$\sum_{i=1}^k n_i =n$$ then $$\sum_{i=1}^k n_i^2\leq n^2-(k-1)(2n-k)$$ Like comment, If we apply induction, $i)\ k=2$ $n_1+n_2=n\land n_1,n_2\...
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2answers
35 views

Using the sequential definition of uniform continuity to show $\sin(x)$ is uniformly continuous on $\mathbb{R}$

I want to show $\sin(x)$ is uniformly continuous on $\mathbb{R}$. Let $\{a_{n}\}$ and $\{b_{n}\}$ be sequences such that $\lim_{n\to\infty}[b_{n} - a_{n}] = 0$. Then, we need to show $\lim_{n\to\infty}...
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0answers
25 views

$\|x-y\|_{L^2}^2\geq \|x\|_{L^2}^2-\|y\|_{L^2}^2$?

Is this true? \begin{equation} \|x-y\|_{L^2(\Omega)}^2\geq \|x\|_{L^2(\Omega)}^2-\|y\|_{L^2(\Omega)}^2 \end{equation}
1
vote
1answer
37 views

If $f \in H(\Omega)$ holomorphic, then $|f(x) - f(y)| \geq (1/2)|f'(c)||x-y|$

I was reading Rudin's proof on the Open mapping theorem. He declare $\Omega$ to be a region, which is connected and open. Then he claims there is a neighbourhood of $c$ such that $|f(x) - f(y)| \...
0
votes
1answer
15 views

Equivalent forms for a product notation

Context: See "2 Hoeffding’s Inequality" in : http://www.stat.cmu.edu/~larry/=stat705/Lecture2.pdf My particular question arises within 'section 2 Hoeffding's Inequality' is: $$ e^{-tn\varepsilon }\...
3
votes
1answer
66 views

When does $\Vert AB \Vert = \Vert A \Vert \Vert B \Vert$?

Motivation If $a$ and $b$ are vector, then thinking simply vector 2 norm, $\Vert a \cdot b\Vert = \Vert b\Vert \Vert a\Vert \cos(a,b) $, we know the difference is simply a ratio between the angle of $...
1
vote
2answers
63 views

Significance of assumption in competition inequality questions

Refer to the problem below (IMO 2009 Shortlist) Let $a, b, c$ be positive real numbers such that $$\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = a + b+ c$$ Prove that $$\frac{1}{(2a+b+c)^2} + \...
1
vote
1answer
25 views

$\int|f|^\alpha|g|^{1-\alpha}d\mu\le(\int|f|d\mu)^\alpha(\int|g|d\mu)^{1-\alpha}$

$f,g$ on $(\Omega, \mathcal{A}, \mu)$ and $0<\alpha<1$. Then $$ \int|f|^\alpha |g|^{1-\alpha} d\mu \le \left(\int|f|d\mu\right)^\alpha \left(\int|g|d\mu\right)^{1-\alpha} $$ This ...
0
votes
1answer
35 views

Determine all real numbers $x $ that satisfy the inequality

Let $ r$ be a fixed real number. Determine all real numbers $x$ that satisfy the inequality $\frac{1}{1+x^2}$ $≤ r$ . Can someone help me start this question? I am aware of a method when $r = 0$ , ...
1
vote
0answers
20 views

A notation to represent that all elements of a vector must be less than zero?

I am defining an optimization algorithm. One of the constraints of the length $p$ vector $\boldsymbol{\theta}^-$ is that all elements in that vector must be less than or equal to zero. I could say ...
0
votes
1answer
39 views

Using Jensen inequality to show $|\sum_{i=1}^{n}a_ix_i|\le \ldots \le c\sqrt{\sum_{i=1}^{n}x_i^2}$

My goal is to have a chain of inequalities so that $|\sum_{i=1}^{n}a_ix_i|\le \ldots \le c\sqrt{\sum_{i=1}^{n}x_i^2}$ $|\sum_{i=1}^{n}a_ix_i|\le\sum_{i=1}^{n}|a_ix_i|=\sum_{i=1}^{n}\sqrt{(a_ix_i)^2}\...
3
votes
1answer
35 views

Show that $|y_{n+1} - x_{n+1}| \le \frac{|b-a|}{4^n}$ for $x_{n+1} ={1\over 2}(x_n + y_n)$ and $y_{n+1} = \sqrt{{1\over 2}(x_n^2 + y_n^2)}$

This question is the last part of the problem statement which was not included in this question. Let: $$ \begin{cases} x_{n+1} = {1\over 2}(x_n+y_n)\\ y_{n+1} = \sqrt{{1\over 2}(x_n^2 + y_n^2)} \\...